text: BEST CONSTANTS FOR A FAMILY OF CARLESON SEQUENCES 15 ± ± ∈ Indeed, being constant on I , this weight is in Ad(I ) with [w] ± = 1. Since x Ω 2 Ad(I ) Q 2 and { | ∈ − | ∈ (cid:104) (cid:105) (cid:104) − (cid:105) (cid:54) } Ad,Q(I) = w : w − Ad,Q(I ), w Ad,Q(I+), w w 1 Q , 2 I 2 I+ 2 I I the conclusion follows. This means that the supremum and inﬁmum in the deﬁnitions (2.3) and (2.4) are taken over a non-empty set. (cid:88) ∈ (cid:88) (cid:88) Now, for any w Ad,Q(I), 2 1 1 1 1 1 | | cΦ(w) = | −| cΦ(w) + | | cΦ(w) + f(s (w))R (w), I ∈D J 2 I ∈D − J 2 I+ ∈D J I I J (I) J (I ) J (I+) where we have used notation (2.2). Inequality (4.1) now follows from considering any weight ∗ ∈ − w Ad,Q(I) that realizes (or almost realizes) the supremum in (2.3) on both I and I+ 2 ∗| ∗| (note that w − and w can be chosen independently of each other). I I+ (cid:104) (cid:105) The boundary condition (4.3) holds since the only functions w in Ad(I) satisfying w = (cid:104) − (cid:105) 2 (cid:80) I 1/ w 1 are constants. I Lastly, to show (4.4), take any τ > 0 and replace w with τw in the deﬁnitions (2.3) and (2.4). The class Ad,Q is invariant under this transformation, as is the sum |1| ∈D cΦ(w). 2 I J (I) J Therefore, (cid:112) − B (τx , τ 1x ) = B (x , x ), Q,Φ 1 2 Q,Φ 1 2 (cid:3) and setting τ = x /x ﬁnishes the proof. 2 1 The central point in Bellman analysis on martingales is that if one has any function B on Ω that satisﬁes (4.1) and (4.3), that function is automatically a majorant of B ; a similar Q Q,Φ conclusion applies to b . Speciﬁcally, we have the following lemma, which implements in our Q,Φ setting the main part of any Bellman-function argument, sometimes referred to as “Bellman induction” or “induction on scales.” (cid:20) (cid:21) (cid:16) (cid:17) Lemma 4.2. Let B and b be functions on Ω satisfying Q − − − − − x + x+ (cid:62) 1 − 1 (x x+)2 (x x+)2 (4.6) B B(x ) + B(x+) + Φ(x x ) 1 1 + 2 2 2 2 2 1(cid:20) 2 x2 x2 (cid:21) (cid:16) (cid:17) 1 2 and − − − − − x + x+ (cid:54) 1 − 1 (x x+)2 (x x+)2 (4.7) b b(x ) + b(x+) + Φ(x x ) 1 1 + 2 2 , 2 2 2 1 2 x2 x2 1 2 − ∈ − ∈ respectively, for all points x , x+ Ω such that (x + x+)/2 Ω , as well as Q Q ∀ ∈ ∞ (4.8) B(x , 1/x ) = 0, b(x , 1/x ) = 0, x (0, ). 1 1 1 1 1 Assume that Φ is a non-negative, bounded function on [1, Q]. Then (cid:62) (cid:54) ∀ ∈ B(x) B (x), b(x) b (x), x Ω . Q,Φ Q,Φ Q ∈ Proof. First, we obtain element(cid:114)ary estimates on B and b.(cid:114)Take any x Ω and write − (cid:16) (cid:17) (cid:16) (cid:17) Q x = (x + x+)/2, where, as in (4.5), ± ± − 1 ± ∓ − 1 x = x 1 1 , x = x 1 1 . 1 1 x x 2 2 x x 1 2 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BEST

text: CONSTANTS

text: FOR

text: A

text: FAMILY

text: OF

text: CARLESON

text: SEQUENCES

text: Indeed,

text: being

text: constant

text: on

text: I

text: this

text: weight

text: is

text: in

text: Ad(I

text: with

text: [w]

text: Since

text: x

text: Ω

text: Ad(I

text: Q

text: and

text: Ad,Q(I)

text: w

text: w

text: Ad,Q(I

text: w

text: Ad,Q(I+),

text: w

text: w

text: Q

text: I

text: I+

text: I

text: I

text: the

text: conclusion

text: follows.

text: This

text: means

text: that

text: the

text: supremum

text: and

text: inﬁmum

text: in

text: the

text: deﬁnitions

text: and

text: are

text: taken

text: over

text: a

text: non-empty

text: set.

text: Now,

text: for

text: any

text: w

text: Ad,Q(I),

text: cΦ(w)

text: cΦ(w)

text: cΦ(w)

text: f(s

text: (w))R

text: (w),

text: I

text: ∈D

text: J

text: I

text: ∈D

text: J

text: I+

text: ∈D

text: J

text: I

text: I

text: J

text: (I)

text: J

text: (I

text: J

text: (I+)

text: where

text: we

text: have

text: used

text: notation

text: Inequality

text: now

text: follows

text: from

text: considering

text: any

text: weight

text: w

text: Ad,Q(I)

text: that

text: realizes

text: (or

text: almost

text: realizes)

text: the

text: supremum

text: in

text: on

text: both

text: I

text: and

text: I+

text: (note

text: that

text: w

text: and

text: w

text: can

text: be

text: chosen

text: independently

text: of

text: each

text: other).

text: I

text: I+

text: The

text: boundary

text: condition

text: holds

text: since

text: the

text: only

text: functions

text: w

text: in

text: Ad(I)

text: satisfying

text: w

text: I

text: w

text: are

text: constants.

text: I

text: Lastly,

text: to

text: show

text: take

text: any

text: τ

text: and

text: replace

text: w

text: with

text: τw

text: in

text: the

text: deﬁnitions

text: and

text: The

text: class

text: Ad,Q

text: is

text: invariant

text: under

text: this

text: transformation,

text: as

text: is

text: the

text: sum

text: ∈D

text: cΦ(w).

text: I

text: J

text: (I)

text: J

text: Therefore,

text: B

text: (τx

text: τ

text: B

text: (x

text: x

text: Q,Φ

text: Q,Φ

text: and

text: setting

text: τ

text: x

text: /x

text: ﬁnishes

text: the

text: proof.

text: The

text: central

text: point

text: in

text: Bellman

text: analysis

text: on

text: martingales

text: is

text: that

text: if

text: one

text: has

text: any

text: function

text: B

text: on

text: Ω

text: that

text: satisﬁes

text: and

text: that

text: function

text: is

text: automatically

text: a

text: majorant

text: of

text: B

text: a

text: similar

text: Q

text: Q,Φ

text: conclusion

text: applies

text: to

text: b

text: Speciﬁcally,

text: we

text: have

text: the

text: following

text: lemma,

text: which

text: implements

text: in

text: our

text: Q,Φ

text: setting

text: the

text: main

text: part

text: of

text: any

text: Bellman-function

text: argument,

text: sometimes

text: referred

text: to

text: as

text: “Bellman

text: induction”

text: or

text: “induction

text: on

text: scales.”

text: Lemma

text: Let

text: B

text: and

text: b

text: be

text: functions

text: on

text: Ω

text: satisfying

text: Q

text: x

text: x+

text: (x

text: (x

text: B

text: B(x

text: B(x+)

text: Φ(x

text: x

text: and

text: x

text: x+

text: (x

text: (x

text: b

text: b(x

text: b(x+)

text: Φ(x

text: x

text: respectively,

text: for

text: all

text: points

text: x

text: x+

text: Ω

text: such

text: that

text: (x

text: Ω

text: as

text: well

text: as

text: Q

text: Q

text: B(x

text: b(x

text: x

text: Assume

text: that

text: Φ

text: is

text: a

text: non-negative,

text: bounded

text: function

text: on

text: Q].

text: Then

text: B(x)

text: B

text: (x),

text: b(x)

text: b

text: (x),

text: x

text: Ω

text: Q,Φ

text: Q,Φ

text: Q

text: Proof.

text: First,

text: we

text: obtain

text: estimates

text: on

text: B

text: and

text: any

text: x

text: Ω

text: and

text: write

text: Q

text: x

text: (x

text: where,

text: as

text: in

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:0) (cid:1) 16 LEONID SLAVIN − − − − − − − We ha(cid:107)ve(cid:107) x1 x2 = x+1 x+2 = 1 and (x1 x+1 )2(cid:16)/x21 +−(x2 (cid:17) x+2 )2/x22 = 8 1 1/(x1x2) . Let M = Φ ∞ . Using (4.6) and noting that B(x ) = B(x+) = 0 by (4.8), we obtain L ([1,Q]) (cid:62) − 1 (cid:62) B(x) 8Φ(x x ) 1 0. (cid:16) 1 2 (cid:17) (cid:16) (cid:17) x x 1 2 Similarly, (cid:54) − 1 (cid:54) − 1 b(x) 8Φ(x x ) 1 8M 1 . 1 2 x x (cid:0) x x (cid:1) 1 2 1 2 ∈ ∈ Now, again take any x Ω and any interval I. Let w Ad,Q(I) be any weight such that (cid:104) (cid:105) (cid:104) − (cid:105) Q ∈ D (cid:104) (cid:105)2 (cid:104) − (cid:105) ∈ w = x and w 1 = x . For any J (I), let x = w , w 1 ; since w Ad,Q(I), ∈I 1 I 2 J J J 2 x Ω . Applying (4.6) repeatedly, we have J Q (cid:62) 1 1 (cid:88) (cid:88) (cid:88) B(x) B(x −) + B(x ) + Φ(x x ) R (w) 2 I 2 I+ 1 2 I − n 1 (cid:62) 1 | | | | J B(x ) + f(s )R (w). J J J I ∈D ∈D J n(I) k=0 J n(I) (cid:62) → ∞ Since B 0, we can drop the ﬁrst sum and then take the limit as n . Taking the ∈ (cid:104) (cid:105) (cid:104) − (cid:105) supremum over all w Ad,Q(I) with w = x and w 1 = x proves the lemma for B. 2 (cid:88) I 1 (cid:88) (cid:88) I 2 The argument for b is similar. Using (4.7) repeatedly, we get − n 1 (cid:54) 1 | | (4.9) b(x) | | (cid:80) J b(xJ) +(cid:16) (cid:17) f(sJ)RJ(w). I ∈D ∈D J n(I) k=0 J n(I) | | − → ∞ The ﬁrst sum is bounded by 8|M| ∈D J 1 1 , which converges to 0 as n I J n(I) s2 J by the Lebesgue diﬀerentiation theorem and the Lebesgue dominated convergence theorem. → ∞ Taking the limit as n and then the inﬁmum over all appropriate w completes the (cid:3) proof. Remark 4.3. The reader may wonder if the condition that Φ be bounded is necessary in this lemma. It is clearly not needed for the conclusion about B. For the lower candidate b, however, some technical assumption is needed in order to disregard the ﬁrst sum in the right-hand side of (4.9). An alternative would be to assume that b is continuous on Ω , as Q one can then run the induction only for dyadically-simple weights (for which the induction is ﬁnite) and then approximate an arbitrary weight by dyadically-simple ones. In any case, this distinction is inconsequential: ﬁrst, in all cases of Theorem 2.11 Φ is a√utomatically bounded on any ﬁnite interval; and second, our only lower candidate, b(x) = a ( x x ), is continuous f 1 2 on Ω . Q We now turn to ﬁnding candidates B and b satisfying the hypotheses of Lemma 4.2. Let us focus on the upper candidate B; for the lower candidate, the steps below are exactly the same, except all inequality signs are reversed. (cid:20) (cid:21) If B is suﬃciently diﬀerentiable, condition (4.6) yield(cid:16)s the(cid:17)follow(cid:16)ing d(cid:17)iﬀerential inequality: 1 dx 2 dx 2 (cid:54) (4.10) [dx dx ] d2B(x) [dx dx ]T + Φ(x x ) 1 + 2 0, 1 2 1 2 1 2 8 x x 1 2 where d2B is the Hessian of B. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: LEONID

text: SLAVIN

text: We

text: and

text: Let

text: M

text: Φ

text: Using

text: and

text: noting

text: that

text: B(x

text: B(x+)

text: by

text: we

text: obtain

text: L

text: B(x)

text: x

text: x

text: x

text: Similarly,

text: b(x)

text: x

text: x

text: x

text: x

text: x

text: Now,

text: again

text: take

text: any

text: x

text: Ω

text: and

text: any

text: interval

text: I.

text: Let

text: w

text: Ad,Q(I)

text: be

text: any

text: weight

text: such

text: that

text: Q

text: D

text: w

text: x

text: and

text: w

text: x

text: For

text: any

text: J

text: (I),

text: let

text: x

text: w

text: w

text: since

text: w

text: Ad,Q(I),

text: ∈I

text: I

text: J

text: J

text: J

text: x

text: Ω

text: Applying

text: repeatedly,

text: we

text: have

text: J

text: Q

text: B(x)

text: B(x

text: B(x

text: Φ(x

text: x

text: R

text: (w)

text: I

text: I+

text: I

text: n

text: J

text: B(x

text: f(s

text: )R

text: (w).

text: J

text: J

text: J

text: I

text: ∈D

text: ∈D

text: J

text: n(I)

text: J

text: n(I)

text: Since

text: B

text: we

text: can

text: drop

text: the

text: ﬁrst

text: sum

text: and

text: then

text: take

text: the

text: limit

text: as

text: n

text: Taking

text: the

text: supremum

text: over

text: all

text: w

text: Ad,Q(I)

text: with

text: w

text: x

text: and

text: w

text: x

text: proves

text: the

text: lemma

text: for

text: B.

text: I

text: I

text: The

text: argument

text: for

text: b

text: is

text: similar.

text: Using

text: repeatedly,

text: we

text: get

text: n

text: b(x)

text: J

text: b(xJ)

text: f(sJ)RJ(w).

text: I

text: ∈D

text: ∈D

text: J

text: n(I)

text: J

text: n(I)

text: The

text: ﬁrst

text: sum

text: is

text: bounded

text: by

text: ∈D

text: J

text: which

text: converges

text: to

text: as

text: n

text: I

text: J

text: n(I)

text: J

text: by

text: the

text: Lebesgue

text: diﬀerentiation

text: theorem

text: and

text: the

text: Lebesgue

text: dominated

text: convergence

text: theorem.

text: Taking

text: the

text: limit

text: as

text: n

text: and

text: then

text: the

text: inﬁmum

text: over

text: all

text: appropriate

text: w

text: completes

text: the

text: proof.

text: Remark

text: The

text: reader

text: may

text: wonder

text: if

text: the

text: condition

text: that

text: Φ

text: be

text: bounded

text: is

text: necessary

text: in

text: this

text: lemma.

text: It

text: is

text: clearly

text: not

text: needed

text: for

text: the

text: conclusion

text: about

text: B.

text: For

text: the

text: lower

text: candidate

text: b,

text: however,

text: some

text: technical

text: assumption

text: is

text: needed

text: in

text: order

text: to

text: disregard

text: the

text: ﬁrst

text: sum

text: in

text: the

text: right-hand

text: side

text: of

text: An

text: alternative

text: would

text: be

text: to

text: assume

text: that

text: b

text: is

text: continuous

text: on

text: Ω

text: as

text: Q

text: one

text: can

text: then

text: run

text: the

text: induction

text: only

text: for

text: dyadically-simple

text: weights

text: (for

text: which

text: the

text: induction

text: is

text: ﬁnite)

text: and

text: then

text: approximate

text: an

text: arbitrary

text: weight

text: by

text: dyadically-simple

text: ones.

text: In

text: any

text: case,

text: this

text: distinction

text: is

text: inconsequential:

text: ﬁrst,

text: in

text: all

text: cases

text: of

text: Theorem

text: Φ

text: is

text: a√utomatically

text: bounded

text: on

text: any

text: ﬁnite

text: interval;

text: and

text: second,

text: our

text: only

text: lower

text: candidate,

text: b(x)

text: a

text: x

text: x

text: is

text: continuous

text: f

text: on

text: Ω

text: Q

text: We

text: now

text: turn

text: to

text: ﬁnding

text: candidates

text: B

text: and

text: b

text: satisfying

text: the

text: hypotheses

text: of

text: Lemma

text: Let

text: us

text: focus

text: on

text: the

text: upper

text: candidate

text: B;

text: for

text: the

text: lower

text: candidate,

text: the

text: steps

text: below

text: are

text: exactly

text: the

text: same,

text: except

text: all

text: inequality

text: signs

text: are

text: reversed.

text: If

text: B

text: is

text: suﬃciently

text: diﬀerentiable,

text: condition

text: inequality:

text: dx

text: dx

text: [dx

text: dx

text: [dx

text: dx

text: ]T

text: Φ(x

text: x

text: x

text: x

text: where

text: is

text: the

text: Hessian

text: of

text: B.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 24 LEONID SLAVIN Theorem 2.11 gives the formula for lower Bellman functions b when h(s) = Φ(s2)/s2 is ∞ Q,Φ increasing on [1, ). However, Φ˜(s2)/s2 does not have this property – in fact, it is decreasing (cid:54) (cid:62) − for s s and increasing for s s . We could ﬁnd the Bellman function for (Φ Φ )χ{ (cid:62) } 0 0 s0 s s0 as a substitute, but, as a willing reader can verify, the resulting formula would be much more cumbersome than (6.2) and the improvement in the estimate, numerically quite small. In any case, such manipulations cannot produce a sharp estimate, because, as we demonstrate in Section below, the optimizing weights for the Bellman candidates A and a are essentially L,f f unique – and diﬀerent. This implies that an estimate of the form (6.3), combining sharp estimates that involve both A and a , cannot itself be sharp. L,f f If one desires, as we do, to ﬁnd the actual function B in the case when h is con- Q,Φ − cave, one has to understand the nature of the extremizing split x = (x + x+)/2 in the ± main inequality (4.1), i.e., the choice of x that turns the inequality into an equality or ap- proximate equality. The candidates A and a were derived under the assumption that ± L,f f such x are inﬁnitesimally close to x, meaning (4.6) is equivalent to its inﬁnitesimal ver- sion, (4.10). However, for concave h, A fails the main inequality at every point, as is clear L,f from formula (5.10) in the proof of Lemma 5.1. For similar reasons, a does not work either. f Therefore, the optimizing split is not inﬁnitesimal in this case. An approach that seems promising is to consider piecewise-linear concave h, and then (cid:88) approximate an arbitrary concave h by such functions. One then would look for the Bellman A candidate such that A(cid:48)(cid:48) − (s) = 16h(s) + c δ (s), k sk k (cid:48) where s are the points of discontinuity of h , δ are Dirac masses at s , and c are the neg- k sk k kA ative coeﬃcients chosen so that the main inequality for the resulting candidate is satisﬁed on the whole domain. The presence of c would entail a non-inﬁnitesimal optimal split at the k point x = (s , s ). The nature of the dependence of the values of c on the local behavior of k k k h is the subject of further study. 7. Optimizers In this section we prove the converse inequalities to those established in Lemma 5.1, thus completing the proof of Theorems 2.10 and 2.11. This is done through the construction of two optimizing sequences – one for a and one for A . Importantly, these sequences do not f L,f depend on f, meaning that the upper Bellman function B is linear with respect to Φ for Q,Φ all increasing Φ such that h(s) = Φ(s2)/s2 is convex and, separately, for all decreasing Φ. Likewise, the lower Bellman function b is linear in Φ on the class of all Φ such that h is Q,Φ increasing. Without loss of generality, we will deﬁne our optimizers almost everywhere on I = (0, 1). ∈ { } Fix a point x Ω . We say that a sequence of functions wx on (0, 1) is an optimizing Q A { } n sequence for a Bellman candidate at x, if wx satisﬁes the following three conditions: n ∀ ∈ (7.1) n, wx Ad,Q((0, 1)); (cid:88)∀ (cid:104) n(cid:16)(cid:105) 2 (cid:104) (cid:17)− (cid:105) (cid:0) (cid:1) (7.2) n, wx = x , (wx) 1 = x ; n (0,1) 1 n (0,1) 2 √ | | (cid:104) (cid:105) (cid:104) − (cid:105) −→ A → ∞ (7.3) J Φ wx (wx) 1 R (wx) x x , as n . n J n J J n 1 2 ∈D J (0,1) ##LTLine##

text: LEONID

text: SLAVIN

text: Theorem

text: gives

text: the

text: formula

text: for

text: lower

text: Bellman

text: functions

text: b

text: when

text: h(s)

text: is

text: Q,Φ

text: increasing

text: on

text: However,

text: does

text: not

text: have

text: this

text: property

text: in

text: fact,

text: it

text: is

text: decreasing

text: for

text: s

text: s

text: and

text: increasing

text: for

text: s

text: s

text: We

text: could

text: ﬁnd

text: the

text: Bellman

text: function

text: for

text: (Φ

text: Φ

text: )χ{

text: s

text: as

text: a

text: substitute,

text: but,

text: as

text: a

text: willing

text: reader

text: can

text: verify,

text: the

text: resulting

text: formula

text: would

text: be

text: much

text: more

text: cumbersome

text: than

text: and

text: the

text: improvement

text: in

text: the

text: estimate,

text: numerically

text: quite

text: small.

text: In

text: any

text: case,

text: such

text: manipulations

text: cannot

text: produce

text: a

text: sharp

text: estimate,

text: because,

text: as

text: we

text: demonstrate

text: in

text: Section

text: below,

text: the

text: optimizing

text: weights

text: for

text: the

text: Bellman

text: candidates

text: A

text: and

text: a

text: are

text: essentially

text: L,f

text: f

text: unique

text: and

text: diﬀerent.

text: This

text: implies

text: that

text: an

text: estimate

text: of

text: the

text: form

text: combining

text: sharp

text: estimates

text: that

text: involve

text: both

text: A

text: and

text: a

text: cannot

text: itself

text: be

text: sharp.

text: L,f

text: f

text: If

text: one

text: desires,

text: as

text: we

text: do,

text: to

text: ﬁnd

text: the

text: actual

text: function

text: B

text: in

text: the

text: case

text: when

text: h

text: is

text: con-

text: Q,Φ

text: cave,

text: one

text: has

text: to

text: understand

text: the

text: nature

text: of

text: the

text: extremizing

text: split

text: x

text: (x

text: in

text: the

text: main

text: inequality

text: i.e.,

text: the

text: choice

text: of

text: x

text: that

text: turns

text: the

text: inequality

text: into

text: an

text: equality

text: or

text: ap-

text: proximate

text: equality.

text: The

text: candidates

text: A

text: and

text: a

text: were

text: derived

text: under

text: the

text: assumption

text: that

text: L,f

text: f

text: such

text: x

text: are

text: inﬁnitesimally

text: close

text: to

text: x,

text: meaning

text: is

text: equivalent

text: to

text: its

text: inﬁnitesimal

text: ver-

text: sion,

text: However,

text: for

text: concave

text: h,

text: A

text: fails

text: the

text: main

text: inequality

text: at

text: every

text: point,

text: as

text: is

text: clear

text: L,f

text: from

text: formula

text: in

text: the

text: proof

text: of

text: Lemma

text: For

text: similar

text: reasons,

text: a

text: does

text: not

text: work

text: either.

text: f

text: Therefore,

text: the

text: optimizing

text: split

text: is

text: not

text: inﬁnitesimal

text: in

text: this

text: case.

text: An

text: approach

text: that

text: seems

text: promising

text: is

text: to

text: consider

text: piecewise-linear

text: concave

text: h,

text: and

text: then

text: approximate

text: an

text: arbitrary

text: concave

text: h

text: by

text: such

text: functions.

text: One

text: then

text: would

text: look

text: for

text: the

text: Bellman

text: A

text: candidate

text: such

text: that

text: (s)

text: c

text: δ

text: (s),

text: k

text: sk

text: k

text: where

text: s

text: are

text: the

text: points

text: of

text: discontinuity

text: of

text: h

text: δ

text: are

text: Dirac

text: masses

text: at

text: s

text: and

text: c

text: are

text: the

text: neg-

text: k

text: sk

text: k

text: kA

text: ative

text: coeﬃcients

text: chosen

text: so

text: that

text: the

text: main

text: inequality

text: for

text: the

text: resulting

text: candidate

text: is

text: satisﬁed

text: on

text: the

text: whole

text: domain.

text: The

text: presence

text: of

text: c

text: would

text: entail

text: a

text: non-inﬁnitesimal

text: optimal

text: split

text: at

text: the

text: k

text: point

text: x

text: (s

text: s

text: The

text: nature

text: of

text: the

text: dependence

text: of

text: the

text: values

text: of

text: c

text: on

text: the

text: local

text: behavior

text: of

text: k

text: k

text: k

text: h

text: is

text: the

text: subject

text: of

text: further

text: study.

text: Optimizers

text: In

text: this

text: section

text: we

text: prove

text: the

text: converse

text: inequalities

text: to

text: those

text: established

text: in

text: Lemma

text: thus

text: completing

text: the

text: proof

text: of

text: Theorems

text: and

text: This

text: is

text: done

text: through

text: the

text: construction

text: of

text: two

text: optimizing

text: sequences

text: one

text: for

text: a

text: and

text: one

text: for

text: A

text: Importantly,

text: these

text: sequences

text: do

text: not

text: f

text: L,f

text: depend

text: on

text: f,

text: meaning

text: that

text: the

text: upper

text: Bellman

text: function

text: B

text: is

text: linear

text: with

text: respect

text: to

text: Φ

text: for

text: Q,Φ

text: all

text: increasing

text: Φ

text: such

text: that

text: h(s)

text: is

text: convex

text: and,

text: separately,

text: for

text: all

text: decreasing

text: Φ.

text: Likewise,

text: the

text: lower

text: Bellman

text: function

text: b

text: is

text: linear

text: in

text: Φ

text: on

text: the

text: class

text: of

text: all

text: Φ

text: such

text: that

text: h

text: is

text: Q,Φ

text: increasing.

text: Without

text: loss

text: of

text: generality,

text: we

text: will

text: deﬁne

text: our

text: optimizers

text: almost

text: everywhere

text: on

text: I

text: Fix

text: a

text: point

text: x

text: Ω

text: We

text: say

text: that

text: a

text: sequence

text: of

text: functions

text: wx

text: on

text: is

text: an

text: optimizing

text: Q

text: A

text: n

text: sequence

text: for

text: a

text: Bellman

text: candidate

text: at

text: x,

text: if

text: wx

text: satisﬁes

text: the

text: following

text: three

text: conditions:

text: n

text: n,

text: wx

text: n,

text: wx

text: x

text: (wx)

text: x

text: n

text: n

text: A

text: J

text: Φ

text: wx

text: (wx)

text: R

text: (wx)

text: x

text: x

text: as

text: n

text: n

text: J

text: n

text: J

text: J

text: n

text: ∈D

text: J

text: ##LTLine##


text: BEST CONSTANTS FOR A FAMILY OF CARLESON SEQUENCES 29 − Figure 2. The optimizer for A : normal splits x = 1(x + x+) inside the L,f 2 domain; tangential split on the boundary. − Our recursive construction for w(k) directly implies that for k = 1, ..., 2N+n 1, n 1 1 Σ = Σ − + Σ + 8h(s ) ∆2. k 2 k 1 2 k+1 k n − On the other hand, Σ = 0, while (7.11)implies Σ = Σ + 8(L2 s2)h(L). We caneasily 2N+n (cid:88) 0 1 (cid:88)1 solve this tridiagonal linear system:   − 2N+n 1 k − − − − − Σ = (2N+n k)8(L2 s2)h(L) + 16∆2 (2N+n j) h(s ) (k j) h(s ) , k 1 n j j j=1 j=1 for k = 0, ..., 2N+m (where for k = 0 the second sum in brackets is zero by convention). Since − − − − for any i and j, (i j)∆ = s s , and(cid:88)L2 s2 = ∆ (2L ∆ )(cid:88), we have n j i  1 n n  − 2N+n 1 k − − − − − Σ = 8(s 1)(2L ∆ )h(L) + 16 (s 1) h(s ) ∆ (s s ) h(s ) ∆ . k k n j j n j k j n (cid:20)j=(cid:90)1 (cid:90) j=1 (cid:21) → ∞ Setting k = m2n and s = s, and letting n , we get k −−−→ − f(L) L − − L − Σ 16(s 1) + 16 (z 1) h(z) dz (z s) h(z) dz , m2n →∞ (cid:0) (cid:1) n L 1 s which is the same thing as A (s) from (4.19). √ L,f (cid:62) ∈ √ So far, we have proved that B (x , x ) A x x for all (x , x ) Ω such that Q,Φ 1 2 L,f 1 2 ∈1 2 Q x x is a dyadically-rational point in the interval [1, L]. Now, any s (1, L) can be written 1 2 as s = lim →∞ s , where each s is dyadically-rational. Observe that B (s, s) is continuous n n n Q,Φ on (1, L) as a function of s, since it is concave on [1, L] by (5.1). Since A is also continuous L,f on [1, L], we can take the limit in the inequality (cid:62) B (s , s ) A (s ), Q,Φ n n L,f n (cid:3) thus completing the proof. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BEST

text: CONSTANTS

text: FOR

text: A

text: FAMILY

text: OF

text: CARLESON

text: SEQUENCES

label: caption

text: Figure

text: The

text: optimizer

text: for

text: A

text: normal

text: splits

text: x

text: x+)

text: inside

text: the

text: L,f

text: domain;

text: tangential

text: split

text: on

text: the

text: boundary.

text: Our

text: recursive

text: construction

text: for

text: w(k)

text: directly

text: implies

text: that

text: for

text: k

text: n

text: Σ

text: Σ

text: Σ

text: k

text: k

text: k

text: n

text: On

text: the

text: other

text: hand,

text: Σ

text: while

text: Σ

text: Σ

text: We

text: caneasily

text: solve

text: this

text: tridiagonal

text: linear

text: system:

text: k

text: Σ

text: j)

text: h(s

text: (k

text: j)

text: h(s

text: k

text: n

text: j

text: j

text: for

text: k

text: (where

text: for

text: k

text: the

text: second

text: sum

text: in

text: brackets

text: is

text: zero

text: by

text: convention).

text: Since

text: for

text: any

text: i

text: and

text: j,

text: (i

text: j)∆

text: s

text: s

text: we

text: have

text: n

text: j

text: i

text: n

text: n

text: k

text: Σ

text: )h(L)

text: (s

text: h(s

text: (s

text: s

text: h(s

text: k

text: k

text: n

text: j

text: j

text: n

text: j

text: k

text: j

text: n

text: Setting

text: k

text: and

text: s

text: s,

text: and

text: letting

text: n

text: we

text: get

text: k

text: f(L)

text: L

text: L

text: Σ

text: (z

text: h(z)

text: dz

text: (z

text: s)

text: h(z)

text: dz

text: n

text: L

text: s

text: which

text: is

text: the

text: same

text: thing

text: as

text: A

text: (s)

text: from

text: L,f

text: So

text: far,

text: we

text: have

text: proved

text: that

text: B

text: (x

text: x

text: A

text: x

text: x

text: for

text: all

text: (x

text: x

text: Ω

text: such

text: that

text: Q,Φ

text: L,f

text: ∈1

text: Q

text: x

text: x

text: is

text: a

text: dyadically-rational

text: point

text: in

text: the

text: interval

text: L].

text: Now,

text: any

text: s

text: L)

text: can

text: be

text: written

text: as

text: s

text: lim

text: s

text: where

text: each

text: s

text: is

text: dyadically-rational.

text: Observe

text: that

text: B

text: (s,

text: s)

text: is

text: continuous

text: n

text: n

text: n

text: Q,Φ

text: on

text: L)

text: as

text: a

text: function

text: of

text: s,

text: since

text: it

text: is

text: concave

text: on

text: L]

text: by

text: Since

text: A

text: is

text: also

text: continuous

text: L,f

text: on

text: L],

text: we

text: can

text: take

text: the

text: limit

text: in

text: the

text: inequality

text: B

text: (s

text: s

text: A

text: (s

text: Q,Φ

text: n

text: n

text: L,f

text: n

text: thus

text: completing

text: the

text: proof.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 LEONID SLAVIN (cid:90) (3) If Φ is decreasing, then Q Φ(y) K (Q) = 8 dy. Φ y 1 Remark 2.2. As the formulation above suggests, we do not know the actual function K in the Φ case when h is concave. That is because the solution of the corresponding extremal problem diﬀers in structure from that for other Φ; in particular, it does not arise as the solution of a diﬀerential equation. Instead, we provide an upper estimate. As explained in Section , the estimate given above can be slightly improved, but the result is far less transparent and likely still not sharp. The estimate in Part (2) is the best known and easy to use, especially when Φ is speciﬁed to be a power function (see below). Our second theorem concerns lower estimates. Theorem 2.3. (cid:90) (1) If h is increasing, then Q Φ(y) k (Q) = 8 dy. Φ (cid:16) y (cid:17) 1 (2) If h is decreasing, then − 1 k (Q) = 8Φ(Q) 1 . Φ Q Note that we make no explicit assumptions of pointwise diﬀerentiability, or even continuity, on the part ofΦ or h. Let us restate these results for the case Φ(t) = tα. (cid:0) (cid:1) (cid:0) (cid:3) (cid:2) (cid:1)  Corollary 2.4.    8(2α + 1) − 32α − 8 ∈ ∪ ∪ ∞  Qα − Qα 1/2 + − , α 0, 1 1, 1 3, ; α 2α 1 α(2α 1) 2 2 2  − − 32Q1/2 8 log Q 32, α = 1/2;  Kα(Q) =   8 log Q, α = 0; 8 − (Qα 1), α < 0; α (cid:2) (cid:3) √ (cid:2) (cid:3) √ √ (cid:0) (cid:1) (cid:54) · − − − − − − − K (Q) 8 32 2α ( Q 1)(8Q Q 1)2α 2(3 Q 1)3 2α α = K (Q) 3−2α K (Q) 2α−2, α ∈ 1, 3 ;  1 3/2 2  8(cid:0) − (cid:1) (cid:62) (Qα 1), α 1; k (Q) = α α − − 8 Qα Qα 1 , α < 1. ∈ ≈ Remark 2.5. Thus, for α > 0, α / (1, 3/2), the sharp order of growth is K (Q) 8(2+1/α)Qα ∈ α for large Q. On the other hand, for α (1, 3/2) using the formula from Part (2) of Theorem 2.1 amounts simply to log-linear interpolation between the sharp results for α = 1 and α = 3/2, or, equivalently, to an application of H¨older’s inequality. The logarithmic terms in the cases α = 0 and α = 1/2 correspond to the blow-up of the Nazarov–Volberg estimate (1.2) for (cid:107){ }(cid:107) these values of α. Lastly, note that we get the same constant for c(0)(w) C as Wittwer did (cid:107){ }(cid:107) − J for a (w) C in (1.3), even though c(0)(w) = a (w) + a (w 1). This reﬂects the fact that J − J J J the quantities a (w) and a (w 1) cannot be too large at the same time. J J ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: LEONID

text: SLAVIN

text: If

text: Φ

text: is

text: decreasing,

text: then

text: Q

text: Φ(y)

text: K

text: (Q)

text: dy.

text: Φ

text: y

text: Remark

text: As

text: the

text: formulation

text: above

text: suggests,

text: we

text: do

text: not

text: know

text: the

text: actual

text: function

text: K

text: in

text: the

text: Φ

text: case

text: when

text: h

text: is

text: concave.

text: That

text: is

text: because

text: the

text: solution

text: of

text: the

text: corresponding

text: extremal

text: problem

text: diﬀers

text: in

text: structure

text: from

text: that

text: for

text: other

text: Φ;

text: in

text: particular,

text: it

text: does

text: not

text: arise

text: as

text: the

text: solution

text: of

text: a

text: diﬀerential

text: equation.

text: Instead,

text: we

text: provide

text: an

text: upper

text: estimate.

text: As

text: explained

text: in

text: Section

text: the

text: estimate

text: given

text: above

text: can

text: be

text: slightly

text: improved,

text: but

text: the

text: result

text: is

text: far

text: less

text: transparent

text: and

text: likely

text: still

text: not

text: sharp.

text: The

text: estimate

text: in

text: Part

text: is

text: the

text: best

text: known

text: and

text: easy

text: to

text: use,

text: especially

text: when

text: Φ

text: is

text: speciﬁed

text: to

text: be

text: a

text: power

text: function

text: (see

text: below).

text: Our

text: second

text: theorem

text: concerns

text: lower

text: estimates.

text: Theorem

text: If

text: h

text: is

text: increasing,

text: then

text: Q

text: Φ(y)

text: k

text: (Q)

text: dy.

text: Φ

text: y

text: If

text: h

text: is

text: decreasing,

text: then

text: k

text: (Q)

text: Φ

text: Q

text: Note

text: that

text: we

text: make

text: no

text: explicit

text: assumptions

text: of

text: pointwise

text: diﬀerentiability,

text: or

text: even

text: continuity,

text: on

text: the

text: part

text: ofΦ

text: or

text: h.

text: Let

text: us

text: restate

text: these

text: results

text: for

text: the

text: case

text: Φ(t)

text: tα.

text: Corollary

text: 8(2α

text: Qα

text: Qα

text: α

text: α

text: 2α

text: α(2α

text: log

text: Q

text: α

text: Kα(Q)

text: log

text: Q,

text: α

text: (Qα

text: α

text: α

text: K

text: (Q)

text: 2α

text: Q

text: Q

text: Q

text: 2α

text: α

text: K

text: (Q)

text: 3−2α

text: K

text: (Q)

text: α

text: (Qα

text: α

text: k

text: (Q)

text: α

text: α

text: Qα

text: Qα

text: α

text: Remark

text: Thus,

text: for

text: α

text: α

text: the

text: sharp

text: order

text: of

text: growth

text: is

text: K

text: (Q)

text: α

text: for

text: large

text: Q.

text: On

text: the

text: other

text: hand,

text: for

text: α

text: using

text: the

text: formula

text: from

text: Part

text: of

text: Theorem

text: amounts

text: simply

text: to

text: log-linear

text: interpolation

text: between

text: the

text: sharp

text: results

text: for

text: α

text: and

text: α

text: or,

text: equivalently,

text: to

text: an

text: application

text: of

text: H¨older’s

text: inequality.

text: The

text: logarithmic

text: terms

text: in

text: the

text: cases

text: α

text: and

text: α

text: correspond

text: to

text: the

text: blow-up

text: of

text: the

text: Nazarov–Volberg

text: estimate

text: for

text: these

text: values

text: of

text: α.

text: Lastly,

text: note

text: that

text: we

text: get

text: the

text: same

text: constant

text: for

text: C

text: as

text: Wittwer

text: did

text: J

text: for

text: a

text: (w)

text: C

text: in

text: even

text: though

text: a

text: (w)

text: a

text: (w

text: This

text: reﬂects

text: the

text: fact

text: that

text: J

text: J

text: J

text: J

text: the

text: quantities

text: a

text: (w)

text: and

text: a

text: (w

text: cannot

text: be

text: too

text: large

text: at

text: the

text: same

text: time.

text: J

text: J

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: BEST CONSTANTS FOR A FAMILY OF CARLESON SEQUENCES 5 Let us record several related results, all of which are proved in the next section. First, the lower estimates of Theorem 2.3 allow us to obtain a range of equivalent deﬁnitions of Ad in { } → ∞ → ∞ 2 terms of sequences cΦ for all increasing Φ such that Φ(t) as t . We note that J this limit condition, as well as the inverted lower estimate in Part (ii) have direct analogs in a recent paper [5] concerning equivalent deﬁnitions of BMO. Theorem 2.6. Assume that Φ is increasing. (cid:0) (cid:1) (cid:0) (cid:1) ∈ { } (i) If w Ad, then the sequence cΦ(w) is Carleson and 2 J (cid:107){ }(cid:107) (cid:54) cΦ(w) C 8Φ [w] log [w] . J Ad Ad 2 2 { } (cid:0) ∞ (cid:1) ∈ (ii) If cΦ(w) is Carleson and lim Φ(t) = , then w Ad and J →∞ 2 t (cid:54) (cid:107){ }(cid:107) (cid:90) [w] v cΦ(w) C , Ad J 2 ∞ → ∞ 8 Q (cid:62) where v : [0, ) [1, ) is the inverse to the function u(Q) = Φ(t) dt, Q 1. Q (cid:54) ∞ ∈ { 1 } (iii) If lim Φ(t) = , then there exists a weight w / Ad such that cΦ(w) is Carleson. →∞ 2 J t ∞ The main thrust of this theorem is that the condition lim →∞ Φ(t) = is necessary and { } ⇒ ∈ t suﬃcient for the implication “ cΦ(w) is Carleson” w Ad. The quantitative estimates J 2 are stated so that they work for all increasing Φ; as such, they are sharp only on the class of all such Φ. If one has a speciﬁc Φ that falls under one of the cases in both Theorem 2.1 and Theorem 2.3, one can, in general, do better by using the estimates from those theorems. In particular, we have the following corollary. Corollary 2.7. { } (cid:0) (cid:1) ∈ (i) If α > 0 and c(α)(w) is a Carleson sequence and, then w Ad and J 2 (cid:54) − (cid:107){ }(cid:107) [w] k 1 c(α)(w) C , Ad α J 2 − ∞ → ∞ where k 1 : [0, ) [1, ) is the inverse to k given in Theorem 2.13. This estimate α α is sharp. (cid:54) ∈ { } (ii) If α 0, there exists a weight w / Ad such that c(α)(w) is a Carleson sequence. 2 J { } The reader will notice that the sequence cΦ(w) does not change if we replace w with τw J for τ > 0. This zero-degree homogeneity is of central importance in our sharp proofs, but if we are willing to slightly compromise on sharpness, we can easily extend our main theorems to settings with diﬀerent homogeneity. To illustrate this point, let us examine the sequence | | (cid:104) (cid:105) (cid:104) − (cid:105) c(α,β)(w) = J w α w 1 β R (w) J J J J (cid:54) for α = β. For this sequence, we can obtain a range of inequalities of the ﬂavor studied in [2]. ∈(cid:88) (cid:0) (cid:1) Corollary 2.8. For any w Ad and any α, β such that α > β, 2 1 (cid:54) (cid:104) − (cid:105) | | c(α,β)(w) e K [w] wα β . I ∈D J α Ad2 I J (I) − Remark 2.9. Of course, the same conclusion holds if we interchange w and w 1. In addition, − (cid:54) (cid:104) − (cid:105) (cid:104) (cid:105) − if α β 1, we can replace wα β with w α β by H¨older’s inequality. I I ##LTLine## ##LTLine##

text: BEST

text: CONSTANTS

text: FOR

text: A

text: FAMILY

text: OF

text: CARLESON

text: SEQUENCES

text: Let

text: us

text: record

text: several

text: related

text: results,

text: all

text: of

text: which

text: are

text: proved

text: in

text: the

text: next

text: section.

text: First,

text: the

text: lower

text: estimates

text: of

text: Theorem

text: allow

text: us

text: to

text: obtain

text: a

text: range

text: of

text: equivalent

text: deﬁnitions

text: of

text: Ad

text: in

text: terms

text: of

text: sequences

text: cΦ

text: for

text: all

text: increasing

text: Φ

text: such

text: that

text: Φ(t)

text: as

text: t

text: We

text: note

text: that

text: J

text: this

text: limit

text: condition,

text: as

text: well

text: as

text: the

text: inverted

text: lower

text: estimate

text: in

text: Part

text: (ii)

text: have

text: direct

text: analogs

text: in

text: a

text: recent

text: paper

text: concerning

text: equivalent

text: deﬁnitions

text: of

text: BMO.

text: Theorem

text: Assume

text: that

text: Φ

text: is

text: increasing.

text: (i)

text: If

text: w

text: Ad,

text: then

text: the

text: sequence

text: cΦ(w)

text: is

text: Carleson

text: and

text: J

text: cΦ(w)

text: C

text: 8Φ

text: [w]

text: log

text: [w]

text: J

text: Ad

text: Ad

text: (ii)

text: If

text: cΦ(w)

text: is

text: Carleson

text: and

text: lim

text: Φ(t)

text: then

text: w

text: Ad

text: and

text: J

text: t

text: [w]

text: v

text: cΦ(w)

text: C

text: Ad

text: J

text: Q

text: where

text: v

text: is

text: the

text: inverse

text: to

text: the

text: function

text: u(Q)

text: Φ(t)

text: dt,

text: Q

text: Q

text: (iii)

text: If

text: lim

text: Φ(t)

text: then

text: there

text: exists

text: a

text: weight

text: w

text: Ad

text: such

text: that

text: cΦ(w)

text: is

text: Carleson.

text: J

text: t

text: The

text: main

text: thrust

text: of

text: this

text: theorem

text: is

text: that

text: the

text: condition

text: lim

text: Φ(t)

text: is

text: necessary

text: and

text: t

text: suﬃcient

text: for

text: the

text: implication

text: cΦ(w)

text: is

text: Carleson”

text: w

text: Ad.

text: The

text: quantitative

text: estimates

text: J

text: are

text: stated

text: so

text: that

text: they

text: work

text: for

text: all

text: increasing

text: Φ;

text: as

text: such,

text: they

text: are

text: sharp

text: only

text: on

text: the

text: class

text: of

text: all

text: such

text: Φ.

text: If

text: one

text: has

text: a

text: speciﬁc

text: Φ

text: that

text: falls

text: under

text: one

text: of

text: the

text: cases

text: in

text: both

text: Theorem

text: and

text: Theorem

text: one

text: can,

text: in

text: general,

text: do

text: better

text: by

text: using

text: the

text: estimates

text: from

text: those

text: theorems.

text: In

text: particular,

text: we

text: have

text: the

text: following

text: corollary.

text: Corollary

text: (i)

text: If

text: α

text: and

text: c(α)(w)

text: is

text: a

text: Carleson

text: sequence

text: and,

text: then

text: w

text: Ad

text: and

text: J

text: [w]

text: k

text: c(α)(w)

text: C

text: Ad

text: α

text: J

text: where

text: k

text: is

text: the

text: inverse

text: to

text: k

text: given

text: in

text: Theorem

text: This

text: estimate

text: α

text: α

text: is

text: sharp.

text: (ii)

text: If

text: α

text: there

text: exists

text: a

text: weight

text: w

text: Ad

text: such

text: that

text: c(α)(w)

text: is

text: a

text: Carleson

text: sequence.

text: J

text: The

text: reader

text: will

text: notice

text: that

text: the

text: sequence

text: cΦ(w)

text: does

text: not

text: change

text: if

text: we

text: replace

text: w

text: with

text: τw

text: J

text: for

text: τ

text: This

text: zero-degree

text: homogeneity

text: is

text: of

text: central

text: importance

text: in

text: our

text: sharp

text: proofs,

text: but

text: if

text: we

text: are

text: willing

text: to

text: slightly

text: compromise

text: on

text: sharpness,

text: we

text: can

text: easily

text: extend

text: our

text: main

text: theorems

text: to

text: settings

text: with

text: diﬀerent

text: homogeneity.

text: To

text: illustrate

text: this

text: point,

text: let

text: us

text: examine

text: the

text: sequence

text: c(α,β)(w)

text: J

text: w

text: α

text: w

text: β

text: R

text: (w)

text: J

text: J

text: J

text: J

text: for

text: α

text: β.

text: For

text: this

text: sequence,

text: we

text: can

text: obtain

text: a

text: range

text: of

text: inequalities

text: of

text: the

text: ﬂavor

text: studied

text: in

text: Corollary

text: For

text: any

text: w

text: Ad

text: and

text: any

text: α,

text: β

text: such

text: that

text: α

text: β,

text: c(α,β)(w)

text: e

text: K

text: [w]

text: wα

text: β

text: I

text: ∈D

text: J

text: α

text: I

text: J

text: (I)

text: Remark

text: Of

text: course,

text: the

text: same

text: conclusion

text: holds

text: if

text: we

text: interchange

text: w

text: and

text: w

text: In

text: addition,

text: if

text: α

text: β

text: we

text: can

text: replace

text: wα

text: β

text: with

text: w

text: α

text: β

text: by

text: H¨older’s

text: inequality.

text: I

text: I

text: ##LTLine##

text: ##LTLine##


text: 6 LEONID SLAVIN We now return to Theorems 2.1 and 2.3. To explain where they come from, we need to (cid:62) deﬁne the Bellman functions for our problem. Fix Q 1 and let { (cid:54) (cid:54) } Ω = (x , x ) : 1 x x Q . (cid:8) Q 1 2 1 2 (cid:9) ∈ ∈ D For each x = (x , x ) Ω and each I , let 1 2 Q ∈ (cid:104) (cid:105) (cid:104) − (cid:105) E = w : w Ad,Q(I), w = x , w 1 = x . Q,x,I 2(cid:110) (cid:88) I 1 I (cid:111)2 The upper and lower Bellman functions are deﬁned, respectively, by 1 ∈ (2.3) B (x , x ) = sup | | cΦ(w) : w E Q,Φ 1 2 I ∈D J Q,x,I (cid:110) J(cid:88)(I) (cid:111) and 1 ∈ (2.4) b (x , x ) = inf | | cΦ(w) : w E . Q,Φ 1 2 I ∈D J Q,x,I J (I) If Φ is a power function, Φ(t) = tα, we will write B and b for B and b , Q,α Q,α Q,Φ Q,Φ respectively. One immediately observes that the functions B and b are independent of the interval Q,Φ Q,Φ ∈ ∈ D I that formally enters into their deﬁnitions. In addition, for any w Ad,Q and any I , we (cid:104) (cid:105) (cid:104) − (cid:105) ∈ 2 have ( w , w 1 ) Ω , thus, the functions B and b are deﬁned, at least formally, I I Q Q,Φ (cid:62) Q,Φ ∈ D on Ω . The fact that E is nonempty for every Q 1, every interval I , and every ∈ Q Q,x,I x Ω is subsumed in the(cid:20)statemen(cid:90)ts of Theor(cid:21)ems 2.10 and(cid:90)2.11 below. Q (cid:62) ∞ For a number L 1 and a non-negative function f on [1, ) formally deﬁne: f(L) L f(z) − − s f(z) − (2.5) A (s) = 16 + dz(cid:90) (s 1) 16 (s z) dz, L,f L z2 z2 1 1 s f(z) (2.6) a (s) = 16 dz. f z 1 For reasons that will soon be made clear, we will often refer to these two functions as Bellman candidates. We are now in a position to state the main theorems about the functions B Q,Φ and b , which are used in the next section to prove Theorems 2.1 and 2.3. Q,Φ Theorem 2.10. In the notation (2.1) and (2.2), (1) If Φ is increasing and h is convex, then √ BQ,Φ(x1, x2)(cid:0)= AL,f( x1x2). (cid:1) (cid:0) (cid:1) (2) If Φ is increasing and h i√s concave, then √ √ (cid:54) − − − B (x , x ) 8( x x 1) 4L x x 1 h s ( x x ) , Q,Φ 1 2 1 2 1 2 0 1 2 where − − − 9L2 s2 s 1 s (s) = − − . 0 3(4L s 1) (3) If Φ is decreasing, then √ B (x , x ) = a ( x x ). Q,Φ 1 2 f 1 2 Theorem 2.11. In the notation (2.1) and (2.2), (1) If h is increasing, then √ b (x , x ) = a ( x x ). Q,Φ 1 2 f 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: LEONID

text: SLAVIN

text: We

text: now

text: return

text: to

text: Theorems

text: and

text: To

text: explain

text: where

text: they

text: come

text: from,

text: we

text: need

text: to

text: deﬁne

text: the

text: Bellman

text: functions

text: for

text: our

text: problem.

text: Fix

text: Q

text: and

text: let

text: Ω

text: (x

text: x

text: x

text: x

text: Q

text: Q

text: D

text: For

text: each

text: x

text: (x

text: x

text: Ω

text: and

text: each

text: I

text: let

text: Q

text: E

text: w

text: w

text: Ad,Q(I),

text: w

text: x

text: w

text: x

text: Q,x,I

label: equation

text: I

text: I

text: The

text: upper

text: and

text: lower

text: Bellman

text: functions

text: are

text: deﬁned,

text: respectively,

text: by

text: B

text: (x

text: x

text: sup

text: cΦ(w)

text: w

text: E

text: Q,Φ

text: I

text: ∈D

text: J

text: Q,x,I

text: and

text: b

text: (x

text: x

text: inf

text: cΦ(w)

text: w

text: E

text: Q,Φ

text: I

text: ∈D

text: J

text: Q,x,I

text: J

text: (I)

text: If

text: Φ

text: is

text: a

text: power

text: function,

text: Φ(t)

text: tα,

text: we

text: will

text: write

text: B

text: and

text: b

text: for

text: B

text: and

text: b

text: Q,α

text: Q,α

text: Q,Φ

text: Q,Φ

text: respectively.

text: One

text: immediately

text: observes

text: that

text: the

text: functions

text: B

text: and

text: b

text: are

text: independent

text: of

text: the

text: interval

text: Q,Φ

text: Q,Φ

text: D

text: I

text: that

text: formally

text: enters

text: into

text: their

text: deﬁnitions.

text: In

text: addition,

text: for

text: any

text: w

text: Ad,Q

text: and

text: any

text: I

text: we

text: have

text: w

text: w

text: Ω

text: thus,

text: the

text: functions

text: B

text: and

text: b

text: are

text: deﬁned,

text: at

text: least

text: formally,

text: I

text: I

text: Q

text: Q,Φ

text: Q,Φ

text: D

text: on

text: Ω

text: The

text: fact

text: that

text: E

text: is

text: nonempty

text: for

text: every

text: Q

text: every

text: interval

text: I

text: and

text: every

text: Q

text: Q,x,I

text: x

text: Ω

text: is

text: subsumed

text: in

text: of

text: below.

text: Q

text: For

text: a

text: number

text: L

text: and

text: a

text: non-negative

text: function

text: f

text: on

text: formally

text: deﬁne:

text: f(L)

text: L

text: f(z)

text: s

text: f(z)

text: A

text: (s)

text: (s

text: (s

text: z)

text: dz,

text: L,f

text: L

text: s

text: f(z)

text: a

text: (s)

text: dz.

text: f

text: z

text: For

text: reasons

text: that

text: will

text: soon

text: be

text: made

text: clear,

text: we

text: will

text: often

text: refer

text: to

text: these

text: two

text: functions

text: as

text: Bellman

text: candidates.

text: We

text: are

text: now

text: in

text: a

text: position

text: to

text: state

text: the

text: main

text: theorems

text: about

text: the

text: functions

text: B

text: Q,Φ

text: and

text: b

text: which

text: are

text: used

text: in

text: the

text: next

text: section

text: to

text: prove

text: Theorems

text: and

text: Q,Φ

text: Theorem

text: In

text: the

text: notation

text: and

text: If

text: Φ

text: is

text: increasing

text: and

text: h

text: is

text: convex,

text: then

text: AL,f(

text: If

text: Φ

text: is

text: increasing

text: and

text: h

text: i√s

text: concave,

text: then

text: B

text: (x

text: x

text: x

text: x

text: x

text: x

text: h

text: s

text: x

text: x

text: Q,Φ

text: where

text: s

text: s

text: (s)

text: s

text: If

text: Φ

text: is

text: decreasing,

text: then

text: B

text: (x

text: x

text: a

text: x

text: x

text: Q,Φ

text: f

text: Theorem

text: In

text: the

text: notation

text: and

text: If

text: h

text: is

text: increasing,

text: then

text: b

text: (x

text: x

text: a

text: x

text: x

text: Q,Φ

text: f

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: BEST CONSTANTS FOR A FAMILY OF CARLESON SEQUENCES 7 (cid:16) (cid:17) (2) If h is decreasing, then − 1 b (x , Q/x ) = 8 Φ(Q) 1 . Q,Φ 1 1 Q Remark 2.12. Note that in all cases in these theorems, all integrals involved in the deﬁnitions of A and a are well-deﬁned. L,f f To summarize: in Parts (1) and (3) of Theorem 2.10 and in Part (1) of Theorem 2.11, we ﬁnd the exact Bellman functions at every point of Ω ; in Part (2) of Theorem 2.10, we Q provide a point-wise majorant for B ; and in Part (2) of Theorem 2.11, we ﬁnd b only Q,Φ Q,Φ on the boundary curve x x = Q (which, however, is enough to compute k for this case). 1 2 Φ For Φ(t) = tα, these formulas become  √ (cid:0) (cid:1) (cid:0) (cid:3) (cid:2) (cid:1)  Corollary 2.13. Let s = x1x2.    8 − 32α (cid:0) −(cid:1) − ∈ ∪ ∪ ∞  − (1 s2α) + − L2α 1(s 1), α 0, 1 1, 1 3, ; α(2α 1) 2α 1 2 2 2  − − 32(s 1) 16 log L + 16 s log L , α = 1/2; BQ,α(x1, x2) =  s  16 log s, α = 0; 8 − (cid:2) (s2α 1),(cid:3) (cid:2) (cid:3) α < (cid:0)0; (cid:1) α (cid:54) − − ∈ B (x , x ) B (x , x ) 3 2α B (x , x ) 2α 2, α 1, 3 ; Q,α 1 2 Q,1 1 2 Q,3/2 1 2 2 (cid:0) (cid:1) 8 − (cid:62) b (x , x ) = (s2α 1), α 1; Q,α 1 2 α − − b (x , Q/x ) = 8 L2α 2 L2 1 , α < 1. Q,α 1 1 Before proceeding with the proofs of these theorems and corollaries, we brieﬂy discuss the principal method at play and make several related remarks. The Bellman function technique in analysis consists of ﬁnding special functions with desig- nated size and convexity properties, to aid in induction-based proofs. In martingale settings, such as ours, the ﬁrst Bellman-function proofs can be found in the work of Burkholder [4]. Later, the technique was applied, in a signiﬁcantly diﬀerent form and under the current name, to many problems in harmonic analysis by Nazarov, Treil, and Volberg [11, 10, 12], followed by others; the reader can consult the lecture notes [18] for details. We also note the work of Os¸ekowski on sharp inequalities for martingales [14, 15], which implements what can be termed a mixed Burkholder–Bellman method. An important distinction exists between two kinds of Bellman functions. One kind is the true Bellman functions, which are deﬁned as solutions of extremal problems such as (2.3) and (2.4). They provide complete information about the inequality in question, including the structure of optimizers, but may be diﬃcult to compute. The other and much more prevalent kind are Bellman-type functions, which are useful substitutes with properties similar to those of the true function. These are non-unique and much easier to ﬁnd; however, they rarely produce completely sharp estimates and never the exact optimizers. Inequality (1.2), due to Beznosova and Nazarov–Treil, is a good example of the use of Bellman-type functions. Observe that the bound obtained is sharp in [w] , but sub-optimal in the numerical factor. Ad 2 Recent advances in ﬁnding true Bellman functions have made them viable as stand-alone objects of study, ahead of the inequalities they imply; that perspective is maintained here. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BEST

text: CONSTANTS

text: FOR

text: A

text: FAMILY

text: OF

text: CARLESON

text: SEQUENCES

text: If

text: h

text: is

text: decreasing,

text: then

text: b

text: (x

text: Q/x

text: Φ(Q)

text: Q,Φ

text: Q

text: Remark

text: Note

text: that

text: in

text: all

text: cases

text: in

text: these

text: theorems,

text: all

text: integrals

text: involved

text: in

text: the

text: deﬁnitions

text: of

text: A

text: and

text: a

text: are

text: well-deﬁned.

text: L,f

text: f

text: To

text: summarize:

text: in

text: Parts

text: and

text: of

text: Theorem

text: and

text: in

text: Part

text: of

text: Theorem

text: we

text: ﬁnd

text: the

text: exact

text: Bellman

text: functions

text: at

text: every

text: point

text: of

text: Ω

text: in

text: Part

text: of

text: Theorem

text: we

text: Q

text: provide

text: a

text: point-wise

text: majorant

text: for

text: B

text: and

text: in

text: Part

text: of

text: Theorem

text: we

text: ﬁnd

text: b

text: only

text: Q,Φ

text: Q,Φ

text: on

text: the

text: boundary

text: curve

text: x

text: x

text: Q

text: (which,

text: however,

text: is

text: enough

text: to

text: compute

text: k

text: for

text: this

text: case).

text: Φ

text: For

text: Φ(t)

text: tα,

text: these

text: formulas

text: become

text: Corollary

text: Let

text: s

text: α

text: α(2α

text: 2α

text: log

text: L

text: s

text: log

text: L

text: α

text: s

text: log

text: s,

text: α

text: α

text: α

text: B

text: (x

text: x

text: B

text: (x

text: x

text: 2α

text: B

text: (x

text: x

text: 2α

text: α

text: Q,α

text: b

text: (x

text: x

text: α

text: Q,α

text: α

text: b

text: (x

text: Q/x

text: α

text: Q,α

text: Before

text: proceeding

text: with

text: the

text: proofs

text: of

text: these

text: theorems

text: and

text: corollaries,

text: we

text: brieﬂy

text: discuss

text: the

text: principal

text: method

text: at

text: play

text: and

text: make

text: several

text: related

text: remarks.

text: The

text: Bellman

text: function

text: technique

text: in

text: analysis

text: consists

text: of

text: ﬁnding

text: special

text: functions

text: with

text: desig-

text: nated

text: size

text: and

text: convexity

text: properties,

text: to

text: aid

text: in

text: induction-based

text: proofs.

text: In

text: martingale

text: settings,

text: such

text: as

text: ours,

text: the

text: ﬁrst

text: Bellman-function

text: proofs

text: can

text: be

text: found

text: in

text: the

text: work

text: of

text: Burkholder

text: Later,

text: the

text: technique

text: was

text: applied,

text: in

text: a

text: signiﬁcantly

text: diﬀerent

text: form

text: and

text: under

text: the

text: current

text: name,

text: to

text: many

text: problems

text: in

text: harmonic

text: analysis

text: by

text: Nazarov,

text: Treil,

text: and

text: Volberg

text: followed

text: by

text: others;

text: the

text: reader

text: can

text: consult

text: the

text: lecture

text: notes

text: for

text: details.

text: We

text: also

text: note

text: the

text: work

text: of

text: Os¸ekowski

text: on

text: sharp

text: inequalities

text: for

text: martingales

text: which

text: implements

text: what

text: can

text: be

text: termed

text: a

text: mixed

text: Burkholder–Bellman

text: method.

text: An

text: important

text: distinction

text: exists

text: between

text: two

text: kinds

text: of

text: Bellman

text: functions.

text: One

text: kind

text: is

text: the

text: true

text: Bellman

text: functions,

text: which

text: are

text: deﬁned

text: as

text: solutions

text: of

text: extremal

text: problems

text: such

text: as

text: and

text: They

text: provide

text: complete

text: information

text: about

text: the

text: inequality

text: in

text: question,

text: including

text: the

text: structure

text: of

text: optimizers,

text: but

text: may

text: be

text: diﬃcult

text: to

text: compute.

text: The

text: other

text: and

text: much

text: more

text: prevalent

text: kind

text: are

text: Bellman-type

text: functions,

text: which

text: are

text: useful

text: substitutes

text: with

text: properties

text: similar

text: to

text: those

text: of

text: the

text: true

text: function.

text: These

text: are

text: non-unique

text: and

text: much

text: easier

text: to

text: ﬁnd;

text: however,

text: they

text: rarely

text: produce

text: completely

text: sharp

text: estimates

text: and

text: never

text: the

text: exact

text: optimizers.

text: Inequality

text: due

text: to

text: Beznosova

text: and

text: Nazarov–Treil,

text: is

text: a

text: good

text: example

text: of

text: the

text: use

text: of

text: Bellman-type

text: functions.

text: Observe

text: that

text: the

text: bound

text: obtained

text: is

text: sharp

text: in

text: [w]

text: but

text: sub-optimal

text: in

text: the

text: numerical

text: factor.

text: Ad

text: Recent

text: advances

text: in

text: ﬁnding

text: true

text: Bellman

text: functions

text: have

text: made

text: them

text: viable

text: as

text: stand-alone

text: objects

text: of

text: study,

text: ahead

text: of

text: the

text: inequalities

text: they

text: imply;

text: that

text: perspective

text: is

text: maintained

text: here.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 LEONID SLAVIN (cid:104) ∗(cid:105) (cid:104) ∗ − (cid:105) ∗ − ∗ We compute: w = x , (w ) 1 = Q/x , and R (w ) = 8(1 1/Q); also note that w I ∈ D1 −(cid:88)∪ D I 1 I∗ (cid:16) (cid:17) is constant on each J (I ) (I+). Therefore, [w ] = Q and Ad,Q(I) 2 (cid:54) 1 ∗ 1 | | ∗ ∗ − 1 b (x , Q/x ) | | cΦ(w ) = | | I f(s (w ))R (w ) = 8Φ(Q) 1 . Q,Φ 1 1 I ∈D J I I I Q J (I) (cid:3) (cid:62) Proof of Theorem 2.1. We will derive this theorem as a corollary of Theorem 2.10. Take Q 1 ∈ (cid:88) (cid:0) ∈ D (cid:1) and a w Ad such that Q = [w] . For any I , by the deﬁnition of B , 2 Ad Q,Φ 2 1 (cid:54) (cid:104) (cid:105) (cid:104) − (cid:105) (cid:54) | | cΦ(w) B w , w 1 sup B (x , x ). I J∈D(I) J Q,Φ I I x∈ΩQ Q,Φ 1 2 (cid:107){ }(cid:107) (cid:54) Therefore, cΦ(w) C sup ∈ B (x , x ) and, ﬁnally, J x ΩQ Q,Φ 1 2 (cid:107){ }(cid:107) (cid:54) K (Q) = sup cΦ(w) C sup B (x , x ). Φ J ∈ Q,Φ 1 2 [w]Ad2=Q x ΩQ √ The right-hand side expressions in all three statements of Theorem 2.10 attain their maxima when s = Q = L. For Parts (1) and (3), this is because both A and, respectively, a are (cid:0) (cid:1) L,f f increasing functions. For Part (2), we have − − − − − − 8(s 1)(4L s 1)3 8(s 1) 4L s 1 h(s (s)) = − − − f(s (s)). 0 9(9L2 s2 s 1)2 0 It is easy to verify that the fraction in front of f is increasing in s, as is the function s ; on 0 the other hand, f is increasing by assumption. (cid:54) √ Therefore, we obtain the three statements of Theorem 2.1 — but with “ ” instead of “=” in Parts (1) and (3) — by setting s = Q in the three respective statements of Theorem 2.10 and changing the variable in the integrals. To prove that the ﬁrst and third statements hold with equality, note that by the deﬁnition (cid:62) of B , for any Q 1, any x > 0 and any interval I, there exists a sequence of Ad(I)- Q,Φ{ } 1 (cid:88) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) − (cid:105) 2 weights w , such that for each n [w ] = Q, w = x , w w 1 = Q, and n n Ad2(I) n I 1 n I n I 1 lim | | cΦ(w ) = B (x , Q/x ). n→∞ I ∈D J n Q,Φ 1 1 J (I) R Let us extend each w to all of periodically by replicating it on each dyadic interval of | | n length I ; keep the name w for the extension. Clearly, [w ] = [w ] , which means that n n Ad n Ad(I) √ √ 2 2 the left-hand side does not exceed K (Q). On the other hand, by Theorem 2.10 the right-hand A Φ (cid:3) side equals ( Q) in Part (1) and a ( Q) in Part (3). The proof is complete. L,f f Proof Theorem 2.3. We will assume the statements of Theorem 2.11, Lemma 7.1, and Re- mark 7.2. The last two come from Section , which itself is self-contained. (cid:62) ∈ Proof of Part (1). Take any Q 1 and any w Ad such that [w] = Q. There exists a (cid:88){ } (cid:104) (cid:105) (cid:104) − (cid:105) → 2 (cid:16)→(cid:113)∞ Ad2 (cid:17) sequence of intervals I such that w w 1 Q as n . By the deﬁnition of b , n In In Q,Φ 1 (cid:62) (cid:104) (cid:105) (cid:104) − (cid:105) (cid:104) (cid:105) (cid:104) − (cid:105) | | cΦ(w) b ( w , w 1 ) = a w w 1 . In ∈D J Q,Φ In In f In In J (In) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: LEONID

text: SLAVIN

text: We

text: compute:

text: w

text: x

text: (w

text: Q/x

text: and

text: R

text: (w

text: also

text: note

text: that

text: w

text: I

text: D

text: I

text: I∗

text: is

text: constant

text: on

text: each

text: J

text: (I

text: (I+).

text: Therefore,

text: [w

text: Q

text: and

text: Ad,Q(I)

text: b

text: (x

text: Q/x

text: cΦ(w

text: I

text: f(s

text: (w

text: ))R

text: (w

text: Q,Φ

text: I

text: ∈D

text: J

text: I

text: I

text: I

text: Q

text: J

text: (I)

text: Proof

text: of

text: Theorem

text: We

text: will

text: derive

text: this

text: theorem

text: as

text: a

text: corollary

text: of

text: Theorem

text: Take

text: Q

text: D

text: and

text: a

text: w

text: Ad

text: such

text: that

text: Q

text: [w]

text: For

text: any

text: I

text: by

text: the

text: deﬁnition

text: of

text: B

text: Ad

text: Q,Φ

text: cΦ(w)

text: B

text: w

text: w

text: sup

text: B

text: (x

text: x

text: I

text: J∈D(I)

text: J

text: Q,Φ

text: I

text: I

text: x∈ΩQ

text: Q,Φ

text: Therefore,

text: cΦ(w)

text: C

text: sup

text: B

text: (x

text: x

text: and,

text: ﬁnally,

text: J

text: x

text: ΩQ

text: Q,Φ

text: K

text: (Q)

text: sup

text: cΦ(w)

text: C

text: sup

text: B

text: (x

text: x

text: Φ

text: J

text: Q,Φ

text: x

text: ΩQ

text: The

text: right-hand

text: side

text: expressions

text: in

text: all

text: three

text: statements

text: of

text: Theorem

text: attain

text: their

text: maxima

text: when

text: s

text: Q

text: L.

text: For

text: Parts

text: and

text: this

text: is

text: because

text: both

text: A

text: and,

text: respectively,

text: a

text: are

text: L,f

text: f

text: increasing

text: functions.

text: For

text: Part

text: we

text: have

text: s

text: s

text: h(s

text: (s))

text: f(s

text: (s)).

text: s

text: It

text: is

text: easy

text: to

text: verify

text: that

text: the

text: fraction

text: in

text: front

text: of

text: f

text: is

text: increasing

text: in

text: s,

text: as

text: is

text: the

text: function

text: s

text: on

text: the

text: other

text: hand,

text: f

text: is

text: increasing

text: by

text: assumption.

text: Therefore,

text: we

text: obtain

text: the

text: three

text: statements

text: of

text: Theorem

text: but

text: with

text: instead

text: of

text: in

text: Parts

text: and

text: by

text: setting

text: s

text: Q

text: in

text: the

text: three

text: respective

text: statements

text: of

text: Theorem

text: and

text: changing

text: the

text: variable

text: in

text: the

text: integrals.

text: To

text: prove

text: that

text: the

text: ﬁrst

text: and

text: third

text: statements

text: hold

text: with

text: equality,

text: note

text: that

text: by

text: the

text: deﬁnition

text: of

text: B

text: for

text: any

text: Q

text: any

text: x

text: and

text: any

text: interval

text: I,

text: there

text: exists

text: a

text: sequence

text: of

text: Ad(I)-

text: Q,Φ{

text: weights

text: w

text: such

text: that

text: for

text: each

text: n

text: [w

text: Q,

text: w

text: x

text: w

text: w

text: Q,

text: and

text: n

text: n

text: n

text: I

text: n

text: I

text: n

text: I

text: lim

text: cΦ(w

text: B

text: (x

text: Q/x

text: n→∞

text: I

text: ∈D

text: J

text: n

text: Q,Φ

text: J

text: (I)

text: R

text: Let

text: us

text: extend

text: each

text: w

text: to

text: all

text: of

text: periodically

text: by

text: replicating

text: it

text: on

text: each

text: dyadic

text: interval

text: of

text: n

text: length

text: I

text: keep

text: the

text: name

text: w

text: for

text: the

text: extension.

text: Clearly,

text: [w

text: [w

text: which

text: means

text: that

text: n

text: n

text: Ad

text: n

text: Ad(I)

text: the

text: left-hand

text: side

text: does

text: not

text: exceed

text: K

text: (Q).

text: On

text: the

text: other

text: hand,

text: by

text: Theorem

text: the

text: right-hand

text: A

text: Φ

text: side

text: equals

text: Q)

text: in

text: Part

text: and

text: a

text: Q)

text: in

text: Part

text: The

text: proof

text: is

text: complete.

text: L,f

text: f

text: Proof

text: Theorem

text: We

text: will

text: assume

text: the

text: statements

text: of

text: Theorem

text: Lemma

text: and

text: Re-

text: mark

text: The

text: last

text: two

text: come

text: from

text: Section

text: which

text: itself

text: is

text: self-contained.

text: Proof

text: of

text: Part

text: Take

text: any

text: Q

text: and

text: any

text: w

text: Ad

text: such

text: that

text: [w]

text: Q.

text: There

text: exists

text: a

text: sequence

text: of

text: intervals

text: I

text: such

text: that

text: w

text: w

text: Q

text: as

text: n

text: By

text: the

text: deﬁnition

text: of

text: b

text: n

text: In

text: In

text: Q,Φ

text: cΦ(w)

text: b

text: w

text: w

text: a

text: w

text: w

text: In

text: ∈D

text: J

text: Q,Φ

text: In

text: In

text: f

text: In

text: In

text: J

text: (In)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: V mc/V V mc/V )zH( htdiw 6 (a) (a) (b) 12 000 loading trap precision trap 10 000 moderator 8 000 e - 6 000 FEP particles 4 000 10 2 000 e+ e+ (b) potential on axis “unblocked” 0 e - 5 0 2 4 6 0 2 4 6 hours hours 0 Figure 9. Electron (dashed lines) and positron (solid lines) -5 loading rates for the two electrode potential conﬁgurations -10 shown in ﬁgure 8. Figure (a) shows the conﬁguration where (c) electric field on axis secondary electrons are unblocked and (b) shows the conﬁg- 20 uration where secondary electrons are blocked. When sec- 10 ondary electrons are prevented from entering the trapping re- 0 gion, (b), the large asymmetry in electron and positron load- -10 ing rates, (a), is greatly reduced. -20 10 0 2 4 6 8 (d) potential on axis “blocked” centimeters 5 strikes the surface of the moderator, and the formation 0 of positronium at the moderator surface depends upon a layer of adsorbed gas on the moderator surface. This -5 phenomenon was observed previously when antiprotons -10 struck such a moderator22 and when a laser was used (e) 20 to deliberately heat the moderator23. Thermal cycling 10 of the apparatus restores the adsorbed gas layer and re- 0 stores the loading rate. -10 -20 Our maximum positron loading rate manifests itself as a 687 (10) Hz/hr increase in the measured resonance 0 2 4 6 8 width, corresponding to about 70-140 e+/hr being accu- cm mulated in the trap. For our 6.3 µCi 22Na source, this gives a loading rate of 3-6 e+/s/mCi, in good agreement Figure 8. The voltage and electric ﬁeld proﬁles are shown for with previous work13. the two main loading conﬁgurations used. Solid lines denote the potential and ﬁeld used for positron loading and dashed VI. CONCLUSION lines denote the potential and ﬁeld used for electron loading. Figures (b) and (c) show the conﬁguration where secondary electrons are unblocked (not prevented from entering the trap- Field ionization of slow positronium accumulates the ping region) while (d) and (e) show the conﬁguration where positrons needed to measure the positron magnetic mo- secondary electrons are blocked (prevented from entering the trapping region). A cross section of the electrode stack is ment at the 3 parts in 1013 precision to which we have shown (top). measured the electron magnetic moment. Compared to the positron accumulation used for previous magnetic moment measurements, an orders of magnitude smaller source accumulates trapped positrons at a rate per source loading rates as seen in Fig. 9. This is as expected for activity that is orders of magnitude larger. The tiny 6.3 the loading mechanism discussed. µCi source that provides positrons to a 100 mK Penning Additional factors also aﬀect the loading rate as ex- trap is small enough to avoid special licensing and han- pected. For example, moving the source 3.8 cm (1.5 in) dling methods, provides a dose rate to an unshielded user further from the moderator and loading trap cuts the that is much lower than the average natural background, loading rate of both electrons and positrons in half. Ad- and can be withdrawn from the trap during precise mea- ditionally, ﬁring a ﬁeld emission point (or FEP, located surements. at the bottom of the precision trap and used for quickly loading electrons into either trap) 20 times at approxi- ACKNOWLEDGMENTS mately 1 nA for approximately 1 minute each time de- creased the loading rate by roughly a factor of two. This is because the electron beam from the ﬁeld emission point This work was supported by the NSF. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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