Rodrigo Ba Purdue University Department of Mathematics West - - PowerPoint PPT Presentation

Stability inequalities for martingales and Riesz transforms∗

Rodrigo Ba˜ nuelos†

Purdue University Department of Mathematics West Lafayette, IN. 47906

May 19, 2017

∗With Adam Ose

¸kowski

†Partially supported by NSF

• R. Ba˜

nuelos (Purdue) May 19, 2017

Stability (quantitative/deficit) sharp inequalities

• R. Ba˜

nuelos (Purdue) May 19, 2017

Stability (quantitative/deficit) sharp inequalities

Optimal/sharp inequalities

Suppose you have two functionals E and F on some normed (real) linear space M satisfying the functional inequality E F in the sense that E(x) F(x), ∀x ∈ M. The functional inequality E F is sharp if for all λ < 1 there exist x ∈ M such that E(x) > λF(x) The subset M0 = {x ∈ M : E(x) = F(x)} is called the set of optimizers (extremals) of the inequality. When M0 = ∅, the inequality is said to be optimal. (Note: An optimal functional inequality is sharp but not vice–versa.)

• R. Ba˜

nuelos (Purdue) May 19, 2017

Definition

Let d be a metric on M (not necessarily the norm metric) and Φ a “rate function.” The optimal functional inequality E F is (d, Φ)– stable if F(x) − E(x) Φ(d(x, M0)), ∀x ∈ M In various examples, Φ(t) = ct2 and d(x, y) = x − yM and F(x) − E(x) c inf

z∈M0 x − z2 M.

• R. Ba˜

nuelos (Purdue) May 19, 2017

Definition

Let d be a metric on M (not necessarily the norm metric) and Φ a “rate function.” The optimal functional inequality E F is (d, Φ)– stable if F(x) − E(x) Φ(d(x, M0)), ∀x ∈ M In various examples, Φ(t) = ct2 and d(x, y) = x − yM and F(x) − E(x) c inf

z∈M0 x − z2 M.

Classical Sobolev in Rn (n 3). k2

n = n(n−2) 4

|Sn−1| k2

nf2

2n n−2 ∇f2

2,

∀f ∈ H1

0(Rn) = M,

M0 = {x → c(a + b|x − x0|2)−(n−2)/2, a, b > 0, x0 ∈ Rn, c ∈ R} Optimality: Aubin (1976), Talenti (1976). Stability: Biachi-Egnell (1990) ∇f2

2 − k2 nf2

2n n−2 C inf

g∈M0 ∇(f − g)2 2

• R. Ba˜

nuelos (Purdue) May 19, 2017

More general Sobolev (0 < α < n/2). f

2n n−2α kn,α(−∆)α/2f2

Optimality: Lieb (1983), Stability: Cheng-Frank-Weth (2013) Hardy-Littlewood-Sobolev Optimality: Lieb (1983). Stability: Carlen (2016), Log-Sobolev Gross (1975), Stability Fathi-Indrei-Ledoux (2015), Nash Optimality: Carlen-Loss (1993). Stability: Carlen-Lieb (2017) Housdorff-Young inequality: Sharpness Beckner 1975 (Lieb 1990) Stability: Chris 2015. 1 p 2, q =

p p−1

ˆ fq (Ap)nfp Ap = p1/2pq−1/2q Ap is best contacts. Extremizers are general Gaussians: g(x) = ceQ(x)+x·v.

• R. Ba˜

nuelos (Purdue) May 19, 2017

More general Sobolev (0 < α < n/2). f

2n n−2α kn,α(−∆)α/2f2

Optimality: Lieb (1983), Stability: Cheng-Frank-Weth (2013) Hardy-Littlewood-Sobolev Optimality: Lieb (1983). Stability: Carlen (2016), Log-Sobolev Gross (1975), Stability Fathi-Indrei-Ledoux (2015), Nash Optimality: Carlen-Loss (1993). Stability: Carlen-Lieb (2017) Housdorff-Young inequality: Sharpness Beckner 1975 (Lieb 1990) Stability: Chris 2015. 1 p 2, q =

p p−1

ˆ fq (Ap)nfp Ap = p1/2pq−1/2q Ap is best contacts. Extremizers are general Gaussians: g(x) = ceQ(x)+x·v. Brasco &De Philippis (2016). Torsional rigidity for Brownian motion.

• D∗

Ez(τD∗)dz −

• D

Ez(τD)dz CnA(D)2 (Fraenkel Asymmetry) A(D) := inf{|D△B| |D| : B is a ball with |B| = |D|}. Problem: Prove “it” for stable processes (any subordination of BM).

• R. Ba˜

nuelos (Purdue) May 19, 2017

Martingales: Sharp (but not optimal, i.e., M0 = ∅)) inequalities. (Reference: A. Ose ¸kowski, “Sharp martingale and semimartingale inequalities”, Monografie Matematyczne 72, Birkh¨ auser, 2012.)

Doob

{fn} an Lp, 1 < p ∞ martingale. f ∗ = supn |fn| maximal function. f ∗p p p − 1fp Burkholder (1984), Wang (1991 for dyadic): Inequality is sharp. But M0 = ∅.

Burkholder (1966) S(f) = (

n(fn − fn−1)2)1/2

apfp S(f)p bpfp 1 < p < ∞ Many sharp versions of these exists but none are optimal i.e., M0 = ∅, outside of the trivial case of p = 2. (The first sharp case of these for Brownian martingales/stochastic integrals is due to B. Davis (1976).)

• R. Ba˜

nuelos (Purdue) May 19, 2017

X, Y c´ adl´ ag (right continuous/left limits) martingales: Y is differentially subordinate to X (Y << X), if the process {[X, X]t − [Y, Y ]t}t0 is nonnegative and nondecreasing in t. They are orthogonal (Y ⊥ X) if [X, Y ] = 0. 1 < p < ∞ and . p∗ = max{p, p/(p − 1)}. Set ||X||p = supt0 ||Xt||p

Burkholder (1984) Y << X

||Y ||p (p∗ − 1)||X||p. The constant (p∗ − 1) is best possible. Furthermore, the inequality is always strict unless p = 2. That is, inequality is sharp and M0 = ∅ unless p = 2.

R.B. G. Wang (1995) Y << X and Y ⊥ X

||Y ||p cot π 2p∗

• ||X||p.

The constant cot

• π

2p∗

• is the best possible. Furthermore, the inequality is always

strict unless p = 2 unless p = 2. That is, inequality is sharp and M0 = ∅ unless p = 2.

• R. Ba˜

nuelos (Purdue) May 19, 2017

A careful analysis of proofs reveals that “almost” extremals used to proof sharpness are “almost” eigenfunctions. The dyadic maximal function in Rn (dyadic martingales). Md(f)(x) = sup{ 1 |Q|

• Q

|f(y)|dy : x ∈ Q, Q ∈ [0, 1]n, dyadic cube} Doob (for inequality) Wang (1995) for sharpness: Md(f)p

p p−1fp,

1 < p ∞. (Here we may restrict to non-negative functions.)

• A. Melas (2015): If you take a sequence {fn} of almost externals then

limn Md(fn) −

p p−1fnp = 0

Theorem (Melas 2015)

Fix 2 < p < ∞, ǫ > 0 (small enough). Suppose f 0 (in Lp) is such that Md(f)p

• p

p − 1 − ε

• fp.

Then Md(f) − p p − 1fp cpε1/pfp for some constant cp depending only on p.

• R. Ba˜

nuelos (Purdue) May 19, 2017

Theorem (A.Ose ¸kowski & R.B. 2016: Y << X)

(i) Let 1 < p < 2 and ε > 0. ||Y ||p (

1 p−1 − ε)||X||p. Then

• |Y∞| −

1 (p − 1)|X∞|

• p cpε1/2||X||p.

O(ε1/2) as ε → 0 is sharp. cp = O((2 − p)−1/2) as p ↑ 2 and this is sharp. (ii) Let 2 < p < ∞ and ε > 0. ||Y ||p (p − 1 − ε)||X||p.

• |Y∞| − (p − 1)|X∞|
• p cpε1/p||X||p,

O(ε1/p) as ε → 0 is sharp. cp is O((p − 2)−1/p) as p ↓ 2 and O(p) as p → ∞. These orders are sharp. (iii) For p = 2, no c2 and κ exist such that ||Y ||2 (1 − ε)||X||2 implies

• |Y∞| − |X∞|
• 2 c2εκ||X||2. In fact, there exist martingales Y and X,

Y << X, such that Y 2 = X2, and |Y | − |X|2 X2 > 0 (independent of ε)

• R. Ba˜

nuelos (Purdue) May 19, 2017

Theorem (A.Ose ¸kowski & R.B. 2016: Y << X and Y ⊥ X)

(i)

• |Y∞| − tan

π 2p

• |X∞|
• p

cpε1/2||X||p, 1 < p < 2, if X and Y are such that ||Y ||p

• tan

π 2p

• − ε
• ||X||p.

Orders in ε as ε → 0, and cp, as p ↑ 2, are best possible. (ii)

• |Y∞| − cot

π 2p

• |X∞|
• p

cpε1/p||X||p, 2 < p < ∞, if ||Y ||p

• cot π

2p − ε

• ||X||p

(iii) As in previous theorem, no such estimate exists for p = 2.

• R. Ba˜

nuelos (Purdue) May 19, 2017

Beurling-Ahlfors operator in complex plane C Bf(z) = − 1 π p.v.

• C

f(w) (z − w)2 dw. It is a Calderón-Zygmund singular integral and Bfp Cpfp, 1 < p < ∞.

Conjecture T. Iwaniec 1984

The operator norm of B on Lp is p∗ − 1: Bp→p = (p∗ − 1), 1 < p < ∞

Known:

(p∗ − 1) Bp→p 1.575(p∗ − 1) Lower bound O. Lehto (1965) upper bound R.B and P. Janakiraman (2008). Lehto’s functions used to prove the lower bound have the property that |Bf(z)| ≈ (p∗ − 1)|f(z)|. That is, they are “near eigenfunctions.”

• R. Ba˜

nuelos (Purdue) May 19, 2017

• Bf(ξ)

= ξ ξ

• f(ξ) = ξ

2

|ξ|2 ˆ f(ξ) = ξ2

1 − 2iξ1ξ2 − ξ2 2

|ξ|2 ˆ f(ξ) ⇒ B = R2

1 − R2 2 + 2iR1R2 = Re(B) + i Im(B)

where R1 and R2 are the Riesz transforms in R2.

1

• R. B. Wang (1995): Both Re(B) and Im(B) have norm (p∗ − 1) (Proving

Bp 4(p∗ − 1))

2

Nazarov and Volberg (2004): R2

j − R2 kp→p (p∗ − 1)

2RjRkp→p (p∗ − 1) (Proving Bp,p 2(p∗ − 1))

3

Geiss, Montgomery-Smith and Saksman (2009): R2

j − R2 kp→p = (p∗ − 1),

2RjRkp→p = (p∗ − 1), j = k

• R. Ba˜

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Theorem (A.Ose

¸kowski & R.B. 2016) T either Re(B) or Im(B) or more generally, R2

j − R2 k or 2RjRk, j = k in Rn.

(i) Let 1 < p < 2, ε > 0. If f ∈ Lp(Rn) is such that ||Tf||p ((p − 1)−1 − ε)||f||p, then |Tf| − (p − 1)−1|f|p cpε1/2||f||p. The order O(ε1/2), as ε → 0, is best possible as is the order O((2 − p)−1/2), as p ↑ 2 for cp. (cp is explicit and independent of dimension) (ii) Let 2 < p < ∞, ε > 0. If f ∈ Lp(Rn) is such that ||Tf||p (p − 1 − ε)||f||p, then |Tf| − (p − 1)|f|p cpε1/pfp, (iii) For p = 2, there is no stability result of the above type. That is, there are no finite constants c2 and κ > 0 such that |Tf| − |f|p c2εκ||f||L2(Rd)

• R. Ba˜

nuelos (Purdue) May 19, 2017

First order Riesz transforms Calderón-Zygmund singular integrals and also Fourier multipliers with

• Rjf(ξ) = −iξj

|ξ| , j = 1, 2, . . . , n. For n = 1 this is just the Hilbert transform. n = 1: Pichorides (1972). n > 1: Iwaniec-Martin (1995) ||Rjf||Lp(p cot π 2p∗ ||f||p, j = 1, 2, . . . , n and this is sharp. R.B. Wang (1995) obtained it from Orthogonal martingales.

Theorem (A.Ose

¸kowski & R.B. 2016) Same type of stability results hold for Rj. (i) Let 1 < p < 2, ε > 0. ||Rjf||p

• tan

π 2p

• − ε
• ||f||p ⇒ |Rjf| − tan

π 2p

• |f|p cpε1/2||f||p,
• R. Ba˜

nuelos (Purdue) May 19, 2017

Burkholder’s inequality fn =

n

• k=1

dk, g =

n

• k=1

ek, |ek| |dk|, a.s. ∀k Burkholder’s inequality: considers the function V : R × R → R defined by Vp(x, y) = |y|p − (p∗ − 1)p|x|p. The goal is then to show that EV (fn, gn) 0. Burkholder then “introduces” the function Up(x, y) = βp (|y| − (p∗ − 1)|x|) (|x| + |y|)p−1 , where βp = p

• 1 − 1

p∗ p−1 and proves that this function satisfies the following properties: Vp(x, y) U(x, y) for all x, y ∈ R and EUp(fn, gn) EU(fn−1, gn−1) · · · EU(f0, g0) = 0

• R. Ba˜

nuelos (Purdue) May 19, 2017

Lemma (G. Wang (1995))

Let U : R × R → R which is ”smooth” (outside the origin with certain bounds, . . . ) and satisfying (for all h, k ∈ R) Uxx(x, y)|h|2 + 2Uxy(x, y)hk + Uyy(x, y)|k|2 −ci(x, y)(|h|2 − |k|2). Then if X, Y are two martingales such that Y << X, there is a nondecreasing sequence (τn)n1 of stopping times converging to infinity such that EU(Xτn∧t, Yτn∧t) EU(X0, Y0), n = 1, 2, . . . . Example satisfying Wang’s Lemma is Burkholder’s function: Up(x, y) = p

• 1 − 1

p p−1 ((p − 1)|y| − |x|)(|x| + |y|)p−1, 1 < p < 2

Lemma

The function also has (1 < p < 2) Up(x, y) (p − 1)p|y|p − |x|p +

• 1 − p
• 1 − 1

p p−1 ((p − 1)|y| − |x|)2 (|x| + |y|)2−p

• R. Ba˜

nuelos (Purdue) May 19, 2017

• 1 − p
• 1 − 1

p p−1 E((p − 1)|Y∞| − |X∞|)2 (|X∞| + |Y∞|)2−p ||X||p

p − (p − 1)p||Y ||p p

(1 − (1 − (p − 1)ε)p) ||X||p

p

p(p − 1)ε||X||p

p.

This, H¨

• lder inequality, and Burkholder’s estimate yield

||(p − 1)|Y∞| − |X∞|||p

• E

((p − 1)|Y∞| − |X∞|)2 (|X∞| + |Y∞|)2−p 1/2 |||X∞| + |Y∞|||

(2−p) 2

p

• 

  p(p − 1)ε 1 − p

• 1 − 1

p

p−1   

1/2

||X||p/2

p

·

• p

p − 1||X||p (2−p)

2

.

• R. Ba˜

nuelos (Purdue) May 19, 2017

2 < p < ∞ we consider Up(x, y) =        p

• 1 − 1

p p−1 (|y| − (p − 1)|x|)(|x| + |y|)p−1 if |y| (p − 2)|x|, −(p − 1)2p−2 pp−2 |x|p if |y| < (p − 2)|x|.

Lemma (Up satisfies Wang’s Lemma and)

Up(x, y) |y|p − (p − 1)p|x|p + αp

• |y| − (p − 1)|x|
• p,

αp = p − 2 p − 1 1 2 − 1 e

• .

αp

• |Y∞| − (p − 1)|X∞|
• p

p (p − 1)p||X||p p − ||Y ||p p

• (p − 1)p − (p − 1 − ε)p

||X||p

p

p(p − 1)p−1ε||X||p

p.

• R. Ba˜

nuelos (Purdue) May 19, 2017

Thank You!

• R. Ba˜

nuelos (Purdue) May 19, 2017