Suppose that $(X,d,μ)$ is a space of homogeneous type, with upper dimension $μ$, in the sense of R. R. Coifman and G. Weiss. Let $η$ be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space $h^{*,p}(X)$ via local grand maximal functions and also characterize $h^{*,p}(X)$ via local radial maximal functions, local non-tangential maximal functions, locally atoms and local Littlewood--Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of $h^{*,p}(X)$. As an application, the authors give the dual spaces of $h^{*,p}(X)$ when $p\in(ω/(ω+η),1)$, which further completes the result of G. Dafni and H. Yue on the dual space of $h^{*,1}(X)$. This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of $H^1_{\mathrm{cw}}(X)$ when $μ(X)<\infty$.
We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…
Let $\mathcal X$ be a complete space of homogeneous type. In this note, we prove that the weak$^*$-convergence is true in the Hardy space $H^1(\mathcal X)$ of Coifman and Weiss.
The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies--Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calderón--Zygmund decomposition on product spaces, which is of independent interest, and use it to study the interpolation of these product Hardy spaces. We then show that under the assumption of generalized Gaussian estimates, the product Hardy spaces coincide with the Lebesgue spaces, for an appropriate range of~$p$.
We establish a characterization of the Hardy spaces on the homogeneous groups in terms of the Littlewood-Paley functions. The proof is based on vector-valued inequalities shown by applying the Peetre maximal function.
The purpose of this paper is to introduce and investigate some basic properties of mixed homogeneous Herz-Hardy spaces $H\dot{K}_{\vec{p}}^{α, q}(\mathbb{R}^n)$ and mixed non-homogeneous Herz-Hardy spaces $HK_{\vec{p}}^{α, q}(\mathbb{R}^n)$. Furthermore, we establish the atom and molecular decompositions for $H\dot{K}_{\vec{p}}^{α, q}(\mathbb{R}^n)$ and $HK_{\vec{p}}^{α, q}(\mathbb{R}^n)$, by which the boundedness for a wide class of sublinear operators on mixed Herz-Hardy spaces is obtained. As a byproduct, the dual spaces of mixed homogeneous Herz-Hardy spaces are deduced.
Let $({\mathcal X},d,μ)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.
In the setting of spaces of homogeneous type, we study some Hardy type inequalities, which notably appeared in the proofs of local T(b) theorems as in [AR]. We give some suffi cient conditions ensuring their validity, related to the geometry and distribution of points in the homogeneous space. We study the relationships between these conditions and give some examples and counterexamples in the complex plane.
Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, a…
We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.
Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup $\{e^{-tL}\}_{t>0}$ is bounded on $L^q(\mathbb{R}^n)$. In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces $H_L^p(\mathbb{R}^n)$ for all $p\in(0,\,p_+(L))$, which, when $p=1$, answers a question asked by Deng et al. in [J. Funct. Anal. 263 (2012), 604-674]. Moreover, the authors characterize $H_L^p(\mathb…
We show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot)}$ over a space of homogeneous type $(X,d,μ)$ if and only if it is bounded on its dual space $L^{p'(\cdot)}$, where $1/p(x)+1/p'(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type $(X,d,μ)$.
This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective which means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. To be precise, in this book, the authors first study some basic properties of generalized Herz spaces and obtain boundedness and compactness characterizations of commutators on them. Then the authors introduce the associated Herz-Hardy spaces, local Herz-…
The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. The smallest constant $C$ satisfying this property is called the Hardy constant of the mean $M$. In this paper we determine the Hardy constant in the cases when the mean $M$ is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.
In this paper we introduce Hardy-Lorentz spaces with variable exponents associated to dilation in ${\Bbb R}^n$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.
We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only addresses an open point in the literature, but also gives a complete answer to the question posed by Coifman and Weiss in the case of finite measure. We then apply our results to give maximal function characterizations for Hardy spaces associated to second order elliptic operators with Neumann and Dirichlet boundary conditions, Schrödinger operators with Dirichlet …
Let $(X,d,μ)$ be a space of homogeneous type, with the upper dimension $ω$, in the sense of R. R. Coifman and G. Weiss. Assume that $η$ is the smoothness index of the wavelets on $X$ constructed by P. Auscher and T. Hytönen. In this article, when $p\in(ω/(ω+η),1]$, for the atomic Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal function, the radial maximal function, the non-tangential maximal functions, the various Littlewood-Paley functions and wavelet functions…
In this paper we introduce molecules associated to Hardy spaces with pointwise variable anisotropy, and prove that each molecule can be represented as a sum of atoms.
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characteri…
In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson in…
Let $\mathcal{E}(X,d,μ)$ be a Banach function space over a space of homogeneous type $(X,d,μ)$. We show that if the Hardy-Littlewood maximal operator $M$ is bounded on the space $\mathcal{E}(X,d,μ)$, then its boundedness on the associate space $\mathcal{E}'(X,d,μ)$ is equivalent to a certain condition $\mathcal{A}_\infty$. This result extends a theorem by Andrei Lerner from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type.
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition, $q\in(0,\infty]$ and $A$ be a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the anisotropic variable Hardy-Lorentz space $H_A^{p(\cdot),q}(\mathbb R^n)$ associated with $A$, via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain characterizations of $H_A^{p(\cdot),q}(\mathbb R^n)$, respectively, in terms of the atom and th…
In this paper we introduce variable exponent local Hardy spaces associated with a non-negative self-adjoint operator L. We define them by using an area square integral involving the heat semigroup associated to L. A molecular characterization is established and as an aplication of the molecular characterization we prove that our local Hardy space coincides with the (global) variable exponent Hardy space associated to L, provided that 0 does not belong to the spectrum of L. Also, we show that it coincides with the global variable exponent Hardy space associated to L+I.
In this paper, we prove the Hardy-Leray inequality and related inequalities in variable Lebesgue spaces. Our proof is based on a version of the Stein-Weiss inequality in variable Lebesgue spaces derived from two weight inequalities due to Melchiori and Pradolini. We also discuss an application of our results to establish an existence result for the degenerate $\pp$-Laplace operator.
In this paper, the author obtain the continuity of a class of linear operators on variable anisotropic Hardy-Lorentz spaces. In addition, the author also obtain that the dual space of variable anisotropic Hardy-Lorentz spaces is the anisotropic BMO-type spaces with variable exponents. This result is still new even when the exponent function $p(\cdot)$ is $p$.
Let $(\mathcal{X},d,μ)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in\rm{\widetilde{RBMO}(μ)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(μ)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(μ)$. The b…
In this paper we prove sharp weighted Hardy-type inequalities on Carnot groups with the homogeneous norm $N=u^{1/(2-Q)}$ associated to Folland's fundamental solution $u$ for the sub-Laplacian $Δ_{\mathbb{G}}$. We also prove uncertainty principle, Caffarelli-Kohn-Nirenberg and Rellich inequalities on Carnot groups.
Let $({\mathcal X}, d, μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(μ)$ and prove that its dual space is the known space ${\rm RBMO}(μ)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(μ)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón--Zygmund operators from $H^1(μ)$ to $L^1(μ)$.
In this paper, by using the atomic decomposition theory of Hardy space and weak Hardy space, we discuss the boundedness of parameterized Littlewood-Paley operator with variable kernel on these spaces.
We establish the boundedness of the multilinear Calderón-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multil…
In this paper, by using the atomic decomposition theory of Hardy space and weak Hardy space, the author establishes the boundedness of parameterized Marcinkiewicz integral with variable kernel on these spaces, under the Dini condition or Hörmander condition imposed on kernel.
Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of…
In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterizations in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show that the bo…
Let $({\mathcal X},d,μ)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $ρ\in (1,\infty)$, $0<p\le1\le q\le\infty$, $p\neq q$, $γ\in[1,\infty)$ and $ε\in(0,\infty)$. In this article, the authors introduce the atomic Hardy space ${\widetilde H_{\mathrm{atb},\,ρ}^{p,\,q,\,γ}(μ)}$ and the molecular Hardy space ${\widetilde H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,ε}(μ)}$ via the discrete coefficient $\widetilde{K}^{(ρ),\,p}_{B,\,S}$, and prove that the Calderón-Zygmund operator is bounded from ${\widetilde H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,δ}(μ)}$ (or $…
In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this…
In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory. We also study continuity properties of commutators of multilinear fractional type operators on product of certain Hardy spaces with variable exponents.
Let $λ>0$, $p\in((2\lz+1)/(2\lz+2), 1]$, and $\triangle_λ\equiv-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0, \infty), dm_λ)$ associated with $\triangle_λ$ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood-Paley $g$-function and the Lusin-area function, where $dm_λ(x)\equiv x^{2λ}\,dx$. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.
In this paper, we first introduce $L^{σ_1}$-$(\log L)^{σ_2}$ conditions satisfied by the variable kernels $Ω(x,z)$ for $0\leqσ_1\leq1$ and $σ_2\geq0$. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators $T_Ω$, fractional integrals $T_{Ω,α}$ and parametric Marcinkiewicz integrals $μ^ρ_Ω$ with variable kernels on the Hardy spaces $H^p(\mathbb R^n)$ and weak Hardy spaces $WH^p(\mathbb R^n)$. Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy…
Let $p(\cdot):\mathbb R^n\rightarrow(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of para-product operators $π_b$ on variable Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$, where $b\in BMO(\mathbb R^n)$. As an application, we show that non-convolution type Calderón-Zygmund operators $T$ are bounded on $H^{p(\cdot)}(\mathbb R^n)$ if and only if $T^\ast1=0$, where $\frac{n}{n+ε}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $ε$ is the regular exponent of kernel of $T$. Our a…
The Marcinkiewicz integral is essentially a Littlewood-Paley $g$-function, which plays a important role in harmonic analysis. In this article, by using the atomic decomposition theory of weighted Hardy spaces and homogeneous weighted Herz-type Hardy spaces, we will obtain some weighted weak type estimates for Marcinkiewicz integrals on these spaces.
We present a unified strategy to derive Hardy-Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. The idea also applies to the more general context of variable exponent Sobolev spaces. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.
Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function satisfying that there exists a constant $p_0\in(0,p_-)$, where $p_-:=\mathop{\mathrm {ess\,inf}}_{x\in \mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(\cdot)/p_0}(\mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A…
It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend …
We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including a "finite" decomposition; this decomposition is more like the decomposition for weighted Hardy spaces due to Stromberg and Torchinsky than the classical atomic decomposition. As an application of the atomic decomposition we show that singular integral operators are bounded on variable Hardy spaces with minimal regularity assumptions on the exponent function…
Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is b…
We study a family of fractional integral operator defined on an homogeneous space with a "rectangle doubling" measure. As a result, we give an extension of the classical Hardy-Littlewood-Sobolev theorem to a multi-parameter setting.
We study the boundedness of certain fractional integral operators from Hp(.) into Lq(.). We also obtain the Hp(.)- Hq(.) boundedness of the Riesz potential.
