H p boundedness of Calder on-Zygmund operators on product domains - - PowerPoint PPT Presentation

Hp boundedness of Calder´

• n-Zygmund operators
• n product domains

Ming-Yi Lee

National Central University, TAIWAN

Symposium on Probability and Analysis 2010 Institute of Mathematics, Academia Sinica, Taipei, Taiwan

• This talk is based on joint work with Yongsheng Han,

Chin-Cheng Lin, and Ying-Chieh Lin. All results presented here appeared in J. Funct. Anal. 258 (2010), 2834 – 2861.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Calder´

• n-Zygmund operators play a central role in modern

harmonic analysis. The simplest and most important examples are the Hilbert transform H Hf (x) := p.v.

• R

f (x) x − y dy and the Riesz transforms Rj, j = 1, 2, · · · , n, Rjf (x) := p.v.

• Rn

xj − yj |x − y|n+1 f (y)dy.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Calder´

• n-Zygmund operators for 1-parameter

A singular integral operator T is a continuous linear operator from D(Rn) into its dual associated to a kernel K(x, y), a continuous function defined on Rn × Rn \ {x = y}, satisfying

1 |K(x, y)| ≤ C|x − y|−n; 2 |K(x, y) − K(x, y′)| ≤ C |y−y′|ε

|x−y|n+ε

if |y − y′| ≤ |x−y|

2

;

3 |K(x, y) − K(x′, y)| ≤ C |x−x′|ε

|x−y|n+ε

if |x − x′| ≤ |x−y|

2

. The smallest constant C is denoted by |K|CZ. Moreover, the

• perator T can be represented by

Tf , g =

• Rn
• Rn K(x, y)f (y)g(x)dydx

for all f , g ∈ D(Rn) with supp(f ) ∩ supp(g) = ∅.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Calder´

• n-Zygmund operators for 1-parameter (cont.)

If a singular integral operator T is bounded on L2(Rn), then we say that this T is a Calder´

• n-Zygmund operator and its norm is

defined by TCZ = TL2→L2 + |K|CZ. From Calder´

• n-Zygmund operator theory, every Calder´
• n-Zygmund
• perator T is bounded on Lp(Rn), p > 1, and bounded from

L∞(Rn) to BMO(Rn). However, T is not bounded on Lp(Rn) for p ≤ 1. Instead of Lp(Rn), one can consider the boundedness of T

• n Hardy spaces Hp(Rn) when

n n+ε < p ≤ 1.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Calder´

• n-Zygmund operators for 1-parameter (cont.)

More precisely, every Calder´

• n-Zygmund operator T is bounded on

Hp(Rn),

n n+ε < p ≤ 1 if and only if T ∗1 = 0. The range of p

depends on the cancellation of T and smooth condition of its associated kernel. For convolution operator T, K(x, y) = K(x − y), we do not need T ∗1 = 0 to get Hp-boundedness of T.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Lp boundedness on product domain Rn × Rm

In 1982, R. Fefferman and Stein [Adv. in Math.] investigated the Lp(Rn+m) boundedness of convolution operators. Theorem Suppose that K is integrable on Rn × Rm and satisfies certain “cancellation” and “size” properties. Then f ∗ KLp(Rn+m) ≤ Cpf Lp(Rn+m), 1 < p < ∞.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Journ´ e’s class

A singular integral operator T defined on Rn × Rm is said to be in Journ´ e’s class if Tf (x1, x2) =

• Rn×Rm K(x1, x2, y1, y2)f (y1, y2)dy1dy2,

where the kernel K(x1, x2, y1, y2) satisfies the following conditions. For each x1, y1 ∈ Rn, K 1(x1, y1) is a Calder´

• n-Zygmund operator
• n Rm with the kernel K 1(x1, y1)(x2, y2) = K(x1, x2, y1, y2).

Similarly, K 2(x2, y2)(x1, y1) = K(x1, x2, y1, y2). Moreover,

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Journ´ e’s class (cont.)

1 T is bounded on L2(Rn+m), 2 K 1(x1, y1)CZ ≤

C |x1−y1|n ,

K 1(x1, y1) − K 1(x1, y′

1)CZ ≤ C|y1−y′

1|ε

|x1−y1|n+ε

if |y1 − y′

1|≤ |x1−y1| 2

, K 1(x1, y1) − K 1(x′

1, y1)CZ ≤ C|x1−x′

1|ε

|x1−y1|n+ε

if |x1 − x′

1|≤ |x1−y1| 2

,

3 K 2(x2, y2)CZ ≤

C |x2−y2|m ,

K 2(x2, y2) − K 2(x2, y′

2)CZ ≤ C|y2−y′

2|ε

|x2−y2|m+ε

if |y2 − y′

2|≤ |x2−y2| 2

, K 2(x2, y2) − K 2(x′

2, y2)CZ ≤ C|x2−x′

2|ε

|x2−y2|m+ε

if |x2 − x′

2|≤ |x2−y2| 2

.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Journ´ e’s class (cont.)

K 1(x1, y1)CZ ≤

C |x1−y1|n means that

K 1(x1, y1) is boubded on L2(Rm) and its associated kernel K 1(x1, y1)(x2, y2) = K(x1, x2, y1, y2) satisfies |K 1(x1, y1)(x2, y2)| ≤ K 1(x1, y1)CZ

1 |x2−y2|m ,

|K 1(x1, y1)(x2, y2) − K 1(x1, y1)(x2, y′

2)|

≤ K 1(x1, y1)CZ

|y2−y′

2|ε

|x2−y2|m

if |y2 − y′

2|≤ |x2−y2| 2

, |K 1(x1, y1)(x2, y2) − K 1(x1, y1)(x′

2, y2)|

≤ K 1(x1, y1)CZ

|x2−x′

2|ε

|x2−y2|m

if |x2 − x′

2|≤ |x2−y2| 2

,

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Let C ∞

0,0(Rn) =

• ψ ∈ C ∞

c (Rn) :

• Rn ψ(y)yα dy = 0

for 0 ≤ |α| ≤ Np,n

• .

Let n1 = n, n2 = m, ψi ∈ C ∞

0,0(Rni) supported in the unit ball of

Rni, and ψi satisfy condition ∞

• ψi(tiξi)
• 2 dti

ti = 1 for all ξi = 0, i = 1, 2. For ti > 0 and (x1, x2) ∈ Rn × Rm, set ψi

ti(xi) = t−ni i

ψ(xi/ti) and ψt1t2(x1, x2) = ψ1

t1(x1)ψ2 t2(x2).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Product Hardy spaces Hp(Rn × Rm)

The product Littlewood-Paley g-function of f ∈ S ′(Rn+m) is defined by g(f )(x1, x2) = ∞ ∞

• ψt1,t2 ∗ f (x1, x2)
• 2 dt1

t1 dt2 t2 1/2 . For 0 < p ≤ 1, the product Hardy space is defined by Hp(Rn × Rm) =

• f ∈ S ′(Rn+m) : g(f ) ∈ Lp(Rn+m)
• with f Hp(Rn×Rm) := g(f )Lp(Rn+m).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

The multiparameter product Hardy spaces Hp(Rn × Rm) are much more complicated than the classical one-parameter Hardy spaces Hp(Rn). It was conjectured that Conjecture The product H1(R × R) could be characterized by rectangle atoms; that is, H1(R × R) =

• f : f =
• λkak, where ak are rectangle atoms

and

• |λk| < ∞
• .

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Here a rectangle atom is a function a(x, y) supported on a rectangle R = I × J satisfying a2 ≤ |R|1/2 and

• I

a(x, y)dx =

• J

a(x, y)dy = 0. Carleson disproved this conjecture by constructing a counter-example of a measure satisfying the product form of the Carleson measure which is bounded on

• f : f = λkak, ak rectangle atoms and

k |λk| < ∞

• ,

but is not bounded on H1(R × R).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Atomic decomposition for Hp(Rn × Rm)

In 1980, Chang and R. Fefferman [Ann. of Math.] proved f ∈ Hp(Rn × Rm) ⇐ ⇒ f =

• λkak,

where |λk|p < ∞ and ak(x, y) are atoms; that is,

1 ak is supported in an open set Ω in Rn × Rm with |Ω| < ∞; 2 ak2 ≤ |Ω|1/2−1/p; 3 ak(x, y) =

R∈M(Ω) aR(x, y), each “pre-atom” aR satisfies:

• supp(aR) ⊂ 4R, R := I × J (I a dyadic cube in Rn, J a dyadic

cube in Rm);

• M(Ω) is the collection of all maximal dyadic rectangles

contained in Ω, and

R∈M(Ω) aR2 2

1/2 ≤ |Ω|1/2−1/p;

• aR(x, y)xαdx =
• aR(x, y)y βdy = 0 for 0 ≤ |α| ≤ Np,n and

0 ≤ |β| ≤ Np,m.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Hp(Rn × Rm) − Lp(Rn+m) boundedness

• R. Fefferman [Proc. Natl. Acad. Sci. USA (1986)] proved the

following remarkable result. Theorem Let 0 < p ≤ 1 and T be a bounded linear operator on L2(Rn+m). Suppose that there exist constants C > 0 and δ > 0 such that, for any Hp(Rn × Rm) rectangle atom a supported on R,

• (γR)c |Ta(x, y)|pdxdy ≤ Cγ−δ

∀ γ ≥ 2. Then T extends to a bounded operator from Hp(Rn × Rm) to Lp(Rn+m).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Hp(Rn × Rm) − Lp(Rn+m) boundedness (cont.)

Here a rectangle atom a is a function supported on a rectangle R = I × J (I a cube in Rn, J a cube in Rm) such that

1 a2 ≤ |R|1/2−1/p; 2

I aR(x, y)xαdx =

• J aR(x, y)yβdy = 0 for 0 ≤ |α| ≤ Np,n

and 0 ≤ |β| ≤ Np,m.

• R. Fefferman further proved that product singular integrals in

Journ´ e’s class satisfy the above estimate and hence these operators are bounded from Hp(Rn × Rm) to Lp(Rn+m). The boundedness of these operators on Hp(Rn × Rm) is still open.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Recall the 1-parameter case. We say T ∗(1) = 0 if φ, T ∗(1) = T(φ), 1 =

• Rn
• Rn K(x, y)φ(y)dydx = 0.

For the 2-parameter case, we say that T ∗

1 (1) = 0 if

• Rn
• Rn×Rm K(x1, x2, y1, y2)φ(1)(y1)φ(2)(y2)dy1dy2dx1 = 0

for all φ(1) ∈ C ∞

0,0(Rn) and φ(2) ∈ C ∞ 0,0(Rm). Similarly, T ∗ 2 (1) = 0 if

• Rm
• Rn×Rm K(x1, x2, y1, y2)φ(1)(y1)φ(2)(y2)dy1dy2dx2 = 0.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Main result

Theorem 1 Let T be a singular integral operator in Journ´ e’s class with regularity exponent ε. Then T is bounded on Hp(Rn × Rm) for max{

n n+ε, m m+ε} < p ≤ 1 if and only if T ∗ 1 (1) = T ∗ 2 (1) = 0.

Remark Theorem 1 still holds for any given p ∈ (0, 1] if the kernel of T satisfies more regularity conditions and T satisfies high order cancellation conditions.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof for “only if ” part

By the maximal function characterization of Hp(Rn × Rm), if f (x1, x2) ∈ Hp(Rn × Rm), then

• f (·, x2) ∈ Hp(Rn)

for fixed x2 f (x1, ·) ∈ Hp(Rm) for fixed x1 . For ϕ1 ∈ C ∞

0,0(Rn) and ϕ2 ∈ C ∞ 0,0(Rm),

= ⇒ g(x1, x2) := ϕ1(x1)ϕ2(x2) is a multiple of an Hp(Rn × Rm) atom = ⇒ g(x1, x2) ∈ Hp(Rn × Rm) = ⇒ Tg(·, x2) ∈ Hp(Rn) (by Hp(Rn × Rm) boundedness of T) This implies

• K(x1, x2, y1, y2)ϕ1(y1)ϕ2(y2)dy1dy2dx1 =
• Tg(x1, x2)dx1 = 0,

which means T ∗

1 (1) = 0. Similarly, T ∗ 2 (1) = 0.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 1

Step 1. Reduce the Hp(Rn × Rm) boundedness of T to Hp(Rn × Rm) − Lp

H(Rn+m) boundedness of L:

Define the Hilbert space H by H =

• {ht,s}t,s>0 :
• {ht,s}
• H =

∞ ∞ |ht,s|2 dt t ds s 1/2 < ∞

• .

By the definition of Hp(Rn × Rm), Tf Hp = g(Tf )p =

• ∞

∞

• ψt1,t2∗Tf (x1, x2)
• 2 dt1

t1 dt2 t2 1/2

• p

. Set an H-valued operator L which maps f into {Tt,s(f )}t,s>0 by Tt,s(f ) = ψt,s ∗ Tf . Then Tf Hp =

• {Tt,s(f )}H
• p = Lf Lp

H. Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 1(cont.)

By the Calder´

• n identity, for f ∈ L2(Rn+m) ∩ Hp(Rn × Rm),

f (x1, x2) = ∞ ∞ ψt′,s′ ∗ ψt′,s′ ∗ f (x1, x2)dt′ t′ ds′ s′ in L2(Rn+m), where ψt1,t2(x1, x2) = t−n

1 t−m 2

ψ1(x1/t1)ψ2(x2/t2) and ψi ∈ C ∞

0,0

satisfies ∞

• ψi(tiξi)
• 2 dti

ti = 1 for all ξ = 0, i = 1, 2. Thus, Tt,s(f )(x1, x2) = ψt,s∗T ∞ ∞ ψt′,s′∗ψt′,s′∗f (·, ·)dt′ t′ ds′ s′

• (x1, x2).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 2

Hence the kernel of Tt,s is Tt,s(x1, x2, y1, y2) = ∞ ∞

• Rn×Rm
• Rn×Rm ψt,s(x1 − u1, x2 − u2)K(u1, u2, v1, v2)

×ψt′,s′ ∗ ψt′,s′(v1 − y1, v2 − y2)du1du2dv1dv2 dt′ t′ ds′ s′ . Step 2. The L2(Rn+m) − L2

H(Rn+m) boundedness of L:

Note that the L2(Rn+m) boundedness of T and the product Littlewood-Paley square function g imply that L is bounded from L2(Rn+m) to L2

H(Rn+m):

Lf L2

H =

• ∞

∞

• ψt1,t2 ∗ Tf (x1, x2)
• 2 dt1

t1 dt2 t2 1/2

• 2

= g(Tf )2 ≤ CTf 2 ≤ Cf 2.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 3

Step 3. The Hp − Lp

H boundedness of L:

By a special atomic decomposition, we have Proposition A Let L be a bounded operator from L2(Rn+m) to L2

H(Rn+m). For

0 < p ≤ 1, L extends to be a bounded operator from Hp(Rn × Rm) to Lp

H(Rn+m) if and only if

L(a)Lp

H(Rn+m) ≤ C

∀ Hp(Rn × Rm)-atom a.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 3 (cont.)

Using the almost orthogonality, we have Lemma B Let T be a singular integral operator in Journ´ e’s class. If |y1 − y′

1| ≤ |x1 − y1|/2 and |y2 − y′ 2| ≤ |x2 − y2|/2, then

• Tt,s(x1, x2, y1, y2) − Tt,s(x1, x2, y′

1, y2)

• −
• Tt,s(x1, x2, y1, y′

2) − Tt,s(x1, x2, y′ 1, y′ 2)

• t,s>0
• H

≤ C |y1 − y′

1|ε′

|x1 − y1|n+ε′ |y2 − y′

2|ε′

|x2 − y2|m+ε′ for all ε′ < ε.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 3 (cont.)

The definition TCZ = TL2→L2 + |K|CZ and properties of Jour´ ne’s class yield Lemma C Let T be a singular integral operator in Journ´ e’s class. For ε′ < ε, (i) if |y1 − xI| ≤ |x1 − xI|/2, then

• Rm
• Tt,s(x1, ·, y1, y2)−Tt,s(x1, ·, xI, y2)
• f (y2)dy2
• L2

H(Rm)

≤ C |y1 − xI|ε′ |x1 − xI|n+ε′ f 2; (ii) if |y2 − yJ| ≤ |x2 − yJ|/2, then

• Rn
• Tt,s(·, x2, y1, y2) − Tt,s(·, x2, y1, yJ)
• f (y1)dy1
• L2

H(Rn)

≤ C |y2 − yJ|ε′ |x2 − yJ|m+ε′ f 2.

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Proof sketch for “if ” part: Step 3 (cont.)

Using R. Fefferman’s idea [Harmonic analysis on product spaces,

• Ann. of Math., 1987], coupled with Lemmas B and C, we obtain
• {Tt,s(a)}t,s>0
• Lp

H(Rn+m) ≤ C

∀ Hp(Rn × Rm)-atom a. Proposition A shows that L extends to be a bounded operator from Hp(Rn × Rm) to Lp

H(Rn+m).

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s

Thank you!

Ming-Yi Lee Hp(Rn × Rm) boundedness of CZO’s