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1.
Let p∈(0,1] and s≥[n(1/p−1)], where [n(1/p−1)] denotes the maximal integer no more than n(1/p−1). In this paper, the authors prove that a linear operator T extends to a bounded linear operator from the Hardy space H
p
(ℝ
n
) to some quasi-Banach space ℬ if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of ℬ.
相似文献
2.
Let ${{\mathcal X}}Let X{{\mathcal X}} be an RD-space with m(X)=¥{\mu({\mathcal X})=\infty}, which means that X{{\mathcal X}} is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this
paper, we characterize the atomic Hardy spaces Hpat(X){H^p_{\rm at}({\mathcal X})} of Coifman and Weiss for p ? (n/(n+1),1]{p\in(n/(n+1),1]} via the radial maximal function, where n is the “dimension” of X{{\mathcal X}}, and the range of index p is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this
setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos
et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on
RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized
Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces Hpat(M){H^p_{\rm at}(M)} via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator Δ on complete noncompact connected
manifolds M having a doubling property and supporting a scaled Poincaré inequality for all p ? (n/(n+a),1]{p\in(n/(n+\alpha),1]}, where α represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ. 相似文献
3.
Eiichi Nakai 《数学学报(英文版)》2008,24(8):1243-1268
Let X = (X, d,μ) The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P(X) and prove that the generalized Hardy spaces have the same property as H^P(X). Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let (X, δ, μ) be the normalized space of (X, d, μ) in the sense of Macias and Segovia. We also study the relations of our function spaces for (X, d, μ) and (X, δ,μ). 相似文献
4.
Yong Ding Senhua Lan 《分析论及其应用》2006,22(4):339-352
Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn). 相似文献
5.
Yong DING Shan Zhen LU Qing Ying XUE 《数学学报(英文版)》2007,23(9):1537-1552
In this paper, the authors prove that if Ω satisfies a class of the integral Dini condition, then the parametrized area integral μΩ,S^ρ is a bounded operator from the Hardy space H1 (R^n) to L1 (R^n) and from the weak Hardy space H^1,∞ (R^n) to L^1,∞ (R^n), respectively. As corollaries of the above results, it is shown that μΩ,S^ρ is also an operator of weak type These conclusions are substantial improvement and (1, 1) and of type (p,p) for 1 〈 p 〈 2, respectively extension of some known results. 相似文献
6.
Ferenc Weisz 《分析论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded
from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H
1
#
(T×T), L1(T2)), where the Hardy space H
1
#
(T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H
1
#
(T×T)⊃LlogL(T
2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces
Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.
This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt
Foundation. 相似文献
7.
Let(X,d,μ) be an RD-space with "dimension" n,namely,a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse doubling condition.Using the Calder'on reproducing formula,the authors hereby establish boundedness for paraproduct operators from the product of Hardy spaces H p(X) × H q(X) to the Hardy space H r(X),where p,q,r ∈(n/(n + 1),∞) satisfy 1/p + 1/q = 1/r.Certain endpoint estimates are also obtained.In view of the lack of the Fourier transform in this setting,the proofs are based on the derivation of appropriate kernel estimates. 相似文献
8.
Ferenc Weisz 《逼近论及其应用》2000,16(1):52-65
The two-dimensional classical Hardy space Hp(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space Hp(T×T) to Lp(T2) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H 1 # (T×T), L1(T2)), where the Hardy space H 1 # (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H 1 # (T×T)⊃LlogL(T 2) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces Hp(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈Hp(T×T), the (C, α, β) means convergs to f in Hp(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. 相似文献
9.
Bin Han 《Advances in Computational Mathematics》2006,24(1-4):375-403
In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented.
Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function
vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz
space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167)
in the univariate setting to the multivariate setting.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C20, 41A25, 39B12.
Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant
G121210654. 相似文献
10.
A. Bonilla F. Pérez-González A. Stray R. Trujillo-González 《Journal d'Analyse Mathématique》1997,73(1):65-89
This paper is concerned with several approximation problems in the weighted Hardy spacesH
p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH
p(Ω) coincides, moduloH
p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan
sets forH
p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer
functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH
p(D), 1 ⪯p < ∞.
Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday. 相似文献
11.
Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L... 相似文献
12.
For 0<p<∞, let Hp(R
n) denote the Lebesgue space for p>1 and the Hardy space for p ≤1. In this paper, the authors study Hp(R
n)×Hq(R
n)→Hr(R
n) mapping properties of bilinear operators given by finite sums of the products of the standard fractional integrals or the
standard fractional integral with the Calderón-Zygmund operator. The authors prove that such mapping properties hold if and
only if these operators satisfy certain cancellation conditions.
Supported by the NNSF and the National Education Comittee of China. 相似文献
13.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
14.
The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H
1 (ℝ
n
× ℝ
m
) to the Lebesgue space L
1(ℝ
n
× ℝ
m
) and their commutators with Lipschitz functions from the Hardy space H
1 (ℝ
n
× ℝ
m
) to the Lebesgue space L
q
(ℝ
n
× ℝ
m
) for some q > 1. 相似文献
15.
Marcinkiewicz Integrals with Non-Doubling Measures 总被引:2,自引:0,他引:2
Let μ be a positive Radon measure on
which may be non doubling. The only condition that μ must satisfy is μ(B(x, r)) ≤ Cr
n
for all
, r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some H?rmander-type
condition, and assume that it is bounded on L
2(μ). We then establish its boundedness, respectively, from the Lebesgue space L
1(μ) to the weak Lebesgue space L
1,∞(μ), from the Hardy space H
1(μ) to L
1(μ) and from the Lebesgue space L
∞(μ) to the space RBLO(μ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L
p
(μ) with p ∈ (1,∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(μ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger H?rmander-type condition, respectively,
from L
p
(μ) with p ∈ (1,∞) to itself, from the space L log L(μ) to L
1,∞(μ) and from H
1(μ) to L
1,∞(μ). Some of the results are also new even for the classical Marcinkiewicz integral.
The third (corresponding) author was supported by National Science Foundation for Distinguished Young Scholars (No. 10425106)
and NCET (No. 04-0142) of China. 相似文献
16.
References: 《高校应用数学学报(英文版)》2007,22(1):29-36
In this note, the regularity of Poisson equation -△u = f with f lying in logarithmic function space Lp(LogL)a(Ω)(1<p <∞, a ∈ R) is studied. The result of the note generalizes the W2,p estimate of Poisson equation in Lp(Ω). 相似文献
17.
In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB
π, p
(R
n
), 1≤p<∞, i.e., for 1<p<∞,B
π, p
(R
n
) is isomorphic tol
p
(Z
n
), and forp=1,B
π, 1
(R
n
) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z
n
).
This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution
of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan. 相似文献
18.
Let (X, μ) be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz
interpolation theorem for a quasilinear operator T in Lorentz spaces L
p,q
(X) with p, q ∈ (0,∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.: Classical Fourier Analysis, Second Edition, Graduate
Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ∈ [1,∞) and q ∈ [1,∞), was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press,
Princeton, N.J., 1971]. 相似文献
19.
In this paper we establish a discrete Calderón’s identity which converges in both L
q
(ℝ
n+m
) (1<q<∞) and Hardy space H
p
(ℝ
n
×ℝ
m
) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H
p
(ℝ
n
×ℝ
m
) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L
q
(ℝ
n+m
) for some 1<q<∞, is bounded from H
p
(ℝ
n
×ℝ
m
) to L
p
(ℝ
n+m
) if and only if T is bounded uniformly on all (p,q)-product atoms in L
p
(ℝ
n+m
). The similar result from H
p
(ℝ
n
×ℝ
m
) to H
p
(ℝ
n
×ℝ
m
) is also obtained. 相似文献
20.
Zeng Jian LOU Shou Zhi YANG Dao Jin SONG 《数学学报(英文版)》2005,21(4):949-954
We give a decomposition of the Hardy space Hz^1(Ω) into "div-curl" quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1(Ω) into Jacobians det Du, u ∈ W0^1,2 (Ω,R^2) for Ω in R^2. This partially answers a well-known open problem. 相似文献