rodrigo banuelos~1auffing/banuelos.pdf · not completely crazy martingale techniques have been...

Hardy-Littlewood-Sobolev/Burkholder-Gundy, Doob

Rodrigo Banuelos1

Department of MathematicsPurdue University

May 6, 2016

1Supported in part by NSFR. Banuelos (Purdue) HLS/BG-D May 6, 2016

f ∈ C∞0 (Rd).

‖f ‖2d/(d−2) ≤ C‖∇f ‖2, d ≥ 3. (classical Sobolev)

More general: d ≥ 2, 1 ≤ p < d ,

‖f ‖pd/(d−p) ≤ C‖∇f ‖p, (also classical)

Best constants found Aubin and Talenti in mid 70’s. Many other proofs over theyears

Without care for best constants, general follows from “isoperimetric” case p = 1.

‖f ‖d/(d−1) ≤ C‖∇f ‖1

applied to f β for β = (d−1)pd−p . Set q = p/(p − 1).

‖f β‖d/(d−1) ≤ βC∫Rd

|f (x)|β−1|∇f (x)| dx ≤ βC‖f β−1‖q‖∇f ‖p,

R. Banuelos (Purdue) HLS/BG-D May 6, 2016

Sobolev, log-sobolev and heat kernel decay (d ≥ 3)

‖f ‖2d/(d−2) ≤ C‖∇f ‖2 =[C

(∫Rd

−f ∆f

)1/2

= C√E(f )

](S)

∫Rd

f 2 log f dx ≤ ε∫Rd

|∇f |2 dx + β(ε)‖f ‖22 + ‖f ‖22 log ‖f ‖2, f > 0, (Log-S)

β(ε) = C + d4 log 1

ε .

pt(x , y) =1

(4πt)d/2e−|x−y|2

4t ≤ C

td/2(Heat(d))

S⇒ Log-S⇒ Heat(d)⇒ S

S⇒ Log-S: Take ‖f ‖2 = 1. p = 2dd−2 . Jensen gives∫

Rd

f 2 log f =1

p − 2

∫Rd

log(f p−2)f 2dx ≤ 1

p − 2log

∫Rd

f pdx =p

2(p − 2)log ‖f ‖2p

use log(a) ≤ εa + log 1ε and apply (S)


The Hardy-Littlewood-Sobolev (HLS) inequality–(HL-1932, S-1938)

Iα(f )(x) =Γ( d−α

2 )

2απd/2Γ(α2 )

∫Rd

f (y)

|x − y |d−αdy .

Let 0 < α < d , 1 < p < dα and set 1

q = 1p −

αd or q = pd

d−pα .

‖Iα(f )‖ pdd−pα

= ‖Iα(f )‖q ≤ Cp,d,α‖f ‖p. (HLS)

If d > 2, p = 2, α = 1,‖f ‖2d/d−2 ≤ cd‖∇f |‖2 (S)

The relation 1q = 1

p −αd comes from scaling.

Indeed: With τδf (x) = f (δx) one sees that τδ−1Iατδ = δ−αIα, δ > 0.

δ−α‖Iαf ‖q = ‖τδ−1Iατδf ‖q = δd/q‖Iα(τδf )‖q≤ δd/qC‖τδf ‖p = Cδd/qδ−d/p‖f ‖p

⇒ α + d/q − d/p = 0


Iα(f )(x) =Γ( d−α

2 )

2απd/2Γ(α2 )

∫Rd

f (y)

|x − y |d−αdy .

Our Aim: Write Iα(f ) as a martingale transform and obtain (HLS) from theBurkholder-Gundy and Doob inequalities.

Why in the world would we want to do this?

Motivation: Find a new proof of Eliot Lieb’s (1983) sharp (HLS) on Rd andextend it to other geometric settings?

Not completely crazy

Martingale techniques have been extremely successful in obtaining sharpinequalities for various singular integral operators and Fourier multipliers,including Riesz transforms, the Beurling-Ahlfors operator on Rd , Lie groups andmanifolds under curvature assumptions. Large literature on this subject.


Burkholder 1984

Bt d–dimensional BM. Set

Xt =

∫ t

0

Hs · dBs

and Ms an d × d predictable matrix with

sups>0‖M(s, ω)‖∞ ≤ 1.

Set

M ∗ Xt =

∫ t

0

MsHs · dBs .

For any 1 < p <∞,‖M ∗ X‖p ≤ (p∗ − 1)‖X‖p,

p∗ = max{p, p

p − 1}.

The inequality is sharp but never attained unless p = 2.


Theorem (E. Lieb: Sharp constants on Hardy-Littlewood-Sobolev and relatedinequalities (1983))

There is “an” extremal function h for HLS for all p, q as above.

When p = 2, or q = 2 or p = qq−1 (conjugate exponent of q) he identifies

the extremal explicitly and computes the constant explicitly. That is hecomputes the constant explicitly in three cases.

(i) ‖Iα(f )‖2d/d−2α ≤ B1‖f ‖2. (α = 1, Aubin (1976), Talenti (1976). S. G.Bobkov & M. Ledoux (2010) from Brunn–Minkowski.)

(ii) ‖Iα(f )‖2 ≤ B2‖f ‖2d/d+2α

(iii) ‖Iα(f )‖2d/d−α ≤ B3‖f ‖2d/d+α

For the above cases the, extremal is (up to multiplication by constants anddilations) unique and has form

h(x) = (1 + |x |2)−µ

Very large literature on sharp Sobolev!

R. Frank and E. Lieb–2010, 2012: Sharp version for Heisenberg group(rearrangement free)


Proof of HLS: L. Hedberg (1972).

Fix δ > 0.∫{|x−y |≤δ}

|f (y)||x − y |d−α

dy =∞∑k=0

∫{δ2−k−1<|x−y |≤δ2−k}

|f (y)||x − y |d−α

dy

≤ C∞∑k=0

(δ2−k)α1

(δ2−k)d

∫{|x−y |≤δ2−k}

|f (y)|dy

≤ ≤ CδαM(f )(x)∞∑k=0

2−kα = CαδαMf (x),

Mf Hardy-Littlewood maximal function: Mf (x) = supr>01

|B(x,r)|∫B(x,r)

|f (y)|dy .

Apply Holder (p > 1)∫{|x−y |>δ}

|f (y)||x − y |d−α

dy ≤ Cd‖f ‖p(∫ ∞

δ

rd−1

rp′(d−α)

)1/p′

≤ C‖f ‖pδα−d/p

|Iα(f )(x)| ≤ Cα,d [δαMf (x) + ‖f ‖pδα−d/p] (a Peter-Paul inequality)

Minimizing in δ,|Iα(f )(x)| ≤ CMf (x)1−pα/d‖f ‖pα/dp


1

q=

1

p− α

d⇒ p

q= 1− pα

d, &

pα

d= 1− p

q=

q − p

q

⇒ |Iα(f )(x)| ≤ CMf (x)1−pα/d‖f ‖pα/dp = CMf (x)p/q‖f ‖q−pq

p

⇒∫Rd

|Iα(f )(x)|qdx ≤ C

(∫Rd

Mf (x)pdx

)‖f ‖(q−p)p

≤ C(‖f ‖pp

)‖f ‖(q−p)p = ‖f ‖qp.

Used fact:‖Mf ‖p ≤ C‖f ‖p, 1 < p <∞.

Remark

The “classical” proof (E. Stein “Singular Integrals . . . ” page 120) uses the“off diagonal” version of the Marcinkiewicz interpolation and not theHardy-Littlewood Max Function M.


Heat semigroup on Rd

Tt f (x) =

∫Rd

pt(x − y)f (y)dy , pt(x) =1

(4πt)d/2e−|x|

2/4t

By Fubini and doing the integral in t, one finds that

Iα(f )(x) =1

Γ(α/2)

∫ ∞0

tα/2−1Tt f (x) dt,

Taking Fourier transform:

Iα(f )(ξ) =

(1

Γ(α/2)

∫ ∞0

tα/2−1e−t|ξ|2

dt

)f (ξ)

=

(1

Γ(α/2)(|ξ|2)−α/2

∫ ∞0

tα/2−1e−t dt

)f (ξ)

= (|ξ|2)−α/2 f (ξ).

Thus (Riesz potentials)

Iα(f )(x) = (−∆)−α/2f (x).


Properties of Semigroup

(i) Semigroup is symmetric Markovian, contraction on all Lp, 1 ≤ p ≤ ∞.

(ii) Semigroup is ultracontractive: ‖Tt f ‖∞ ≤ Ctd/2‖f ‖1 ∀t > 0. In fact, Jensen’s

shows for any 1 ≤ p <∞,

‖Tt f ‖∞ ≤C

td/2p‖f ‖p, ∀t > 0

(ii) If f ∗(x) = supt>0 |Tt f (x)|, then ‖f ∗‖p ≤ cp‖f ‖p, 1 < p ≤ ∞.Follows from: pt is an approximation to the identity, radial, decreasing.

Theorem (E. Stein 1961 “On the maximal ergodic theorem.”)

‖f ∗‖p ≤ cp‖f ‖p, 1 < p <∞ holds for any symmetric Markovian semigroup.

Remark (Carl Herz )

Stein follows from Doob and holds with cp = pp−1 . (I learned the proof from

Doob and Burkholder about 30 years ago. Other proofs.)


Hedberg, 2nd time

|Iα(f )(x)| ≤ C1

(∫ δ

0

tα/2−1 dt

)f ∗(x)

+‖f ‖p

Γ(α/2)(4π)d/(2p)

∫ ∞δ

t [α2 −1]−

d2p dt

≤ Cd,α

(δα/2f ∗(x) + δ[

α2 −

d2p ]‖f ‖p

).

(Used the fact that α− dp = − d

q < 0.) Minimize by picking

δ =

(‖f ‖pf ∗

)2p/d

|Iα(f )(x)| ≤ Cd,α (f ∗)1−αp/d ‖f ‖αp/dp ≤ Cd,α (f ∗)p/q ‖f ‖αp/dp .

and as before‖Iα(f )‖q ≤ C‖f ‖p ⇔

‖(−∆)−α/2f (x)‖q ≤ C‖f ‖p.


Hedberg 3rd time: “Hedberg’s 1972 proof of Varopoulos 1985 theorem”

Same as ”Hedberg 2nd time” gives: (S ,S, µ) measure space, Tt a symmetricMarkovian semigroup with dimension n in the sense of Varopoulos, i.e., Tt is

Tt f (x) =

∫S

pt(x , y)f (y)µ(dy),

there exists C > 0 so that for all t > 0, x , y ∈ S , pt(x , y) ≤ Ct−n2 . (n does not

have to be an integer, many examples exist.) Define

Iα(f )(x) =1

Γ(α/2)

∫ ∞0

tα/2−1Tt f (x) dt,

Let −A be the (self-adjoint) infinitesimal generator of (Tt , t ≥ 0).

Iα(f )(x) = A−α2 f ,

(f ∈ Dom(A−

α2 ) ∩ L1(S)

)Theorem (N. Varopoulos 1985)

Let 0 < α < n, 1 < p < nα and set 1

q = 1p −

αn .

||Iα(f )||q ≤ Cp,n,α||f ||p.


Can keep track of constants:

‖A−α2 f ‖ npn−pα

≤(

p

p − 1

)1−α/n2nCα/n

α(n − pα)Γ(α/2)‖f ‖p.

Sobolev for any semigroup of dimesnion n > 2.

With n > 2, p = 2, α = 1,

‖f ‖ 2nn−2≤ 21−1/n 2nC 1/n

(n − 2)√π

√E(f ), (E Dirchlet form of semigroup) (*)

where C constant in pt(x , y) ≤ Ct−n/2.

For any semigroup of dimension n > 2, we have (as on Rd)

S⇒ Log-S⇒ Heat(n)⇒ S


Martingale representation–back to heat on Rd .

Let Bt Brownian motion in Rd starting at z , Pz and Ez be the probability and Ethe expectation of the BM stating with Lebesgue measure. ThenFix 0 < a <∞. Consider the martingale

M(f )t = Ta−t f (Bt), 0 ≤ t ≤ a,

Ito:⇒ M(f )t = Taf (z) +

∫ t

0

∇xTa−s f (Bs) · dBs .

Martingale Transform:

∫ a

0

(a− s)α/2∇xTa−s f (Bs) · dBs .

E|∫ a

0

(a− s)α/2∇xTa−s f (Bs) · dBs |p

=

∫Rd

Ez |∫ a

0

(a− s)α/2∇xTa−s f (Bs) · dBs |pdz

≤ cpapα

∫Rd

Ez |f (Ba)|pdz = cpapα

∫Rd

(∫Rd

pa(x − z)|f (x)|pdx)dz

= cpapα

∫Rd

|f (x)|pdx <∞


Fractional Integral as projection of martingale transforms

Theorem (D. Applebaum, R.B–2014)

For f ∈ C∞0 set

Sa,αf (x) = E(∫ a

0

(a− s)α/2∇(Ta−s f )(Bs) · dBs | Ba = x

).

Then

Sa,αf (x) = −∫ a

0

sα/2Ts(∆Ts f )(x)ds

=

(−1

2aα/2T2af (x) +

α

4

∫ a

0

sα/2−1T2s f (x)ds

)and as a→∞,

lima→∞

Sa,αf (x)→ Γ(α/2)

2Iα(f )(x), a.e.


Burkholder-Gundy: there is a Cp independent of a such that

‖Sa,αf (x)‖q ≤∥∥∥∫ a

0

(a− s)α/2∇(Ta−s f )(Bs) · dBs

∥∥∥q

≤ Cq

∥∥∥(∫ a

0

(a− s)α|∇(Ta−s f )(Bs)|2 ds)1/2 ∥∥∥

q

Lemma

(∫ a

0

(a− s)α|∇(Ta−s f )(Bs)|2 ds)1/2

≤ Cp,α,d

(sup

0<s<2a|(T(2a−s)|f |)(Bs)|

)p/q

||f ||q−pq

p

Raise to q, take E expectation and apply Doob to first term.

A “Hedberg 1972 proof of Applebaum-Banuelos 2014 Theorem”


Several weakness in above martingale proof (outside of best constant). Does notwork for general symmetric Markovian semigroups

(i) Proof is “very” Rd : Uses the gradient.

(ii) Proof uses a simple estimate on gradient on Rd :

|∇xTt f (x)| ≤ Cd√tT2t |f |(x)

which follows from the fact that for pt(x) Gausian on Rd ,

|∇xpt(x)| ≤ 2d+42

1√tp2t(x).

Proof can be extended to manifolds of non-negative Ricci curvature . . .


Different martingale representation via Poisson (Cauchy) semigroup on Rd

Pt f (x) = Cn

∫Rd

f (y)

(t2 + |x − y |2)n+1/2dy =

∫ ∞0

Ts f (x)ηt(ds),

ηt(ds) = t2√πe−t

2/4ss−3/2ds. (“half-stable subordination”)

Theorem (R.B. many years ago)

T a,αf (x) = Ea[

∫ τ

0

Y αs

∂uf∂y

(Zs)dYs |Xτ = x ],

uf (x , y) = Py f (x) the harmonic extension of f .

lima→∞

T a,αf =Γ(α + 2)

2α+2Iα(f )

Zs = (Xs ,Ys), Xs B.M. on Rd , Ys B.M. on R+ = (0,∞). τ= first time Ys = 0

Theorem (Daesung Kim (2015))

Formula extend to symmetric Markovian semigroups and most importantly, it canbe used to prove HLS. (Proof: Based on ”Littlewood-Paley” theory. )


Given T : Lp(µ)→ Lq(µ), with ‖Tf ‖q ≤ A‖f ‖p,

λqµ{|Tf | > λ} ≤∫|Tf |qdµ ≤ Aq‖f ‖qp

‖|Tf |‖weak(q) = supλ>0

λ (µ{|Tf | > λ})1q ≤ A‖f ‖p

For cube (0, 1]d ⊂ Rd , define.

||f ||weak(q) = sup

{1

|E |1−1/q

∫E

|f |dx : E ⊂ (0, 1]d , |E | > 0

}.

Theorem (R.B., A. Osekowski-2015: version Iα on dyadic martingales on Rd)

‖Iαf ‖weak(q) ≤ K (α, d , p)‖f ‖p, 1/q = 1/p − α/d

K (α, d , p) =2d−α − 2−α

2d−α − 1

(1 +

(1− 2−α)p′

(1− 2−d)p′−1(2(d−pα)(p′−1) − 1)

)1/p′

p′ conjugate exponent of p. The inequality is sharp.

Proof via Bellman functions. Not a ”Hedberg 1972 proof ofBanuelos-Osekowski 2015 theorem”


Thank you!


rodrigo banuelos~1auffing/banuelos.pdf · not completely crazy martingale techniques have been...

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