Atomic decomposition characterizations of ag Hardy spaces · Hardy spaces Hp F (R n Rm) for 0

Atomic decomposition characterizations of flagHardy spaces

Xinfeng WuUniversity of Kansas

Prairie Analysis SeminarKansas State University

September 25 2015

XW

(Prairie Analysis SeminarKansas State University)Atomic decomposition September 25 2015 1 / 21

Abstract

In this talk, we discuss atomic decomposition characterizations of flagHardy spaces Hp

F (Rn × Rm) for 0 < p ≤ 1. A feature of atoms of suchflag Hardy spaces is that these atoms have only partial cancellationconditions. As an application, we obtain a boundedness criterion foroperators on flag Hardy spaces.

XW (KU) Atomic decomposition September 25 2015 2 / 21

Definition of flag kernels

The flag kernels were introduced by Muller, Ricci and Stein in

Marcinkiewicz multipliers and multi-parameter structure onHeisenberg(-type) groups, I. Invent. Math. 119 (1995), 119–233.

where they proved that Marcinkiewicz multipliers on the Heisenberggroups are singular integrals with flag kernels, that these flag kernels areprojections of product kernels, and obtained the Lp, 1 < p <∞,boundedness of these operators.



Definition (Flag kernels on Rn × Rm)

A flag kernel, associated to the flag (0, 0) ⊂ (0, y) ⊂ R2, is adistribution K which coincides with a C∞ function away from thecoordinate subspace x = 0 and satisfies(i) Differential inequalities: For all (α, β) ∈ Nn × Nm and for x 6= 0,

|∂αx ∂βyK(x, y)| . |x|−n−|α|(|x|+ |y|)−m−|β|;

(ii) Cancellation conditions: Define a distribution K(2)ψr

by〈Kψ,r, ϕ〉 = 〈K, ψr ⊗ ϕ〉, then Kψr is a one-parameter kernel on Rm.Similarly for K(1)

ψr.



A typical example of flag kernels on R× R is

sgn(y)

x√x2 + y2

.

Remark

(i) The flag kernel is more singular than the classical C-Z convolutionkernel, but less singular than the product kernel.(ii) All flag kernels are product kernels.


Lp, 1 < p <∞ boundedness

Nagel, Ricci and Stein studied a class of operators on nilpotent Lie groupsgiven by convolution with flag kernels.

Singular integrals with flag kernels and analysis on quadratic CRmanifolds, J. Func. Anal. 181(2001), 29–118.

They show that product kernels can be written as finite sums of flagkernels and that flag kernels have good regularity, restriction andcomposition properties.


Applications in several complex variables:

(a) Lp regularity for certain derivatives of the relative fundamental solutionof b and for the corresponding Szego projections onto the null space, see

Nagel, Ricci and Stein, Singular integrals with flag kernels and analysis onquadratic CR manifolds, J. Func. Anal. 181(2001), 29–118.

(b) Optimal estimates for solutions of the Kohn-Laplacian for certainclasses of model domains in several complex variables, see

A. Nagel and E. M. Stein, The ∂b-complex on decoupled boundaries in Cn,Ann. of Math. 164 (2006), 649–713.


Further generalizations to homogeneous groups

Nagel-Ricci-Stein-Wainger generalized the above results to homogeneousgroups and proved the Lp, 1 < p <∞, boundedness via Littlewood-Paleytheory.

Singular Integrals with Flag Kernels on Homogeneous Groups: I, Rev.Mat. Iberoam. 28 (2012), 631-722.

P. G lowacki independently obtained L2 boundedness by using differentmethod))Melin calculus, see

Composition and L2-boundedness of flag kernels, Colloq. Math., 118(2010), 581–585.


Hp, 0 < p ≤ 1 results

Han-Lu and Han-Lu-Sawyer developed a theory of flag Hardy spaces HpF

via the discrete Littlewood-Paley theory and discrete Calderon’s identityand proved the Hp

F −HpF and Hp

F − Lp, 0 < p ≤ 1, boundedness of flagsingular integrals.

Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory andmulti-parameter Hardy spaces associated with flag singular integrals,arXiv:0801.1701.

Y. Han, G. Lu, E. Sawyer, Flag Hardy spaces and Marcinkiewiczmultipliers on the Heisenberg group. Anal. PDE 7 (2014), no. 7,1465õ1534.


Hp, 0 < p ≤ 1 results

Han-Lu and Han-Lu-Sawyer developed a theory of flag Hardy spaces HpF

via the discrete Littlewood-Paley theory and discrete Calderon’s identityand proved the Hp

F −HpF and Hp

F − Lp, 0 < p ≤ 1, boundedness of flagsingular integrals.

Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory andmulti-parameter Hardy spaces associated with flag singular integrals,arXiv:0801.1701.

Y. Han, G. Lu, E. Sawyer, Flag Hardy spaces and Marcinkiewiczmultipliers on the Heisenberg group. Anal. PDE 7 (2014), no. 7,1465õ1534.


Definition of flag Hardy spaces

Let ψ(1) ∈ S(Rn × Rm) and ψ(2) ∈ S(Rm) satisfy

supp ψ(1) ⊂ (ξ, η) ∈ Rn × Rm : 1/2 ≤ |(ξ, η)| ≤ 2,∑j∈Z|ψ(1)(2−jξ, 2−jη)|2 = 1 for (ξ, η) ∈ Rn × Rm\(0, 0),

supp ψ(2) ⊂ η ∈ Rm : 1/2 ≤ |η| ≤ 2,∑j∈Z|ψ(2)(2−kη)|2 = 1 for η ∈ Rm\(0),

and let ψj,k(x, y) = (ψ(1)j ∗2 ψ

(2)k )(x, y).



Then we have the following Calderon’s reproducing formula

f =∑j,k∈Z

∑R∈Rj,k

|R|ψj,k ∗ f(xI , yJ)ψj,k(x− xI , y − yJ),

where Rj,k denotes the set of dyadic rectangles R = I × J ∈ Rn × Rmwith sidelength `(I) = 2j and `(J) = 2j∨k.

Definition (Littlewood-Paley square function)

The flag Littlewood-Paley square function gF (f) is defined by

gF (f) := ∑j,k∈Z

∑R∈Rj,k

|ψj,k ∗ f(xI , yJ)|2χR1/2

.



Definition

The product test function class S∞(Rn+m × Rm) is the collection of allfunctions f ∈ S(Rn+m × Rm) with∫

Rn+m

f(x, y, z)xαyβdxdy =

∫Rm

f(x, y, z)zγdz = 0

for all multi-indices α, β, γ of nonnegative integers.



The flag test function is defined to be the projection of correspondingproduct test functions.

Definition (Flag test functions and distributions)

A function f(x, y) defined on Rn × Rm is said to be a test function inSF (Rn × Rm) if there exists a function f# ∈ S∞(Rm+n × Rm) such that

f(x, y) =

∫Rm

f#(x, y − z, z)dz. (2.1)

The norm of f ∈ SF is defined by

‖f‖SF = inf‖f#‖S∞ : for all representations of f in (2.1).

Denote by S ′F the dual of SF .



Definition (Flag Hardy spaces)

For 0 < p <∞, the flag Hardy space HpF (Rn × Rm) is defined by

HpF := f ∈ S ′F : ‖gF (f)‖p <∞.

Remark

For 1 < p <∞, it has been shown that the flag Hardy space HpF coincides

with Lp with their norms equivalent.



Definition (Flag Hardy spaces)

For 0 < p <∞, the flag Hardy space HpF (Rn × Rm) is defined by

HpF := f ∈ S ′F : ‖gF (f)‖p <∞.

Remark

For 1 < p <∞, it has been shown that the flag Hardy space HpF coincides

with Lp with their norms equivalent.


Atomic decompositions

Atomic decomposition is a significant tool in studying various functionspaces and operators arising in harmonic analysis and wavelet analysis

Y.Meyer, Wavelets and operators, Cambridge University Press,Cambridge, 1992.

R. R. Coifman and Y. Meyer , Wavelets, Calderon-Zygmund andmultilinear operators, Cambridge Univ. Press, Cambridge, 1997.



Atomic decomposition is a significant tool in studying various functionspaces and operators arising in harmonic analysis and wavelet analysis

Y.Meyer, Wavelets and operators, Cambridge University Press,Cambridge, 1992.

R. R. Coifman and Y. Meyer , Wavelets, Calderon-Zygmund andmultilinear operators, Cambridge Univ. Press, Cambridge, 1997.


Definition of flag atoms

Suppose 0 < p ≤ 1. A flag p-atom is a L2 function a(x1, x2) on Rn × Rmsupported in some open set Ω of finite measure such that

1 ‖a‖L2(Rn×Rm) ≤ |Ω|1/p−1/2;

2

a =∑R∈Ω

aR,

where aR are flag particles. The flag particles can be written as

aR(x1, x2) =

∫a#R(x1, x3;x2 − x3)dx3 =

∫a#R(x1, x2 − x3;x3)dx3,

where a#R and a#

R are ”product particles“ in the sense ofS.-Y. A. Chang and R. Fefferman, A continuous version of dulity ofH1 and BMO on the bidisk, Ann. Math. 112 (1980) 179–201.



Suppose 0 < p ≤ 1. A flag p-atom is a L2 function a(x1, x2) on Rn × Rmsupported in some open set Ω of finite measure such that

1 ‖a‖L2(Rn×Rm) ≤ |Ω|1/p−1/2;

2

a =∑R∈Ω

aR,

where aR are flag particles. The flag particles can be written as

aR(x1, x2) =

∫a#R(x1, x3;x2 − x3)dx3 =

∫a#R(x1, x2 − x3;x3)dx3,

where a#R and a#

R are ”product particles“ in the sense ofS.-Y. A. Chang and R. Fefferman, A continuous version of dulity ofH1 and BMO on the bidisk, Ann. Math. 112 (1980) 179–201.



If 2kR ≥ 2jR , then

(a) supp a#R ⊂ R# := IR × IR × JR, where IR is a cube in Rm centered

at origin with the same sidelength as IR;

(b) a#R satisfies the following moment conditions: For|α|, |β| ≤ lp,n, |γ| ≤ lp,m,∫∫

Rm×Rn

a#R(x1, x3;x2)xα1x

β3dx1dx3 =

∫Rm

a#R(x1, x3;x2)xγ2dx2 = 0.

(c) a#R is Ckpn in x1 and x3, and Ckpm in x2 and satisfies ‖a#

R‖∞ ≤ dR,∥∥∥∥∥∂kna#R

∂xkn1

∥∥∥∥∥∞

,

∥∥∥∥∥∂kna#R

∂xkn3

∥∥∥∥∥∞

≤ dR|IR|k

, and

∥∥∥∥∥∂kma#R

∂xkm2

∥∥∥∥∥∞

≤ dR|JR|k

for each k ≤ kp and∑d2R |R| |IR|2 ≤ A|Ω|1−2/p.







Rm×Rn

a#R(x1, x3;x2)xα1x

β3dx1dx3 =

∫Rm

a#R(x1, x3;x2)xγ2dx2 = 0.


R‖∞ ≤ dR,∥∥∥∥∥∂kna#R

∂xkn1

∥∥∥∥∥∞

,

∥∥∥∥∥∂kna#R

∂xkn3

∥∥∥∥∥∞

≤ dR|IR|k

, and

∥∥∥∥∥∂kma#R

∂xkm2

∥∥∥∥∥∞

≤ dR|JR|k








Rm×Rn

a#R(x1, x3;x2)xα1x

β3dx1dx3 =

∫Rm

a#R(x1, x3;x2)xγ2dx2 = 0.


R‖∞ ≤ dR,∥∥∥∥∥∂kna#R

∂xkn1

∥∥∥∥∥∞

,

∥∥∥∥∥∂kna#R

∂xkn3

∥∥∥∥∥∞

≤ dR|IR|k

, and

∥∥∥∥∥∂kma#R

∂xkm2

∥∥∥∥∥∞

≤ dR|JR|k




Remark

Flag convolution destroys the cancellations so that the flag atoms haveonly partial cancellation conditions:∫

aR(x, y)yβdy = 0.

To overcome the difficulty, we use an idea of Muller, Ricci and Stein in[Invent. Math. 119(1995), 119–233.] to write

aR(x1, x2) =

∫a#R(x1, x3;x2 − x3)dx3 for R “vertical”∫a#R(x1, x2 − x3;x3)dx3 for R “horizontal”

,

Then the resulting product particles in higher dimensions are well localizedand satisfy the desired cancellation conditions.



The atomic decomposition characterizations of the flag Hardy spaces HpF

are as follows.

Theorem

Let 0 < p ≤ 1. Then f ∈ HpF if and only if f =

∑λkak, where each ak is

a flag p-atom. Moreover,

‖f‖HpF≈ inf

(∑k

λpk

) 1p

.

Remark

The results here can be easily extends to the Heisenberg groups, where thetheory of flag Hardy spaces was developed by Han-Lu-Sawyer.



The atomic decomposition characterizations of the flag Hardy spaces HpF

are as follows.

Theorem

Let 0 < p ≤ 1. Then f ∈ HpF if and only if f =

∑λkak, where each ak is

a flag p-atom. Moreover,

‖f‖HpF≈ inf

(∑k

λpk

) 1p

.

Remark

The results here can be easily extends to the Heisenberg groups, where thetheory of flag Hardy spaces was developed by Han-Lu-Sawyer.



The atoms can be viewed as building blocks of functions spaces and theproblem of boundedness on function spaces can simply be reduced to theuniform boundedness of operators on such building blocks “atoms”.Here is a boundedness criterion in the flag setting.

Theorem

Let T be a L2 bounded linear operator. Then T is bounded fromHpF (Rn × Rm) 0 < p ≤ 1 to Lp(Rn × Rm) 0 < p ≤ 1 if and only if‖Ta‖Lp ≤ C uniformly for all Hp

F (Rn × Rm) atoms a, and T is boundedon Hp

F (Rn × Rm) 0 < p ≤ 1 if and only if ‖Ta‖HpF≤ C uniformly for all

HpF (Rn × Rm) atoms a.


Thank you very much


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Atomic decomposition characterizations of ag Hardy spaces · Hardy spaces Hp F (R n Rm) for 0

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