atomic decomposition characterizations of ag hardy spaces · hardy spaces hp f (r n rm) for 0
TRANSCRIPT
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.
XW (KU) Atomic decomposition September 25 2015 3 / 21
Definition of flag kernels
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.
XW (KU) Atomic decomposition September 25 2015 4 / 21
Definition of flag kernels
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.
XW (KU) Atomic decomposition September 25 2015 5 / 21
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.
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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.
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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.
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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.
XW (KU) Atomic decomposition September 25 2015 9 / 21
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.
XW (KU) Atomic decomposition September 25 2015 9 / 21
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).
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Definition of flag Hardy spaces
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
.
XW (KU) Atomic decomposition September 25 2015 11 / 21
Definition of flag Hardy spaces
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.
XW (KU) Atomic decomposition September 25 2015 12 / 21
Definition of flag Hardy spaces
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 .
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Definition of flag Hardy spaces
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.
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Definition of flag Hardy spaces
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.
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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.
XW (KU) Atomic decomposition September 25 2015 15 / 21
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.
XW (KU) Atomic decomposition September 25 2015 15 / 21
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.
XW (KU) Atomic decomposition September 25 2015 16 / 21
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.
XW (KU) Atomic decomposition September 25 2015 16 / 21
Definition of flag atoms
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.
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Definition of flag atoms
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.
XW (KU) Atomic decomposition September 25 2015 17 / 21
Definition of flag atoms
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.
XW (KU) Atomic decomposition September 25 2015 17 / 21
Atomic decompositions
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.
XW (KU) Atomic decomposition September 25 2015 18 / 21
Atomic decompositions
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.
XW (KU) Atomic decomposition September 25 2015 19 / 21
Atomic decompositions
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.
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Atomic decompositions
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.
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Thank you very much
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