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American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2009. [6] Denis Auroux. Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Go¨kova Geom. Topol. GGT, 1:51–91, 2007. [7] Victor V. Batyrev. Higher dimensional toric varieties with ample anticanonical class. PhD thesis, Moscow State University, 1985. Text in Russian. [8] Victor V. Batyrev. Toric degenerations of Fano varieties and constructing mirror manifolds. In The Fano Conference, pages 109–122. Univ. Torino, Turin, 2004. 1 [9] I. N. Bernˇste˘ın, I. M. Gel fand, and V. A. Ponomarev. Coxeter functors, and Gabriel’s theorem. Uspehi Mat. Nauk, 28(2(170)):19–33, 1973. [10] Gavin Brown and Alexander M. Kasprzyk. The Graded Ring Database. Online. Access via http://grdb.lboro.ac.uk/. [11] Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, and Alexander M. Kasprzyk. Mirror symmetry and Fano manifolds. In European Congress of Mathematics Krak´ow, 2–7 July, 2012, pages 285–300, 2014. [12] Tom Coates, Alessio Corti, Sergey Galkin, and Alexander M. Kasprzyk. Quantum periods for 3-dimensional Fano manifolds. arXiv:1303.3288 [math.AG], 2013. p q [13] Alessio Corti and Liana Heuberger. Del Pezzo surfaces with 1 1,1 points. In preparation. 3 ď [14] Dimitrios I. Dais. Classiﬁcation of toric log del Pezzo surfaces having Picard number 1 and index 3. Results Math., 54(3-4):219–252, 2009. [15] V. V. Fock and A. B. Goncharov. Cluster ensembles, quantization and the dilogarithm. II. The intertwiner. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, volume 269 of Progr. Math., pages 655–673. Birkh¨auser Boston, Inc., Boston, MA, 2009. [16] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2):497–529 (elec- tronic), 2002. [17] William Fulton. Introduction to toric varieties, volume 131 of Ann. of Math. Stud. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. [18] Sergey Galkin and Alexandr Usnich. Mutations of potentials. Preprint IPMU 10-0100, 2010. [19] Mark Gross, Paul Hacking, and Sean Keel. Birational geometry of cluster algebras. arXiv:1309.2573 [math.AG], 2013. [20] Mark Gross and Bernd Siebert. From real aﬃne geometry to complex geometry. Ann. of Math. (2), 174(3):1301– 1428, 2011. [21] Mark Gross and Bernd Siebert. An invitation to toric degenerations. In Surveys in diﬀerential geometry. Volume XVI. Geometry of special holonomy and related topics, volume 16 of Surv. Diﬀer. Geom., pages 43–78. Int. Press, Somerville, MA, 2011. ` [22] Christian Haase and Josef Schicho. Lattice polygons and the number 2i 7. Amer. Math. Monthly, 116(2):151–165, 2009. [23] Paul Hacking and Yuri Prokhorov. Smoothable del Pezzo surfaces with quotient singularities. Compos. Math., 146(1):169–192, 2010. [24] Nathan Owen Ilten. Mutations of Laurent polynomials and ﬂat families with toric ﬁbers. SIGMA Symmetry Inte- grability Geom. Methods Appl., 8:47–53, 2012. [25] Alexander M. Kasprzyk. Bounds on fake weighted projective space. Kodai Math. J., 32:197–208, 2009. [26] Alexander M. Kasprzyk, Maximilian Kreuzer, and Benjamin Nill. On the combinatorial classiﬁcation of toric log del Pezzo surfaces. LMS Journal of Computation and Mathematics, 13:33–46, 2010. [27] J. Koll´ar and N. I. Shepherd-Barron. Threefolds and deformations of surface singularities. Invent. Math., 91(2):299– 338, 1988. [28] Maxim Kontsevich and Yan Soibelman. Aﬃne structures and non-Archimedean analytic spaces. In The unity of mathematics, volume 244 of Progr. Math., pages 321–385. Birkh¨auser Boston, Boston, MA, 2006. [29] Jeﬀrey C. Lagarias and Gu¨nter M. Ziegler. Bounds for lattice polytopes containing a ﬁxed number of interior points in a sublattice. Canad. J. Math., 43(5):1022–1035, 1991. [30] Kurt Mahler. Ein U¨bertragungsprinzip fu¨r konvexe K¨orper. Cˇasopis Pˇest. Mat. Fys., 68:93–102, 1939. [31] Markus Perling. Combinatorial aspects of exceptional sequences on (rational) surfaces. arXiv:1311.7349 [math.AG], 2013. [32] Georg Alexander Pick. Geometrisches zur zahlenlehre. Sitzungber Lotos, 19:311–319, 1899. ##LTLine##

text: MINIMALITY

text: AND

text: MUTATION-EQUIVALENCE

text: OF

text: POLYGONS

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text: tronic),

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text: Roever

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text: Sergey

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text: Alexandr

text: Usnich.

text: Mutations

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text: potentials.

text: Preprint

text: IPMU

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text: Gross,

text: Paul

text: Hacking,

text: and

text: Sean

text: Keel.

text: Birational

text: geometry

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text: algebras.

text: [math.AG],

text: Mark

text: Gross

text: and

text: Bernd

text: Siebert.

text: From

text: real

text: aﬃne

text: geometry

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text: geometry.

text: Ann.

text: of

text: Math.

text: Mark

text: Gross

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text: Bernd

text: Siebert.

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text: degenerations.

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text: Geometry

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text: Geom.,

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text: MA,

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text: Haase

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text: Schicho.

text: Lattice

text: polygons

text: and

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text: number

text: Amer.

text: Math.

text: Monthly,

text: Paul

text: Hacking

text: and

text: Yuri

text: Prokhorov.

text: Smoothable

text: del

text: Pezzo

text: surfaces

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text: quotient

text: singularities.

text: Compos.

text: Math.,

text: Nathan

text: Owen

text: Ilten.

text: Mutations

text: of

text: Laurent

text: polynomials

text: and

text: ﬂat

text: families

text: with

text: toric

text: ﬁbers.

text: SIGMA

text: Symmetry

text: Inte-

text: grability

text: Geom.

text: Methods

text: Appl.,

text: Alexander

text: M.

text: Kasprzyk.

text: Bounds

text: on

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text: weighted

text: projective

text: space.

text: Kodai

text: Math.

text: J.,

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text: M.

text: Kasprzyk,

text: Maximilian

text: Kreuzer,

text: and

text: Benjamin

text: Nill.

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text: combinatorial

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text: Pezzo

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text: Computation

text: and

text: Mathematics,

text: J.

text: Koll´ar

text: and

text: N.

text: I.

text: Shepherd-Barron.

text: Threefolds

text: and

text: deformations

text: of

text: surface

text: singularities.

text: Invent.

text: Math.,

text: Maxim

text: Kontsevich

text: and

text: Yan

text: Soibelman.

text: Aﬃne

text: structures

text: and

text: non-Archimedean

text: analytic

text: spaces.

text: In

text: The

text: unity

text: of

text: mathematics,

text: volume

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text: Math.,

text: pages

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text: Boston,

text: Boston,

text: MA,

text: Jeﬀrey

text: C.

text: Lagarias

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text: Gu¨nter

text: M.

text: Ziegler.

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text: Canad.

text: J.

text: Math.,

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text: Ein

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text: fu¨r

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text: K¨orper.

text: Cˇasopis

text: Pˇest.

text: Mat.

text: Fys.,

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text: Lotos,

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text: THE REGULARITY OF SOME VECTOR-VALUED VARIATIONAL INEQUALITIES WITH GRADIENT CONSTRAINTS MOHAMMAD SAFDARI Abstract. We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case. 1. Introduction Let U ⊂ Rn be an open bounded set. Suppose K ⊂ Rn is a balanced (symmetric with respect to the origin) compact convex set whose interior contains 0. Also suppose that η ∈ RN is a ﬁxed nonzero vector. Consider the following problem of minimizing ˆ | | − · (1.1) I(v) := Dv 2 η v dx U over (1.2) K := {v = (v1, · · · , vN) ∈ H1(U; RN) | (cid:107)Dv(cid:107) ≤ 1 a.e.}, 1 0 2,K Where | | (cid:107) (cid:107) Az (1.3) A := sup 2,K (cid:54) γ (z) z=0 K × for an N n matrix A, and γ is the norm associated to K deﬁned by K { | ∈ } (1.4) γ (x) := inf λ > 0 x λK . K As K is a closed convex set and I is coercive, bounded and weakly sequentially 1 lower semicontinuous, this problem has a unique solution u. We will show that under some extra assumptions on K u ∈ C1,1(U; RN). loc Email address: safdari@berkeley.edu. 1 ##LTLine## ##LTLine##

text: THE

label: paragraph

text: REGULARITY

text: OF

text: SOME

text: VECTOR-VALUED

text: VARIATIONAL

text: INEQUALITIES

text: WITH

text: GRADIENT

text: CONSTRAINTS

text: MOHAMMAD

text: SAFDARI

text: Abstract.

text: We

label: abstract

text: prove

text: the

text: optimal

text: regularity

text: for

text: some

text: class

text: of

text: vector-valued

text: variational

text: inequalities

text: with

text: gradient

text: constraints.

text: We

text: also

text: give

text: a

text: new

text: proof

text: for

text: the

text: optimal

text: regularity

text: of

text: some

text: scalar

text: variational

text: inequalities

text: with

text: gradient

text: constraints.

text: In

text: addition,

text: we

text: prove

text: that

text: some

text: class

text: of

text: variational

text: inequalities

text: with

text: gradient

text: constraints

text: are

text: equivalent

text: to

text: an

text: obstacle

text: problem,

text: both

text: in

text: the

text: scalar

text: and

text: vector-valued

text: case.

label: section

text: Introduction

text: Let

text: U

text: Rn

text: be

text: an

text: open

text: bounded

text: set.

text: Suppose

text: K

text: Rn

text: is

text: a

text: balanced

text: (symmetric

text: with

text: respect

text: to

text: the

text: origin)

text: compact

text: convex

text: set

text: whose

text: interior

text: contains

text: Also

text: suppose

text: that

text: η

text: RN

text: is

text: a

text: ﬁxed

text: nonzero

text: vector.

text: Consider

text: the

text: following

text: problem

text: of

text: minimizing

text: ˆ

text: I(v)

text: Dv

text: η

text: v

text: dx

text: U

text: over

text: K

text: {v

text: vN)

text: RN)

text: a.e.},

text: Where

text: Az

text: A

text: sup

text: γ

text: (z)

text: K

text: for

text: an

text: N

text: n

text: matrix

text: A,

text: and

text: γ

text: is

text: the

text: norm

text: associated

text: to

text: K

text: deﬁned

text: by

text: K

text: γ

text: (x)

text: inf

text: λ

text: x

text: λK

text: K

text: As

text: K

text: is

text: a

text: closed

text: convex

text: set

text: and

text: I

text: is

text: coercive,

text: bounded

text: and

text: weakly

text: sequentially

text: lower

text: semicontinuous,

text: this

text: problem

text: has

text: a

text: unique

text: solution

text: u.

text: We

text: will

text: show

text: that

text: under

text: some

text: extra

text: assumptions

text: on

text: K

text: u

text: RN).

text: loc

text: Email

text: address:

text: safdari@berkeley.edu.

text: ##LTLine##

text: ##LTLine##


text: This problem is a generalization to the vector-valued case of the elastic-plastic ˆ torsion problem, which is the problem of minimizing | | − J (v) := Dv 2 ηv dx η U for some η > 0, over { ∈ | | | ≤ } v H1(U) Dv 1 a.e. . 0 The regularity of the elastic-plastic torsion problem has been studied by Brezis and Stampacchia [2], and Caﬀarelli and Rivière [3]. There has been several extensions of their results to more general scalar problems with gradient constraints. See for example Jensen [8], Gerhardt [6], Evans [4], Wiegner [14], Ishii and Koike [7]. To the best of author’s knowledge, the only work on the regularity of vector-valued problems with gradient constraints is Rozhkovskaya [12]. Our approach is to show that the above vector-valued problem is reducible to the scalar problem of minimizing J over 1 { ∈ | | | ∈ ◦ } v H1(U) η Dv K a.e. , 0 ◦ where K is the polar of K (See section 2). Then we show that this scalar problem is equivalent to a double obstacle problem with only Lipschitz obstacles. At the end, we generalize the proof of Caﬀarelli and Rivière [3], to obtain the optimal regularity. We should note that Lieberman [9] proves the regularity of a more general double obstacle problem by diﬀerent methods. In the process described above, we also show that our vector-valued problem with gradient constraint is equivalent to a vector-valued obstacle problem. This result, which is the ﬁrst result of its kind as far as the author knows, is a gen- eralization to the vector-valued case of the equivalence between the elastic-plastic torsion problem and an obstacle problem, proved by Brezis and Sibony [1]. Later Treu and Vornicescu [13] proved that the equivalence holds for a larger class of scalar variational inequalities with gradient constraints. We will further generalize their result. Suppose f : Rn → R and g : R → R are convex functions. Consider the problem of minimizing ˆ (1.5) J(v) := f(Dv(x)) + g(v(x)) dx U over { ∈ | ∈ } (1.6) W := v u + W1,p(U) Dv(x) K a.e. , K 0 0 ∈ where u W1,p(U). We will show that under appropriate assumptions, the 0 minimizer of J over W is the same as its minimizer over K { ∈ | − ≤ ≤ } (1.7) W − := v u + W1,p(U) u (x) v(x) u+(x) a.e. , u ,u+ 0 0 − for some suitable functions u , u+. The diﬀerence of our result with that of Treu and Vornicescu [13] is that we allow f, g to be only convex, and K to have empty interior. Some of our results has been proved using diﬀerent means by Mariconda and Treu [10]. 2

text: This

text: problem

text: is

text: a

text: generalization

text: to

text: the

text: vector-valued

text: case

text: of

text: the

text: elastic-plastic

text: ˆ

text: torsion

text: problem,

text: which

text: is

text: the

text: problem

text: of

text: minimizing

text: J

text: (v)

text: Dv

text: ηv

text: dx

text: η

text: U

text: for

text: some

text: η

text: over

text: v

text: Dv

text: a.e.

text: The

text: regularity

text: of

text: the

text: elastic-plastic

text: torsion

text: problem

text: has

text: been

text: studied

text: by

text: Brezis

text: and

text: Stampacchia

text: and

text: Caﬀarelli

text: and

text: Rivière

text: There

text: has

text: been

text: several

text: extensions

text: of

text: their

text: results

text: to

text: more

text: general

text: scalar

text: problems

text: with

text: gradient

text: constraints.

text: See

text: for

text: example

text: Jensen

text: Gerhardt

text: Evans

text: Wiegner

text: Ishii

text: and

text: Koike

text: To

text: the

text: best

text: of

text: author’s

text: knowledge,

text: the

text: only

text: work

text: on

text: the

text: regularity

text: of

text: vector-valued

text: problems

text: with

text: gradient

text: constraints

text: is

text: Rozhkovskaya

text: Our

text: approach

text: is

text: to

text: show

text: that

text: the

text: above

text: vector-valued

text: problem

text: is

text: reducible

text: to

text: the

text: scalar

text: problem

text: of

text: minimizing

text: J

text: over

text: v

text: η

text: Dv

text: K

text: a.e.

text: where

text: K

text: is

text: the

text: polar

text: of

text: K

text: (See

text: section

text: Then

text: we

text: show

text: that

text: this

text: scalar

text: problem

text: is

text: equivalent

text: to

text: a

text: double

text: obstacle

text: problem

text: with

text: only

text: Lipschitz

text: obstacles.

text: At

text: the

text: end,

text: we

text: generalize

text: the

text: proof

text: of

text: Caﬀarelli

text: and

text: Rivière

text: to

text: obtain

text: the

text: optimal

text: regularity.

text: We

text: should

text: note

text: that

text: Lieberman

text: proves

text: the

text: regularity

text: of

text: a

text: more

text: general

text: double

text: obstacle

text: problem

text: by

text: diﬀerent

text: methods.

text: In

text: the

text: process

text: described

text: above,

text: we

text: also

text: show

text: that

text: our

text: vector-valued

text: problem

text: with

text: gradient

text: constraint

text: is

text: equivalent

text: to

text: a

text: vector-valued

text: obstacle

text: problem.

text: This

text: result,

text: which

text: is

text: the

text: ﬁrst

text: result

text: of

text: its

text: kind

text: as

text: far

text: as

text: the

text: author

text: knows,

text: is

text: a

text: gen-

text: eralization

text: to

text: the

text: vector-valued

text: case

text: of

text: the

text: equivalence

text: between

text: the

text: elastic-plastic

text: torsion

text: problem

text: and

text: an

text: obstacle

text: problem,

text: proved

text: by

text: Brezis

text: and

text: Sibony

text: Later

text: Treu

text: and

text: Vornicescu

text: proved

text: that

text: the

text: equivalence

text: holds

text: for

text: a

text: larger

text: class

text: of

text: scalar

text: variational

text: inequalities

text: with

text: gradient

text: constraints.

text: We

text: will

text: further

text: generalize

text: their

text: result.

text: Suppose

text: f

text: Rn

text: R

text: and

text: g

text: R

text: R

text: are

text: convex

text: functions.

text: Consider

text: the

text: problem

text: of

text: minimizing

text: ˆ

text: J(v)

text: f(Dv(x))

text: g(v(x))

text: dx

text: U

text: over

text: W

text: v

text: u

text: Dv(x)

text: K

text: a.e.

text: K

text: where

text: u

text: We

text: will

text: show

text: that

text: under

text: appropriate

text: assumptions,

text: the

text: minimizer

text: of

text: J

text: over

text: W

text: is

text: the

text: same

text: as

text: its

text: minimizer

text: over

text: K

text: W

text: v

text: u

text: u

text: (x)

text: v(x)

text: u+(x)

text: a.e.

text: u

text: ,u+

text: for

text: some

text: suitable

text: functions

text: u

text: u+.

text: The

text: diﬀerence

text: of

text: our

text: result

text: with

text: that

text: of

text: Treu

text: and

text: Vornicescu

text: is

text: that

text: we

text: allow

text: f,

text: g

text: to

text: be

text: only

text: convex,

text: and

text: K

text: to

text: have

text: empty

text: interior.

text: Some

text: of

text: our

text: results

text: has

text: been

text: proved

text: using

text: diﬀerent

text: means

text: by

text: Mariconda

text: and

text: Treu


text: ˆ ≤ − γ (Du (x)) ˆ γ (η (x y)Du(y)) dy K (cid:15) K (cid:15) B(cid:15)(x) − ≤ = η (x y)γ (Du(y)) dy 1, (cid:15) K B(cid:15)(x) where we used Jensen’s inequality in the ﬁrst inequality. Thus ˆ − 1 − · − u(cid:15)(y) u(cid:15)(x) = ˆ Du(cid:15)(x + t(y x)) (y x) dt 0 ≤ 1 − − ≤ − γ (Du (x + t(y x)))γ ◦(y x) dt γ ◦(y x). K (cid:15) K K 0 → Now we can let (cid:15) 0 to obtain − ≤ − u(y) u(x) γ ◦(y x) for a.e. x, y. K We can redeﬁne u on the measure zero set where this relation fails, in a similar way that we extend Lipschitz functions to the closure of their domains. The extension (cid:3) will satisfy this relation everywhere. Lemma 2. Each function in W is Lipschitz continuous. Also, W is bounded in ∞ K K L (U) and in W1,p(U). ∈ ∈ Proof. To see this, let u W . Then u = u + v where v W1,p(U). Thus K 0 0 | | | − | Dv = Du Du < 2R 0 for some R > 0. Now we can extend v by zero to all of Rn, and the extension will satisfy the same gradient bound. Therefore by arguments similar to the previous lemma, we can see that the extension of v, and hence v itself, is Lipschitz with Lipschitz constant 2R. Using the fact that v is zero on the boundary, this also (cid:107) (cid:107) ≤ implies that v ∞ 2RD, where D is the diameter of U. The result for u follows L easily, noting that u is Lipschitz. (cid:107) (cid:107) 0 (cid:107) (cid:107) Now as Du ∞ < C for some constant C independent of u, we have Du < L ∈ Lp C since U is bounded. Noting that all u W have the same boundary value, we (cid:107) (cid:107) K (cid:3) get by Poincare inequality u < C. W1,p ± ± Now we can see that J are bounded on W . As J are linear, they are weakly K continuous. Furthermore W is convex, closed and bounded in W1,p(U). Hence K ± W is compact with respect to sequential weak convergence. These imply that J K − ≤ have minimizers over W . The uniqueness and the fact that u u+ a.e. on U, K follows from a similar argument to the proof of the next lemma. Lemma 3. We have ⊂ W W − . K u ,u+ ∈ ˆ ± ± ≤ ±ˆ Proof. Suppose u W , then J (u ) J (u). Thus K ∓ ± ≤ ∓ u dx u dx, ˆ ˆ ˆ U U so − ≤ ≤ u dx u dx u+ dx. U U U 4

text: ˆ

label: equation

text: γ

text: (Du

text: (x))

text: ˆ

text: γ

text: (η

text: (x

text: y)Du(y))

text: dy

text: K

text: K

text: η

text: (x

text: y)γ

text: (Du(y))

text: dy

text: K

text: where

text: we

text: used

text: Jensen’s

text: inequality

text: in

text: the

text: ﬁrst

text: inequality.

text: Thus

text: ˆ

text: ˆ

text: t(y

text: x))

text: (y

text: x)

text: dt

text: γ

text: (Du

text: (x

text: t(y

text: x)))γ

text: ◦(y

text: x)

text: dt

text: γ

text: ◦(y

text: x).

text: K

text: K

text: K

text: Now

text: we

text: can

text: let

text: to

text: obtain

text: u(y)

text: u(x)

text: γ

text: ◦(y

text: x)

text: for

text: a.e.

text: x,

text: y.

text: K

text: We

text: can

text: redeﬁne

text: u

text: on

text: the

text: measure

text: zero

text: set

text: where

text: this

text: relation

text: fails,

text: in

text: a

text: similar

text: way

text: that

text: we

text: extend

text: Lipschitz

text: functions

text: to

text: the

text: closure

text: of

text: their

text: domains.

text: The

text: extension

text: will

text: satisfy

text: this

text: relation

text: everywhere.

text: Lemma

text: Each

text: function

text: in

text: W

text: is

text: Lipschitz

text: continuous.

text: Also,

text: W

text: is

text: bounded

text: in

text: K

text: K

text: L

text: (U)

text: and

text: in

text: Proof.

text: To

text: see

text: this,

text: let

text: u

text: W

text: Then

text: u

text: u

text: v

text: where

text: v

text: Thus

text: K

text: Dv

text: Du

text: Du

text: for

text: some

text: R

text: Now

text: we

text: can

text: extend

text: v

text: by

text: zero

text: to

text: all

text: of

text: Rn,

text: and

text: the

text: extension

text: will

text: satisfy

text: the

text: same

text: gradient

text: bound.

text: Therefore

text: by

text: arguments

text: similar

text: to

text: the

text: previous

text: lemma,

text: we

text: can

text: see

text: that

text: the

text: extension

text: of

text: v,

text: and

text: hence

text: v

text: itself,

text: is

text: Lipschitz

text: with

text: Lipschitz

text: constant

text: Using

text: the

text: fact

text: that

text: v

text: is

text: zero

text: on

text: the

text: boundary,

text: this

text: also

text: implies

text: that

text: v

text: where

text: D

text: is

text: the

text: diameter

text: of

text: U.

text: The

text: result

text: for

text: u

text: follows

text: L

text: easily,

text: noting

text: that

text: u

text: is

text: Lipschitz.

text: Now

text: as

text: Du

text: C

text: for

text: some

text: constant

text: C

text: independent

text: of

text: u,

text: we

text: have

text: Du

text: L

text: Lp

text: C

text: since

text: U

text: is

text: bounded.

text: Noting

text: that

text: all

text: u

text: W

text: have

text: the

text: same

text: boundary

text: value,

text: we

text: K

text: get

text: by

text: Poincare

text: inequality

text: u

text: C.

text: Now

text: we

text: can

text: see

text: that

text: J

text: are

text: bounded

text: on

text: W

text: As

text: J

text: are

text: linear,

text: they

text: are

text: weakly

text: K

text: continuous.

text: Furthermore

text: W

text: is

text: convex,

text: closed

text: and

text: bounded

text: in

text: Hence

text: K

text: W

text: is

text: compact

text: with

text: respect

text: to

text: sequential

text: weak

text: convergence.

text: These

text: imply

text: that

text: J

text: K

text: have

text: minimizers

text: over

text: W

text: The

text: uniqueness

text: and

text: the

text: fact

text: that

text: u

text: u+

text: a.e.

text: on

text: U,

text: K

text: follows

text: from

text: a

text: similar

text: argument

text: to

text: the

text: proof

text: of

text: the

text: next

text: lemma.

text: Lemma

text: We

text: have

text: W

text: W

text: K

text: u

text: ,u+

text: ˆ

text: ±ˆ

text: Proof.

text: Suppose

text: u

text: W

text: then

text: J

text: (u

text: J

text: (u).

text: Thus

text: K

text: u

text: dx

text: u

text: dx,

text: ˆ

text: ˆ

text: ˆ

text: U

text: U

text: so

text: u

text: dx

text: u

text: dx

text: u+

text: dx.

text: U

text: U

text: U


text: − The inverse ϕ = Φ 1 of Φ provides a system of privileged coordinates at q (see [17]) and in these q q q { · · · } (cid:55)→ coordinates N coincides locally with the set x = = x = 0 . Moreover, the map (q, x) Φ (x) is ∩ k+1 n q smooth on N U, which completes the proof of points (I) and (II). ∈ ∩ (cid:48) ∈ Finally, let us prove (16). Let q U N and q U. We denote respectively by x and y the coordinates of (cid:48) (cid:48) ◦· · ·◦ q centered at q and p, that is, q = Φ (x) = Φ (y). By construction, the point exp (x Z ) exp (x Z ) (q) ∩ q p · · · k k 1 1 belongs to U N, and hence may be written as Φ (z) with z = = z = 0. Therefore, p k+1 n (cid:48) ◦ · · · ◦ ◦ ◦ · · · ◦ q = Φ (x) = exp (x Z ) exp (x Z ) exp (z Z ) exp (z Z ) (p), q n n k+1 k+1 k k 1 1 which implies y = x for i = k + 1, . . . , n, i.e. (16). i i 4.2 Suﬃcient conditions In the sequel, we provide suﬃcient conditions to obtain either local integrability or non integrability S (cid:120) of d QR R around a ﬁxed singular point. These conditions will be of diﬀerent nature (either diﬀerential or dµ metric), depending on whether they are obtained with (14) or (15). Note that by assumption (A), S admits a stratiﬁcation by equisingular submanifolds, hence every sin- gular point belongs to an equisingular submanifold whic(cid:40)h is determined in(cid:12) a unique way.(cid:41)Recall also from Proposition 3.12 that (cid:115)(cid:88) (cid:12)(cid:12) (cid:88) (cid:12) (cid:12) X F I | | ν(q) = ω( I)(q)2 and = = (I , . . . , I ) I = Q . 1 n i R I∈F i 4.2.1 Conditions on nonholonomic orders ∈ ⊂ Consider a singular point p S and an equisingular submanifold N S containing p. Deﬁne { | ∈ ∩ } (cid:37) (p, N) = lim min ord ν q N B(p, (cid:15)) , min → q (cid:15) 0 where ord (h) is the nonholonomic order of the function h at q. Note that by Remark 3.13, ord ν = q X p minI∈F ord (ω( I)), so that p { X | ∈ ∩ I ∈ F} (cid:37) (p, N) = lim min ord (ω( I)) q N B(p, (cid:15)), . min → q (cid:15) 0 S Next result gives an algebraic suﬃcient condition for a neighborhood of p to have inﬁnite QR measure. This condition involves the order of ν at singular points around p ((cid:37) (p, N)), the ﬂag of the distribution min at p (Q(p)) and the ﬂag of the distribution restricted to the singular set at p (Q ). ∈ N Proposition 4.4. Assume (A). Let p S and let N be an equisingular submanifold containing p. If ≥ − S (cid:120) (cid:37) (p, N) Q(p) Q , then d QR R is not µ-locally integrable around p. min N dµ (cid:88) Proof. We will apply condition (15) with the coordinate system ϕ = (y, z) built in Lemma 4.3. Fix R small p enough. By deﬁnition, X ν(Φ (y, δ z))2 = ω( I)(Φ (y, δ z))2, (cid:80) p λ p λ I∈F F I | | where is the family of n-tuples of multi-indices = (I , . . . , I ) such that I = Q . Since p belongs I ∈ F 1 n j j R to the singular set, for any family , the vectors X (p), . . . , X (p) are linearly dependent. Similarly, I1 In for every y near 0, the vectors X (Φ (y, 0)), . . . , X (Φ (y, 0)) are linearly dependent. As a consequence, ≡ I1 p In p ν(Φ (y, 0)) 0. p ∈ {(cid:107) (cid:107) ≤ } Given y y R , denote by (cid:37)(y) the greatest number such that there exists a constant C > 0 y satisfying ≤ ν(Φ (y, δ z)) C λ(cid:37)(y), p λ y 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: inverse

text: ϕ

text: Φ

text: of

text: Φ

text: provides

text: a

text: system

text: of

text: privileged

text: coordinates

text: at

text: q

text: (see

text: and

text: in

text: these

text: q

text: q

text: q

text: coordinates

text: N

text: coincides

text: locally

text: with

text: the

text: set

text: x

text: x

text: Moreover,

text: the

text: map

text: (q,

text: x)

text: Φ

text: (x)

text: is

text: n

text: q

text: smooth

text: on

text: N

text: U,

text: which

text: completes

text: the

text: proof

text: of

text: points

text: (I)

text: and

text: (II).

text: Finally,

text: let

text: us

text: prove

text: Let

text: q

text: U

text: N

text: and

text: q

text: U.

text: We

text: denote

text: respectively

text: by

text: x

text: and

text: y

text: the

text: coordinates

text: of

text: q

text: centered

text: at

text: q

text: and

text: p,

text: that

text: is,

text: q

text: Φ

text: (x)

text: Φ

text: (y).

text: By

text: construction,

text: the

text: point

text: exp

text: (x

text: Z

text: exp

text: (x

text: Z

text: (q)

text: q

text: p

text: k

text: k

text: belongs

text: to

text: U

text: N,

text: and

text: hence

text: may

text: be

text: written

text: as

text: Φ

text: (z)

text: with

text: z

text: z

text: Therefore,

text: p

text: n

text: q

text: Φ

text: (x)

text: exp

text: (x

text: Z

text: exp

text: (x

text: Z

text: exp

text: (z

text: Z

text: exp

text: (z

text: Z

text: (p),

text: q

text: n

text: n

text: k

text: k

text: which

text: implies

text: y

text: x

text: for

text: i

text: k

text: n,

text: i.e.

text: i

text: i

text: Suﬃcient

text: conditions

text: In

text: the

text: sequel,

text: we

text: provide

text: suﬃcient

text: conditions

text: to

text: obtain

text: either

text: local

text: integrability

text: or

text: non

text: integrability

text: S

text: of

text: d

text: QR

text: R

text: around

text: a

text: ﬁxed

text: singular

text: point.

text: These

text: conditions

text: will

text: be

text: of

text: diﬀerent

text: nature

text: (either

text: diﬀerential

text: or

text: dµ

text: metric),

text: depending

text: on

text: whether

text: they

text: are

text: obtained

text: with

text: or

text: Note

text: that

text: by

text: assumption

text: (A),

text: S

text: admits

text: a

text: stratiﬁcation

text: by

text: equisingular

text: submanifolds,

text: hence

text: every

text: sin-

text: gular

text: point

text: belongs

text: to

text: an

text: equisingular

text: submanifold

text: is

text: determined

text: a

text: unique

text: also

text: from

text: Proposition

text: that

text: X

text: F

text: I

text: ν(q)

text: ω(

text: and

text: (I

text: I

text: I

text: Q

text: n

text: i

text: R

text: I∈F

text: i

text: Conditions

text: on

text: nonholonomic

text: orders

text: Consider

text: a

text: singular

text: point

text: p

text: S

text: and

text: an

text: equisingular

text: submanifold

text: N

text: S

text: containing

text: p.

text: Deﬁne

text: (p,

text: N)

text: lim

text: min

text: ord

text: ν

text: q

text: N

text: B(p,

text: min

text: q

text: where

text: ord

text: (h)

text: is

text: the

text: nonholonomic

text: order

text: of

text: the

text: function

text: h

text: at

text: q.

text: Note

text: that

text: by

text: Remark

text: ord

text: ν

text: q

text: X

text: p

text: minI∈F

text: ord

text: (ω(

text: I)),

text: so

text: that

text: p

text: X

text: I

text: F}

text: (p,

text: N)

text: lim

text: min

text: ord

text: (ω(

text: I))

text: q

text: N

text: B(p,

text: min

text: q

text: S

text: Next

text: result

text: gives

text: an

text: algebraic

text: suﬃcient

text: condition

text: for

text: a

text: neighborhood

text: of

text: p

text: to

text: have

text: inﬁnite

text: QR

text: measure.

text: This

text: condition

text: involves

text: the

text: order

text: of

text: ν

text: at

text: singular

text: points

text: around

text: p

text: (p,

text: N)),

text: the

text: ﬂag

text: of

text: the

text: distribution

text: min

text: at

text: p

text: (Q(p))

text: and

text: the

text: ﬂag

text: of

text: the

text: distribution

text: restricted

text: to

text: the

text: singular

text: set

text: at

text: p

text: (Q

text: N

text: Proposition

text: Assume

text: (A).

text: Let

text: p

text: S

text: and

text: let

text: N

text: be

text: an

text: equisingular

text: submanifold

text: containing

text: p.

text: If

text: S

text: (p,

text: N)

text: Q(p)

text: Q

text: then

text: d

text: QR

text: R

text: is

text: not

text: µ-locally

text: integrable

text: around

text: p.

text: min

text: N

text: dµ

text: Proof.

text: We

text: will

text: apply

text: condition

text: with

text: the

text: coordinate

text: system

text: ϕ

text: (y,

text: z)

text: built

text: in

text: Lemma

text: Fix

text: R

text: small

text: p

text: enough.

text: By

text: deﬁnition,

text: X

text: ν(Φ

text: (y,

text: δ

text: ω(

text: I)(Φ

text: (y,

text: δ

text: p

text: λ

text: p

text: λ

text: I∈F

text: F

text: I

text: where

text: is

text: the

text: family

text: of

text: n-tuples

text: of

text: multi-indices

text: (I

text: I

text: such

text: that

text: I

text: Q

text: Since

text: p

text: belongs

text: I

text: F

text: n

text: j

text: j

text: R

text: to

text: the

text: singular

text: set,

text: for

text: any

text: family

text: the

text: vectors

text: X

text: (p),

text: X

text: (p)

text: are

text: linearly

text: dependent.

text: Similarly,

text: In

text: for

text: every

text: y

text: near

text: the

text: vectors

text: X

text: (Φ

text: (y,

text: X

text: (Φ

text: (y,

text: are

text: linearly

text: dependent.

text: As

text: a

text: consequence,

text: p

text: In

text: p

text: ν(Φ

text: (y,

text: p

text: Given

text: y

text: y

text: R

text: denote

text: by

text: the

text: greatest

text: number

text: such

text: that

text: there

text: exists

text: a

text: constant

text: C

text: y

text: satisfying

text: ν(Φ

text: (y,

text: δ

text: z))

text: C

text: p

text: λ

text: y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:98) vol (B(p, r)) 2Q lim H = , (2) r→0 µ(B(p, r)) µˆp(B ) (cid:98) p where B(p, r) is the sub-Riemannian ball centered at p of radius r, B is the unit ball in the nilpotent p approximation at p and µˆp is a measure obtained through a blow-up procedure of µ at p. As a consequence, dvolH is continuous on M and hence locally bounded and locally bounded away from zero on M. With the dµ language of [22, 23], this implies that vol and µ are commensurable and, in particular, that vol is a Radon ∞ H H measure, i.e., vol (K) < for every compact set K. Therefore, when the manifold is equiregular, vol is H H well understood in the sense that it behaves essentially as a smooth volume. Nevertheless, further regularity of dvolH is not granted see [1]. dµ In this paper we study the Hausdorﬀ volume in a non equiregular sub-Riemannian manifold. A point p is (cid:55)→ D D called regular if the growth vector q (dim 1, . . . , dim r(q)) is constant in a neighborhood of p, otherwise q q p is called singular. The natural assumption under which we perform our work is that the manifold is stratiﬁed by equisingular submanifolds, where both the growth vector of the distribution and the growth vector of the distribution restricted to the submanifold are constant. These submanifolds were introduced in [16] and, thanks to their simple Lie algebraic structure, they constitute the fundamental block that allows us to carry through our investigation. When the set S of singular points is not empty, the Hausdorﬀ dimension of M is obviously the maximum \ between the Hausdorﬀ dimension of S and the Hausdorﬀ dimension of the set R = M S of regular points. The ﬁrst question is whether vol is absolutely continuous with respect to a smooth volume µ. It turns H out that this may not be the case and vol may have a singular part. More precisely, under the assumption H that S is µ-negligible, then vol admits the following Lebesgue decomposition H (cid:120) (cid:120) vol = vol +vol , H H R H S (cid:120) (cid:28) (cid:120) ⊥ that is, vol µ and vol µ (see Corollary 3.9). As a consequence, vol is absolutely continuous H R H S H with respect to a smooth volume µ if and only if the Hausdorﬀ dimension of the singular set is smaller than the Hausdorﬀ dimension of the set of regular points. (cid:98) (cid:120) The next problem is to analyze the behavior of the absolutely continuous part vol (which is nontrivial ≥ H R (cid:55)→ if and only if dim R dim S). Thanks to (2), this amounts to study the function q µˆq(B ) near H H q singular points. To this aim, we show that the asymptotics of this function is characterized by the one of the determinants of adapted bases at regular points (see Proposition 3.12). As a direct consequence, we deduce (cid:120) (cid:120) that dvolH R (q) blows up when q approaches the singular set (see Proposition 3.15). In particular, dvolH R dµ (cid:120) dµ is not essentially bounded near the singular set, that is, unlike the equiregular case, µ and vol are no (cid:120) H R longer commensurable. Going further in the regularity analysis, we ﬁnd out that vol may even fail to be (cid:120) H R locally integrable with respect to µ and therefore vol may fail to be a Radon measure. More precisely, H R (cid:120) we exhibit a suﬃcient condition involving the nonholonomic order of dvolH R and the algebraic structure of (cid:120) dµ the distribution at a singular point for non-integrability of vol (see Proposition 4.4), and another one H R involving the usual order of functions and the codimension of the singular set (see Proposition 4.10). For instance, these conditions are satisﬁed when the Hausdorﬀ dimension of R is not greater than the Hausdorﬀ dimension of S or when S topologically splits M (see Corollaries 4.6 and 4.11). We also exhibit a suﬃcient (cid:120) condition for the integrability of vol (see Proposition 4.9), but there is a gap between integrability and H R non-integrability conditions. In Figure 1 we summarize the relations between the Hausdorﬀ volume and µ in all cases. For generic sub-Riemannian manifolds, we apply our technique and we characterize the integrability of the absolutely continuous part by comparing the dimension of the manifold to dimensions of free Lie algebras (see Proposition 6.1). (cid:120) As for the singular part vol , the stratiﬁcation assumption permits to focus on each equisingular ⊂ H S ⊂ submanifold N S. For an equisingular submanifold N S, we give an algebraic characterization of the Hausdorﬀ dimension of N and we compare the Hausdorﬀ volume with a smooth volume on the submanifold | (see Theorem 5.3). We show actually that on the restricted metric space (N, d ) the situation is very N 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: vol

text: (B(p,

text: r))

text: lim

text: H

text: µ(B(p,

text: r))

text: µˆp(B

text: p

text: where

text: B(p,

text: r)

text: is

text: the

text: sub-Riemannian

text: ball

text: centered

text: at

text: p

text: of

text: radius

text: r,

text: B

text: is

text: the

text: unit

text: ball

text: in

text: the

text: nilpotent

text: p

text: approximation

text: at

text: p

text: and

text: µˆp

text: is

text: a

text: measure

text: obtained

text: through

text: a

text: blow-up

text: procedure

text: of

text: µ

text: at

text: p.

text: As

text: a

text: consequence,

text: dvolH

text: is

text: continuous

text: on

text: M

text: and

text: hence

text: locally

text: bounded

text: and

text: locally

text: bounded

text: away

text: from

text: zero

text: on

text: M.

text: With

text: the

text: dµ

text: language

text: of

text: this

text: implies

text: that

text: vol

text: and

text: µ

text: are

text: commensurable

text: and,

text: in

text: particular,

text: that

text: vol

text: is

text: a

text: Radon

text: H

text: H

text: measure,

text: i.e.,

text: vol

text: (K)

text: for

text: every

text: compact

text: set

text: K.

text: Therefore,

text: when

text: the

text: manifold

text: is

text: equiregular,

text: vol

text: is

text: H

text: H

text: well

text: understood

text: in

text: the

text: sense

text: that

text: it

text: behaves

text: essentially

text: as

text: a

text: smooth

text: volume.

text: Nevertheless,

text: further

text: regularity

text: of

text: dvolH

text: is

text: not

text: granted

text: see

text: dµ

text: In

text: this

text: paper

text: we

text: study

text: the

text: Hausdorﬀ

text: volume

text: in

text: a

text: non

text: equiregular

text: sub-Riemannian

text: manifold.

text: A

text: point

text: p

text: is

text: D

text: D

text: called

text: regular

text: if

text: the

text: growth

text: vector

text: q

text: (dim

text: dim

text: r(q))

text: is

text: constant

text: in

text: a

text: neighborhood

text: of

text: p,

text: otherwise

text: q

text: q

text: p

text: is

text: called

text: singular.

text: The

text: natural

text: assumption

text: under

text: which

text: we

text: perform

text: our

text: work

text: is

text: that

text: the

text: manifold

text: is

text: stratiﬁed

text: by

text: equisingular

text: submanifolds,

text: where

text: both

text: the

text: growth

text: vector

text: of

text: the

text: distribution

text: and

text: the

text: growth

text: vector

text: of

text: the

text: distribution

text: restricted

text: to

text: the

text: submanifold

text: are

text: constant.

text: These

text: submanifolds

text: were

text: introduced

text: in

text: and,

text: thanks

text: to

text: their

text: simple

text: Lie

text: algebraic

text: structure,

text: they

text: constitute

text: the

text: fundamental

text: block

text: that

text: allows

text: us

text: to

text: carry

text: through

text: our

text: investigation.

text: When

text: the

text: set

text: S

text: of

text: singular

text: points

text: is

text: not

text: empty,

text: the

text: Hausdorﬀ

text: dimension

text: of

text: M

text: is

text: obviously

text: the

text: maximum

text: between

text: the

text: Hausdorﬀ

text: dimension

text: of

text: S

text: and

text: the

text: Hausdorﬀ

text: dimension

text: of

text: the

text: set

text: R

text: M

text: S

text: of

text: regular

text: points.

text: The

text: ﬁrst

text: question

text: is

text: whether

text: vol

text: is

text: absolutely

text: continuous

text: with

text: respect

text: to

text: a

text: smooth

text: volume

text: µ.

text: It

text: turns

text: H

text: out

text: that

text: this

text: may

text: not

text: be

text: the

text: case

text: and

text: vol

text: may

text: have

text: a

text: singular

text: part.

text: More

text: precisely,

text: under

text: the

text: assumption

text: H

text: that

text: S

text: is

text: µ-negligible,

text: then

text: vol

text: admits

text: the

text: following

text: Lebesgue

text: decomposition

text: H

text: vol

text: vol

text: +vol

text: H

text: H

text: R

text: H

text: S

text: that

text: is,

text: vol

text: µ

text: and

text: vol

text: µ

text: (see

text: Corollary

text: As

text: a

text: consequence,

text: vol

text: is

text: absolutely

text: continuous

text: H

text: R

text: H

text: S

text: H

text: with

text: respect

text: to

text: a

text: smooth

text: volume

text: µ

text: if

text: and

text: only

text: if

text: the

text: Hausdorﬀ

text: dimension

text: of

text: the

text: singular

text: set

text: is

text: smaller

text: than

text: the

text: Hausdorﬀ

text: dimension

text: of

text: the

text: set

text: of

text: regular

text: points.

text: The

text: next

text: problem

text: is

text: to

text: analyze

text: the

text: behavior

text: of

text: the

text: absolutely

text: continuous

text: part

text: vol

text: (which

text: is

text: nontrivial

text: H

text: R

text: if

text: and

text: only

text: if

text: dim

text: R

text: dim

text: S).

text: Thanks

text: to

text: this

text: amounts

text: to

text: study

text: the

text: function

text: q

text: µˆq(B

text: near

text: H

text: H

text: q

text: singular

text: points.

text: To

text: this

text: aim,

text: we

text: show

text: that

text: the

text: asymptotics

text: of

text: this

text: function

text: is

text: characterized

text: by

text: the

text: one

text: of

text: the

text: determinants

text: of

text: adapted

text: bases

text: at

text: regular

text: points

text: (see

text: Proposition

text: As

text: a

text: direct

text: consequence,

text: we

text: deduce

text: that

text: dvolH

text: R

text: (q)

text: blows

text: up

text: when

text: q

text: approaches

text: the

text: singular

text: set

text: (see

text: Proposition

text: In

text: particular,

text: dvolH

text: R

text: dµ

text: dµ

text: is

text: not

text: essentially

text: bounded

text: near

text: the

text: singular

text: set,

text: that

text: is,

text: unlike

text: the

text: equiregular

text: case,

text: µ

text: and

text: vol

text: are

text: no

text: H

text: R

text: longer

text: commensurable.

text: Going

text: further

text: in

text: the

text: regularity

text: analysis,

text: we

text: ﬁnd

text: out

text: that

text: vol

text: may

text: even

text: fail

text: to

text: be

text: H

text: R

text: locally

text: integrable

text: with

text: respect

text: to

text: µ

text: and

text: therefore

text: vol

text: may

text: fail

text: to

text: be

text: a

text: Radon

text: measure.

text: More

text: precisely,

text: H

text: R

text: we

text: exhibit

text: a

text: suﬃcient

text: condition

text: involving

text: the

text: nonholonomic

text: order

text: of

text: dvolH

text: R

text: and

text: the

text: algebraic

text: structure

text: of

text: dµ

text: the

text: distribution

text: at

text: a

text: singular

text: point

text: for

text: non-integrability

text: of

text: vol

text: (see

text: Proposition

text: and

text: another

text: one

text: H

text: R

text: involving

text: the

text: usual

text: order

text: of

text: functions

text: and

text: the

text: codimension

text: of

text: the

text: singular

text: set

text: (see

text: Proposition

text: For

text: instance,

text: these

text: conditions

text: are

text: satisﬁed

text: when

text: the

text: Hausdorﬀ

text: dimension

text: of

text: R

text: is

text: not

text: greater

text: than

text: the

text: Hausdorﬀ

text: dimension

text: of

text: S

text: or

text: when

text: S

text: topologically

text: splits

text: M

text: (see

text: Corollaries

text: and

text: We

text: also

text: exhibit

text: a

text: suﬃcient

text: condition

text: for

text: the

text: integrability

text: of

text: vol

text: (see

text: Proposition

text: but

text: there

text: is

text: a

text: gap

text: between

text: integrability

text: and

text: H

text: R

text: non-integrability

text: conditions.

text: In

text: Figure

text: we

text: summarize

text: the

text: relations

text: between

text: the

text: Hausdorﬀ

text: volume

text: and

text: µ

text: in

text: all

text: cases.

text: For

text: generic

text: sub-Riemannian

text: manifolds,

text: we

text: apply

text: our

text: technique

text: and

text: we

text: characterize

text: the

text: integrability

text: of

text: the

text: absolutely

text: continuous

text: part

text: by

text: comparing

text: the

text: dimension

text: of

text: the

text: manifold

text: to

text: dimensions

text: of

text: free

text: Lie

text: algebras

text: (see

text: Proposition

text: As

text: for

text: the

text: singular

text: part

text: vol

text: the

text: stratiﬁcation

text: assumption

text: permits

text: to

text: focus

text: on

text: each

text: equisingular

text: H

text: S

text: submanifold

text: N

text: S.

text: For

text: an

text: equisingular

text: submanifold

text: N

text: S,

text: we

text: give

text: an

text: algebraic

text: characterization

text: of

text: the

text: Hausdorﬀ

text: dimension

text: of

text: N

text: and

text: we

text: compare

text: the

text: Hausdorﬀ

text: volume

text: with

text: a

text: smooth

text: volume

text: on

text: the

text: submanifold

text: (see

text: Theorem

text: We

text: show

text: actually

text: that

text: on

text: the

text: restricted

text: metric

text: space

text: (N,

text: d

text: the

text: situation

text: is

text: very

text: N

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:113) p such that (y, z) is a system of local coordinates near p. In these coordinates, N is identiﬁed with the set { | } · · · (y, z) z = 0 and ν(z) = z2 + + z2. Thus 1/ν is locally integrable near p, which ends the proof. k+1 n 6.2 Examples In this section we present several examples where assumption (A) is satisﬁed and one can directly tell S whether QR is integrable or not using criteria in Section . (cid:104)· ·(cid:105) Recall that a sub-Riemannian structure (U, , , f) on an n-dimensional manifold M is called almost- Riemannian if the rank of U is n (see [3]). (cid:104)· ·(cid:105) Example 6.2 (Generic almost-Riemannian structures). Let (U, , , f) be a generic almost-Riemannian struc- ture on M. At regular points the structure is Riemannian, i.e., the growth vector is simply (n) and Q = n. S (cid:120) R In particular, n = n˜ and thus the ﬁrst case of Proposition 6.1 applies and d n R is never locally integrable 1 Sdµ(cid:120) (with respect to a smooth volume) around singular points. As a consequence, n is never a Radon measure. R Also, an alternative proof of this fact comes as a consequence of Corollary 4.6 or Corollary 4.11. S We next build examples of almost-Riemannian structures which are not generic and for which n is or fails to be a Radon measure. Example 6.3 (Non generic almost-Riemannian structures). We consider the almost-Riemannian structure on R 3 for which a global generating family is X (x , x , x ) = ∂ , X (x , x , x ) = ∂ , X (x , x , x ) = (x2 + x2)∂ , 1 1 2 3 1 2 1 2 3 2 3 1 2 3 1 2 3 ∧ ∧ { | and the canonical volume form ω = dx dx dx . The singular set S coincides with (x , x , x ) x = } 1 2 3 1 2 3 1 x = 0 and Q = 3. The growth vector at a singular point p is (2, 2, 3), Q(p) = 5, and the growth vector of 2 R R the ﬂag restricted to S is (0, 0, 1), whence Q = 3. Hence the Hausdorﬀ dimension of 3 (endowed with the SS S (cid:120) sub-Riemannian distance) is 3 and vol = 3. Therefore conditions of Corollary 4.6 are satisﬁed and d 3 R H dµ is not integrable near a singular point. In other words, small neighborhoods of singular points have inﬁnite Hausdorﬀ volume. One could also obtain the conclusion by applying Proposition 4.10 since the only adapted basis at regular points is X , X , X . Indeed, 1 2 3 (cid:16) ν(x , x , x ) x2 + x2, 1 2 3 1 2 − and ν is homogeneous of degree 2. Note that (cid:37) (p, S) = (cid:37) (p, S) = 2 = Q(p) Q . min max S Consider now the analogue structure in higher dimension, that is, the almost-Riemannian structure on R 4 for which a global generating family is X (x) = ∂ , X (x) = ∂ , X (x) = ∂ , X (x) = (x2 + x2 + x2)∂ , 1 1 2 2 3 3 4 1 2 3 4 ∧ ∧ ∧ where x = (x , x , x , x ), and the canonical volume form ω = dx dx dx dx . The singular set is { ∈ R |1 2 3 4 } 1 2 3 4 S = x 4 x = x = x = 0 and Q = 4. The growth vector at a singular point p is (3, 3, 4), Q(p) = 6, 1 2 3 R and the growth vector of the ﬂag restricted to S is (0, 0, 1), whence Q = 3. Hence the Hausdorﬀ dimension R SS of 4 (endowed with the sub-Riemannian distance) is 4 and vol = 4. The only adapted basis at regular H points is X , X , X , X , which gives 1 2 3 4 (cid:16) ν(x) x2 + x2 + x2, 1 2 3 − i.e., ν is homogeneous of degree 2 and (cid:37) (p, S) = (cid:37) (p, S) = 2 < Q(p) Q . Then by Proposition 4.9 S min maSx S d 4 is integrable near any singular point. Equivalently, 4 is a Radon measure. dµ R Example 6.4 (the Martinet space). Recall the sub-Riemannian structure of Example 3.14, where M = 3 and a global generating family is x2 X = ∂ , X = ∂ + 1 ∂ . 1 1 2 2 2 3 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: such

text: that

text: (y,

text: z)

text: is

text: a

text: system

text: of

text: local

text: coordinates

text: near

text: p.

text: In

text: these

text: coordinates,

text: N

text: is

text: identiﬁed

text: with

text: the

text: set

text: (y,

text: z)

text: z

text: and

text: ν(z)

text: Thus

text: is

text: locally

text: integrable

text: near

text: p,

text: which

text: ends

text: the

text: proof.

text: n

text: Examples

text: In

text: this

text: section

text: we

text: present

text: several

text: examples

text: where

text: assumption

text: (A)

text: is

text: satisﬁed

text: and

text: one

text: can

text: directly

text: tell

text: S

text: whether

text: QR

text: is

text: integrable

text: or

text: not

text: using

text: criteria

text: in

text: Section

text: Recall

text: that

text: a

text: sub-Riemannian

text: structure

text: (U,

text: f)

text: on

text: an

text: n-dimensional

text: manifold

text: M

text: is

text: called

text: almost-

text: Riemannian

text: if

text: the

text: rank

text: of

text: U

text: is

text: n

text: (see

text: Example

text: (Generic

text: almost-Riemannian

text: structures).

text: Let

text: (U,

text: f)

text: be

text: a

text: generic

text: almost-Riemannian

text: struc-

text: ture

text: on

text: M.

text: At

text: regular

text: points

text: the

text: structure

text: is

text: Riemannian,

text: i.e.,

text: the

text: growth

text: vector

text: is

text: simply

text: (n)

text: and

text: Q

text: n.

text: S

text: R

text: In

text: particular,

text: n

text: n˜

text: and

text: thus

text: the

text: ﬁrst

text: case

text: of

text: Proposition

text: applies

text: and

text: d

text: n

text: R

text: is

text: never

text: locally

text: integrable

text: (with

text: respect

text: to

text: a

text: smooth

text: volume)

text: around

text: singular

text: points.

text: As

text: a

text: consequence,

text: n

text: is

text: never

text: a

text: Radon

text: measure.

text: R

text: Also,

text: an

text: alternative

text: proof

text: of

text: this

text: fact

text: comes

text: as

text: a

text: consequence

text: of

text: Corollary

text: or

text: Corollary

text: S

text: We

text: next

text: build

text: examples

text: of

text: almost-Riemannian

text: structures

text: which

text: are

text: not

text: generic

text: and

text: for

text: which

text: n

text: is

text: or

text: fails

text: to

text: be

text: a

text: Radon

text: measure.

text: Example

text: (Non

text: generic

text: almost-Riemannian

text: structures).

text: We

text: consider

text: the

text: almost-Riemannian

text: structure

text: on

text: R

text: for

text: which

text: a

text: global

text: generating

text: family

text: is

text: X

text: (x

text: x

text: x

text: X

text: (x

text: x

text: x

text: X

text: (x

text: x

text: x

text: and

text: the

text: canonical

text: volume

text: form

text: ω

text: dx

text: dx

text: dx

text: The

text: singular

text: set

text: S

text: coincides

text: with

text: (x

text: x

text: x

text: x

text: x

text: and

text: Q

text: The

text: growth

text: vector

text: at

text: a

text: singular

text: point

text: p

text: is

text: Q(p)

text: and

text: the

text: growth

text: vector

text: of

text: R

text: R

text: the

text: ﬂag

text: restricted

text: to

text: S

text: is

text: whence

text: Q

text: Hence

text: the

text: Hausdorﬀ

text: dimension

text: of

text: (endowed

text: with

text: the

text: SS

text: S

text: sub-Riemannian

text: distance)

text: is

text: and

text: vol

text: Therefore

text: conditions

text: of

text: Corollary

text: are

text: satisﬁed

text: and

text: d

text: R

text: H

text: dµ

text: is

text: not

text: integrable

text: near

text: a

text: singular

text: point.

text: In

text: other

text: words,

text: small

text: neighborhoods

text: of

text: singular

text: points

text: have

text: inﬁnite

text: Hausdorﬀ

text: volume.

text: One

text: could

text: also

text: obtain

text: the

text: conclusion

text: by

text: applying

text: Proposition

text: since

text: the

text: only

text: adapted

text: basis

text: at

text: regular

text: points

text: is

text: X

text: X

text: X

text: Indeed,

text: ν(x

text: x

text: x

text: and

text: ν

text: is

text: homogeneous

text: of

text: degree

text: Note

text: that

text: (p,

text: S)

text: (p,

text: S)

text: Q(p)

text: Q

text: min

text: max

text: S

text: Consider

text: now

text: the

text: analogue

text: structure

text: in

text: higher

text: dimension,

text: that

text: is,

text: the

text: almost-Riemannian

text: structure

text: on

text: R

text: for

text: which

text: a

text: global

text: generating

text: family

text: is

text: X

text: (x)

text: X

text: (x)

text: X

text: (x)

text: X

text: (x)

text: where

text: x

text: (x

text: x

text: x

text: x

text: and

text: the

text: canonical

text: volume

text: form

text: ω

text: dx

text: dx

text: dx

text: dx

text: The

text: singular

text: set

text: is

text: R

text: S

text: x

text: x

text: x

text: x

text: and

text: Q

text: The

text: growth

text: vector

text: at

text: a

text: singular

text: point

text: p

text: is

text: Q(p)

text: R

text: and

text: the

text: growth

text: vector

text: of

text: the

text: ﬂag

text: restricted

text: to

text: S

text: is

text: whence

text: Q

text: Hence

text: the

text: Hausdorﬀ

text: dimension

text: R

text: SS

text: of

text: (endowed

text: with

text: the

text: sub-Riemannian

text: distance)

text: is

text: and

text: vol

text: The

text: only

text: adapted

text: basis

text: at

text: regular

text: H

text: points

text: is

text: X

text: X

text: X

text: X

text: which

text: gives

text: ν(x)

text: i.e.,

text: ν

text: is

text: homogeneous

text: of

text: degree

text: and

text: (p,

text: S)

text: (p,

text: S)

text: Q(p)

text: Q

text: Then

text: by

text: Proposition

text: S

text: min

text: maSx

text: S

text: d

text: is

text: integrable

text: near

text: any

text: singular

text: point.

text: Equivalently,

text: is

text: a

text: Radon

text: measure.

text: dµ

text: R

text: Example

text: (the

text: Martinet

text: space).

text: Recall

text: the

text: sub-Riemannian

text: structure

text: of

text: Example

text: where

text: M

text: and

text: a

text: global

text: generating

text: family

text: is

text: X

text: X

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − and one easily gets (cid:37) (p, S) = 1, (cid:37) (p, S) = 4. Thus (cid:37) (p, S) < Q(p) Q = 3 < (cid:37) (p, S) and neither min max min S max the condition of Proposition 4.4 nor the one of Proposition 4.9 is satisﬁed. However, taking a suﬃciently small neighborhood Uˆof p we haveˆ (cid:112) ˆ (cid:112) S (cid:120) (cid:16) 1 (cid:16) 1 (cid:16) 1 ∞ 7 (U) dx dx dx˜ dx < . R U ν(x) [−1,1]4 ω1(x)2 + ω2(x)2 [−1,1]2 x˜22 + x43 2 3 S S (cid:120) Thus 7 = 7 is a Radon measure. R A Appendix dSQN We show here a technical result which is needed in Section to provide the local equivalent of N . dµ Proposition A.1. Let M be oriented, ω be a volume form on M, K be a compact subset of M, and D →(cid:80) ∞ X , . . . , X be a generating family for ( , g). There exist a constant C > 0 and a function η : K (0, + ) 1 m ∈ such that the following holds for every p K. X { | | (cid:48)| Let = (X , . . . , X ) be a n-tuple in arg max ω (X (cid:48) (p), . . . , X (cid:48) (p)) X (cid:48) , . . . , X (cid:48) s.t. I = } I1 In Y p I1 (cid:88)In I1 In i i Q(p) . Consider another n-tuple of vector ﬁelds = (Y , . . . , Y ) such that, for i = 1, . . . , n, 1 n ∈ { | | ≤ } Y span X : I (cid:96)(Y ) , i.e. Y = Y jX , i Ij j i i i Ij | |≤ Ij (cid:96)(Yi) where all components Y j are smooth functions satisfying, for s ∈ N, i (cid:107) (cid:107) (cid:107) − (cid:107) ≤ Y (s)(p) and (Y (s)) 1(p) 2, (33) (cid:40) (cid:16) (cid:17) Y (s)(p) being the matrix { | | | } (cid:54) ∅ Y j(p) if i, j (cid:96)(Y ) = I = s = Y (s)(p) = i {i,j | (cid:96)(Yi)=|Ij|=s} i j 1 otherwise. ≤ Then, for any (cid:15) η(p) there holds: ⊂ ⊂ BoxiY(p, (cid:15)/C) B(p, (cid:15)) BoxiY(p, C(cid:15)), i = 1, 2, (34) where (cid:40) (cid:32) (cid:33) (cid:41) (cid:88) | | | ≤ Box1Y(p, (cid:15)) = exp y Y (p) y (cid:15)(cid:96)(Yi), i = 1, . . . , n , i i i i { ◦ · · · ◦ | | | ≤ } Box2Y(p, (cid:15)) = exp(z Y ) exp(z Y )(p) z (cid:15)(cid:96)(Yi), i = 1, . . . , n . 1 1 n n i X Y Remark A.2. Notice that any n-tuples and satisfying the hypothesis of the proposition at a point p Y X form a basis adapted to the ﬂag (8) of the distribution at p. Also, = satisﬁes the hypothesis and in this case the inclusions (34) for i = 2 result from the uniform Ball-Box Theorem, [18] and [20, Theorem 4.7]. 29 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: one

text: easily

text: gets

text: (p,

text: S)

text: (p,

text: S)

text: Thus

text: (p,

text: S)

text: Q(p)

text: Q

text: (p,

text: S)

text: and

text: neither

text: min

text: max

text: min

text: S

text: max

text: the

text: condition

text: of

text: Proposition

text: nor

text: the

text: one

text: of

text: Proposition

text: is

text: satisﬁed.

text: However,

text: taking

text: a

text: suﬃciently

text: small

text: neighborhood

text: Uˆof

text: p

text: we

text: haveˆ

text: ˆ

text: S

text: (U)

text: dx

text: dx

text: dx˜

text: dx

text: R

text: U

text: ν(x)

text: S

text: S

text: Thus

text: is

text: a

text: Radon

text: measure.

text: R

text: A

text: Appendix

text: dSQN

text: We

text: show

text: here

text: a

text: technical

text: result

text: which

text: is

text: needed

text: in

text: Section

text: to

text: provide

text: the

text: local

text: equivalent

text: of

text: N

text: dµ

text: Proposition

text: Let

text: M

text: be

text: oriented,

text: ω

text: be

text: a

text: volume

text: form

text: on

text: M,

text: K

text: be

text: a

text: compact

text: subset

text: of

text: M,

text: and

text: D

text: X

text: X

text: be

text: a

text: generating

text: family

text: for

text: g).

text: There

text: exist

text: a

text: constant

text: C

text: and

text: a

text: function

text: η

text: K

text: m

text: such

text: that

text: the

text: following

text: holds

text: for

text: every

text: p

text: K.

text: X

text: Let

text: (X

text: X

text: be

text: a

text: n-tuple

text: in

text: arg

text: max

text: ω

text: (X

text: (p),

text: X

text: (p))

text: X

text: X

text: s.t.

text: I

text: In

text: Y

text: p

text: In

text: i

text: i

text: Q(p)

text: Consider

text: another

text: n-tuple

text: of

text: vector

text: ﬁelds

text: (Y

text: Y

text: such

text: that,

text: for

text: i

text: n,

text: n

text: Y

text: span

text: X

text: I

text: i.e.

text: Y

text: Y

text: jX

text: i

text: Ij

text: j

text: i

text: i

text: i

text: Ij

text: Ij

text: where

text: all

text: components

text: Y

text: j

text: are

text: smooth

text: functions

text: satisfying,

text: for

text: s

text: N,

text: i

text: Y

text: (s)(p)

text: and

text: (Y

text: (s))

text: Y

text: (s)(p)

text: being

text: the

text: matrix

text: Y

text: j(p)

text: if

text: i,

text: j

text: I

text: s

text: Y

text: (s)(p)

text: i

text: {i,j

text: i

text: j

text: otherwise.

text: Then,

text: for

text: any

text: η(p)

text: there

text: holds:

text: BoxiY(p,

text: B(p,

text: BoxiY(p,

text: i

text: where

text: exp

text: y

text: Y

text: (p)

text: y

text: i

text: n

text: i

text: i

text: i

text: i

text: exp(z

text: Y

text: exp(z

text: Y

text: )(p)

text: z

text: i

text: n

text: n

text: n

text: i

text: X

text: Y

text: Remark

text: Notice

text: that

text: any

text: n-tuples

text: and

text: satisfying

text: the

text: hypothesis

text: of

text: the

text: proposition

text: at

text: a

text: point

text: p

text: Y

text: X

text: form

text: a

text: basis

text: adapted

text: to

text: the

text: ﬂag

text: of

text: the

text: distribution

text: at

text: p.

text: Also,

text: satisﬁes

text: the

text: hypothesis

text: and

text: in

text: this

text: case

text: the

text: inclusions

text: for

text: i

text: result

text: from

text: the

text: uniform

text: Ball-Box

text: Theorem,

text: and

text: Theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Y X Y Proof of Proposition A.1. We already know that (34) holds with = and i = 2. Let be a n-tuple of Y vector ﬁelds satisfying the hypothesis of Proposition A.1. We will prove that (34) holds for with i = 1, the proof with i = 2 being similar. (cid:48) Let x and x be the coordinates associated with Box1Y and Box2X respectively, and assume that the change (cid:48) of coordinate formulas are x = ψ(x). From (36) and Lemma A.3, it suﬃces to show that there exists a (cid:48) X constant C > 0, independent of p and of (cid:98), such that 1 (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ (cid:48)(cid:107) (cid:107) ∈ R C(cid:48) x p ψ(x) p C x p for any x n. (37) (cid:32) (cid:33) By deﬁnition of ψ(x), ther(cid:88)e holds − ◦ ◦ · · · ◦ (cid:80)exp xiYi exp(ψ1(x)XI1) exp(ψn(x)XIn)(p) = p. i Remind that Yi = wj≤wi YijXIj. Takingthe homogeneous terms of weighted degree zero in the equality above, we obtain (cid:88) (cid:88)    (cid:98)  (cid:98) (cid:98) (cid:98) (cid:98) − n ◦ ◦ · · · ◦ exp x Y j(p) X exp(ψ (x)X ) exp(ψ (x)X )(p) = p. (cid:80) i i Ij 1 I1 n In { } j=1 i:wi=wj Set y = { } x Y j(p). The Campbell-Hausdorﬀ formula allows to expand the product above as j i:w(cid:88)i=wj i i (cid:88) (cid:88) follows, (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) n − · · · · · · (ψ (x) y )X (p) + cI y y ψ (x) ψ (x)X (p) = 0, (38) j j Ij α,β α1 αs β1 βr I | |≤ j=1 I wn α,β where the second sum is taken over all the multi-indices α = (α , . . . , α ) and β = (β , . . . , β ) such that | | 1 · ·s· 1 r s + r > 1 and w(α) + w(β) = I (we use the notation w(α) for w + + w ). Note that this implies | | α1 αs that all w and w are smaller than I . The structural constants cI do not depend on p but only on αi βi α,β the coeﬃcients of the Campbell-Hausdorﬀ series. More precisely, there exists a constant C depending only | |(cid:80)≤ w¯ on the maximum value w¯ of w on K such th(cid:98)at cI C for every(cid:98)I, α, β appearing in (38) at any point ∈ n α,β w¯ p K. Now, for every multi-index I there holds X (p) = | | | | Xi(p)X (p) where I Ii = I I Ii Xi(p) = ω(XI1, . . . , XIi−1, XI, XIi+1, . . . , XIn)(p). I ω(X , . . . , X )(p) I1 In X | | | | The choice of the family = (cid:98)(X , . . . , X ) ensures that, for every I and i such that I = I we have | | ≤ I1 In i Xi(p) 1. I   Plugging th(cid:88)e expression of X ((cid:88)p) int(cid:88)o (38), we obtain I  (cid:98) (cid:98) (cid:98)  (cid:98) n − · · · · · · ψ (x) y + cI Xi(p)y y ψ (x) ψ (x) X (p) = 0. (39) i i α,β I α1 αs β1 βr Ii | | i=1 I =wi α,β Note that these equalities are in triangular form: all the indices α and β appearing in the ith equality j j are smaller than i. Since moreov(cid:88)er all the coeﬃcients in the abo(cid:88)ve equality are bounded independently of p, (cid:98) we get the following change of coordinates formulas, for i = 1, . . . , n, (cid:48) · · · · · · ψ (x) = a (p)y y = a (p)x x , i α α1 αs α α1 αs α s.t. w(α)=wi α s.t. w(α)=wi 31 ##LTLine## ##LTLine##

text: Y

text: X

text: Y

text: Proof

text: of

text: Proposition

text: We

text: already

text: know

text: that

text: holds

text: with

text: and

text: i

text: Let

text: be

text: a

text: n-tuple

text: of

text: Y

text: vector

text: ﬁelds

text: satisfying

text: the

text: hypothesis

text: of

text: Proposition

text: We

text: will

text: prove

text: that

text: holds

text: for

text: with

text: i

text: the

text: proof

text: with

text: i

text: being

text: similar.

text: Let

text: x

text: and

text: x

text: be

text: the

text: coordinates

text: associated

text: with

text: and

text: respectively,

text: and

text: assume

text: that

text: the

text: change

text: of

text: coordinate

text: formulas

text: are

text: x

text: ψ(x).

text: From

text: and

text: Lemma

text: it

text: suﬃces

text: to

text: show

text: that

text: there

text: exists

text: a

text: X

text: constant

text: C

text: independent

text: of

text: p

text: and

text: of

text: such

text: that

text: R

text: x

text: p

text: ψ(x)

text: p

text: C

text: x

text: p

text: for

text: any

text: x

text: n.

text: By

text: deﬁnition

text: of

text: ψ(x),

text: holds

text: xiYi

text: exp(ψn(x)XIn)(p)

text: p.

text: i

text: Remind

text: that

text: Yi

text: wj≤wi

text: YijXIj.

text: Takingthe

text: homogeneous

text: terms

text: of

text: weighted

text: degree

text: zero

text: in

text: the

text: equality

text: above,

text: we

text: obtain

text: n

text: exp

text: x

text: Y

text: j(p)

text: X

text: exp(ψ

text: (x)X

text: exp(ψ

text: (x)X

text: )(p)

text: p.

text: i

text: i

text: Ij

text: n

text: In

text: i:wi=wj

text: Set

text: y

text: x

text: Y

text: j(p).

text: The

text: Campbell-Hausdorﬀ

text: formula

text: allows

text: to

text: expand

text: the

text: product

text: above

text: as

text: j

text: i

text: i

text: follows,

text: n

text: (ψ

text: (x)

text: y

text: )X

text: (p)

text: cI

text: y

text: y

text: ψ

text: (x)

text: ψ

text: (x)X

text: (p)

text: j

text: j

text: Ij

text: α,β

text: α1

text: αs

text: β1

text: βr

text: I

text: I

text: wn

text: α,β

text: where

text: the

text: second

text: sum

text: is

text: taken

text: over

text: all

text: the

text: multi-indices

text: α

text: (α

text: α

text: and

text: β

text: (β

text: β

text: such

text: that

text: ·s·

text: r

text: s

text: r

text: and

text: w(α)

text: w(β)

text: I

text: (we

text: use

text: the

text: notation

text: w(α)

text: for

text: w

text: w

text: Note

text: that

text: this

text: implies

text: α1

text: αs

text: that

text: all

text: w

text: and

text: w

text: are

text: smaller

text: than

text: I

text: The

text: structural

text: constants

text: cI

text: do

text: not

text: depend

text: on

text: p

text: but

text: only

text: on

text: αi

text: βi

text: α,β

text: the

text: coeﬃcients

text: of

text: the

text: Campbell-Hausdorﬀ

text: series.

text: More

text: precisely,

text: there

text: exists

text: a

text: constant

text: C

text: depending

text: only

text: w¯

text: on

text: the

text: maximum

text: value

text: w¯

text: of

text: w

text: on

text: K

text: such

text: cI

text: C

text: for

text: α,

text: β

text: appearing

text: in

text: at

text: any

text: point

text: n

text: α,β

text: w¯

text: p

text: K.

text: Now,

text: for

text: every

text: multi-index

text: I

text: there

text: holds

text: X

text: (p)

text: Xi(p)X

text: (p)

text: where

text: I

text: Ii

text: I

text: I

text: Ii

text: Xi(p)

text: XI,

text: XIn)(p).

text: I

text: ω(X

text: X

text: )(p)

text: In

text: X

text: The

text: choice

text: of

text: the

text: family

text: X

text: ensures

text: that,

text: for

text: every

text: I

text: and

text: i

text: such

text: that

text: I

text: I

text: we

text: have

text: In

text: i

text: Xi(p)

text: I

text: Plugging

text: expression

text: of

text: X

text: we

text: obtain

text: I

text: n

text: ψ

text: (x)

text: y

text: cI

text: Xi(p)y

text: y

text: ψ

text: (x)

text: ψ

text: (x)

text: X

text: (p)

text: i

text: i

text: α,β

text: I

text: α1

text: αs

text: β1

text: βr

text: Ii

text: I

text: =wi

text: α,β

text: Note

text: that

text: these

text: equalities

text: are

text: in

text: triangular

text: form:

text: all

text: the

text: indices

text: α

text: and

text: β

text: appearing

text: in

text: the

text: ith

text: equality

text: j

text: j

text: are

text: smaller

text: than

text: i.

text: Since

text: all

text: the

text: coeﬃcients

text: in

text: the

text: equality

text: are

text: bounded

text: independently

text: of

text: p,

text: we

text: get

text: the

text: following

text: change

text: of

text: coordinates

text: formulas,

text: for

text: i

text: n,

text: ψ

text: (x)

text: a

text: (p)y

text: y

text: a

text: (p)x

text: x

text: i

text: α

text: α1

text: αs

text: α

text: α1

text: αs

text: α

text: s.t.

text: w(α)=wi

text: α

text: s.t.

text: w(α)=wi

text: ##LTLine##

text: ##LTLine##


text: (cid:98) (cid:98) where B denotes the ball centered at 0 and of radius 1 in T M with respect to d . p p p (cid:98) (cid:98) Remark 3.2. As we will see(cid:98)in Theorem (cid:98)5.3 below, one can interpret the constant 2 in (10) as the diameter of the ball Bp with resp(cid:98)e(cid:98)ct to the dSis(cid:98)tance dp. SiSnc(cid:98)e the nilpotent approximation is a Carnot group here, we then have 2Q = diam (B )Q = Q (B ) (here Q denotes the spherical Hausdorﬀ measure in T M with dp p dp p dp p respect to the distance d ), which gives a clear interpretation to (10). p Theorem 3.1 is a special case of Theorem 5.3 proved in Section . Note that all the statements in Theorem 3.1 are well-known, but they have been proved only fragmentarily in several references: properties (i) and (iii) are stated in [22] (for a rigorous proof see [23]), property (iv) in [1]. Up to the authors’ knowledge, property (ii) has never been stated as is and it is a consequence of basic covering arguments in geometric measure theory. In particular, this property is needed to apply the diﬀerentiation theorem for Radon measures [26, Theorem 4.7 p.24] and ensures that dvdoµlH (p) coincides with lim(cid:15)→0 voµlH(B(B(p(,p(cid:15)),(cid:15)))). H Remark 3.3. Properties (i), (ii), (iii) hold true if we replace volH by dimH M. In particular, applying the H diﬀerentiation theorem for Radon measures [26, Theorem 4.7 p.24] we get that (i) the limit lim → Q(B(p,(cid:15))) H (cid:15) 0 µ(B(p,(cid:15))) exists µ-almost everywhere and it coincides with the Radon-Nikodym derivative of Q with respect to µ. Nevertheless, we do not have an explicit representation of such limit as we have for the spherical Hausdorﬀ case in (10). A ﬁrst consequence of Theorem 3.1 is that the Radon-Nikodym derivative of vol with respect to µ H is continuous on M, see [1, Corollary 2 and Section 4.1]. This is due to the fact that both the nilpotent approximation at q and the tangent measure µˆq depend smoothly on q. Studying higher regularity of dvolH dµ is the main subject in [1], to which (cid:98)we refer the interested reader. A further consequence is that µ and vol are commensurable (see the last remark in [22]). This follows (cid:55)→ H directly by the fact that q µˆq(Bq) is positive and continuous on M, so that dvdoµlH (q) and its inverse dµ (q) are locally bounded. dvolH In conclusion, when the manifold is equiregular, the Hausdorﬀ volume essentially behaves as a smooth measure. This fails when singular points are present, as we see in the next section. 3.2 Non equiregular case D Assume (M, , g) is an oriented sub-Riemannian manifold and µ is a smooth volume on M. The manifold is split into the disjoint union of two sets ∪ M = R S, (cid:55)→ D where R, S denote respectively the set of regular and singular points. Since the functions q dim i are q lower semi-continuous, R is an open and dense subset of M and conversely S is a closed subset of empty interior. We will assume in this paper that µ(S) = 0. This assumption is satisﬁed for a very wide class of sub- Riemannian manifolds: for instance for analytic sub-Riemannian manifolds, for generic sub-Riemannian (cid:55)→ D structures on a given manifold (see Section ), or when q is a distribution of corank 1 [25, Proposi- q (cid:54) tion 3.3]. Nevertheless one can build up sub-Riemannian manifolds where µ(S) = 0 as in the next example. ⊂ R Example 3.4. Let S 4 be any closed set having empty interior and positive Lebesgue measure. By ∈ C∞ R Whitney’s Extension Theorem (see for instance [21, Proposition A.8]), there exists a function f ( 4) − R such that S = f 1(0). Consider the sub-Riemannian structure on 4 for which the following vector ﬁelds are a generating family, x2 X = ∂ , X = ∂ + x ∂ + 1 ∂ , X = f(x)∂ . 1 1 2 2 1 3 2 4 3 4 One easily checks that the singular set coincides with S and thus it is of positive measure. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: B

text: denotes

text: the

text: ball

text: centered

text: at

text: and

text: of

text: radius

text: in

text: T

text: M

text: with

text: respect

text: to

text: d

text: p

text: p

text: p

text: Remark

text: As

text: we

text: will

text: Theorem

text: below,

text: one

text: can

text: interpret

text: the

text: constant

text: in

text: as

text: the

text: diameter

text: of

text: the

text: ball

text: Bp

text: with

text: to

text: the

text: dp.

text: the

text: nilpotent

text: approximation

text: is

text: a

text: Carnot

text: group

text: here,

text: we

text: then

text: have

text: diam

text: (B

text: )Q

text: Q

text: (B

text: (here

text: Q

text: denotes

text: the

text: spherical

text: Hausdorﬀ

text: measure

text: in

text: T

text: M

text: with

text: dp

text: p

text: dp

text: p

text: dp

text: p

text: respect

text: to

text: the

text: distance

text: d

text: which

text: gives

text: a

text: clear

text: interpretation

text: to

text: p

text: Theorem

text: is

text: a

text: special

text: case

text: of

text: Theorem

text: proved

text: in

text: Section

text: Note

text: that

text: all

text: the

text: statements

text: in

text: Theorem

text: are

text: well-known,

text: but

text: they

text: have

text: been

text: proved

text: only

text: fragmentarily

text: in

text: several

text: references:

text: properties

text: (i)

text: and

text: (iii)

text: are

text: stated

text: in

text: (for

text: a

text: rigorous

text: proof

text: see

text: property

text: (iv)

text: in

text: Up

text: to

text: the

text: authors’

text: knowledge,

text: property

text: (ii)

text: has

text: never

text: been

text: stated

text: as

text: is

text: and

text: it

text: is

text: a

text: consequence

text: of

text: basic

text: covering

text: arguments

text: in

text: geometric

text: measure

text: theory.

text: In

text: particular,

text: this

text: property

text: is

text: needed

text: to

text: apply

text: the

text: diﬀerentiation

text: theorem

text: for

text: Radon

text: measures

text: Theorem

text: and

text: ensures

text: that

text: dvdoµlH

text: (p)

text: coincides

text: with

text: H

text: Remark

text: Properties

text: (i),

text: (ii),

text: (iii)

text: hold

text: true

text: if

text: we

text: replace

text: volH

text: by

text: dimH

text: M.

text: In

text: particular,

text: applying

text: the

text: H

text: diﬀerentiation

text: theorem

text: for

text: Radon

text: measures

text: Theorem

text: we

text: get

text: that

text: (i)

text: the

text: limit

text: lim

text: H

text: exists

text: µ-almost

text: everywhere

text: and

text: it

text: coincides

text: with

text: the

text: Radon-Nikodym

text: derivative

text: of

text: Q

text: with

text: respect

text: to

text: µ.

text: Nevertheless,

text: we

text: do

text: not

text: have

text: an

text: explicit

text: representation

text: of

text: such

text: limit

text: as

text: we

text: have

text: for

text: the

text: spherical

text: Hausdorﬀ

text: case

text: in

text: A

text: ﬁrst

text: consequence

text: of

text: Theorem

text: is

text: that

text: the

text: Radon-Nikodym

text: derivative

text: of

text: vol

text: with

text: respect

text: to

text: µ

text: H

text: is

text: continuous

text: on

text: M,

text: see

text: Corollary

text: and

text: Section

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: both

text: the

text: nilpotent

text: approximation

text: at

text: q

text: and

text: the

text: tangent

text: measure

text: µˆq

text: depend

text: smoothly

text: on

text: q.

text: Studying

text: higher

text: regularity

text: of

text: dvolH

text: dµ

text: is

text: the

text: main

text: subject

text: in

text: to

text: which

text: refer

text: the

text: interested

text: reader.

text: A

text: further

text: consequence

text: is

text: that

text: µ

text: and

text: vol

text: are

text: commensurable

text: (see

text: the

text: last

text: remark

text: in

text: This

text: follows

text: H

text: directly

text: by

text: the

text: fact

text: that

text: q

text: µˆq(Bq)

text: is

text: positive

text: and

text: continuous

text: on

text: M,

text: so

text: that

text: dvdoµlH

text: (q)

text: and

text: its

text: inverse

text: dµ

text: (q)

text: are

text: locally

text: bounded.

text: dvolH

text: In

text: conclusion,

text: when

text: the

text: manifold

text: is

text: equiregular,

text: the

text: Hausdorﬀ

text: volume

text: essentially

text: behaves

text: as

text: a

text: smooth

text: measure.

text: This

text: fails

text: when

text: singular

text: points

text: are

text: present,

text: as

text: we

text: see

text: in

text: the

text: next

text: section.

text: Non

text: equiregular

text: case

text: D

text: Assume

text: (M,

text: g)

text: is

text: an

text: oriented

text: sub-Riemannian

text: manifold

text: and

text: µ

text: is

text: a

text: smooth

text: volume

text: on

text: M.

text: The

text: manifold

text: is

text: split

text: into

text: the

text: disjoint

text: union

text: of

text: two

text: sets

text: M

text: R

text: S,

text: D

text: where

text: R,

text: S

text: denote

text: respectively

text: the

text: set

text: of

text: regular

text: and

text: singular

text: points.

text: Since

text: the

text: functions

text: q

text: dim

text: i

text: are

text: q

text: lower

text: semi-continuous,

text: R

text: is

text: an

text: open

text: and

text: dense

text: subset

text: of

text: M

text: and

text: conversely

text: S

text: is

text: a

text: closed

text: subset

text: of

text: empty

text: interior.

text: We

text: will

text: assume

text: in

text: this

text: paper

text: that

text: µ(S)

text: This

text: assumption

text: is

text: satisﬁed

text: for

text: a

text: very

text: wide

text: class

text: of

text: sub-

text: Riemannian

text: manifolds:

text: for

text: instance

text: for

text: analytic

text: sub-Riemannian

text: manifolds,

text: for

text: generic

text: sub-Riemannian

text: D

text: structures

text: on

text: a

text: given

text: manifold

text: (see

text: Section

text: or

text: when

text: q

text: is

text: a

text: distribution

text: of

text: corank

text: Proposi-

text: q

text: tion

text: Nevertheless

text: one

text: can

text: build

text: up

text: sub-Riemannian

text: manifolds

text: where

text: µ(S)

text: as

text: in

text: the

text: next

text: example.

text: R

text: Example

text: Let

text: S

text: be

text: any

text: closed

text: set

text: having

text: empty

text: interior

text: and

text: positive

text: Lebesgue

text: measure.

text: By

text: C∞

text: R

text: Whitney’s

text: Extension

text: Theorem

text: (see

text: for

text: instance

text: Proposition

text: there

text: exists

text: a

text: function

text: f

text: R

text: such

text: that

text: S

text: f

text: Consider

text: the

text: sub-Riemannian

text: structure

text: on

text: for

text: which

text: the

text: following

text: vector

text: ﬁelds

text: are

text: a

text: generating

text: family,

text: X

text: X

text: x

text: X

text: f(x)∂

text: One

text: easily

text: checks

text: that

text: the

text: singular

text: set

text: coincides

text: with

text: S

text: and

text: thus

text: it

text: is

text: of

text: positive

text: measure.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##