Spaces Of Homogeneous Type Research Articles

Weighted endpoint estimates for the composition of Calderón–Zygmund operators on spaces of homogeneous type

Weighted endpoint estimates for the composition of Calderón–Zygmund operators on spaces of homogeneous type

Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.

Weighted Lp estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness

We prove the weighted Lp regularity of the ordinary Bergman and Cauchy-Szegő projections on strongly pseudoconvex domains D in Cn with near minimal smoothness for appropriate generalizations of the Bp/Ap classes. In particular, the Bp/Ap Muckenhoupt type condition is expressed relative to balls in a quasi-metric that arises as a space of homogeneous type on either the interior or the boundary of the domain D.

Real‐variable characterizations of local Hardy spaces on spaces of homogeneous type

AbstractLet be 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 via local grand maximal functions and also characterize via local radial maximal functions, local non‐tangential maximal functions, local 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 . As an application, the authors give the dual spaces of when , which further completes the result of G. Dafni and H. Yue on the dual space of . 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 when .

Wavelet characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications

In this article, the authors establish the wavelet characterization of Besov and Triebel–Lizorkin spaces on a given space ( X , d , μ ) of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel–Lizorkin sequence spaces on X , and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel–Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood–Paley characterizations of Triebel–Lizorkin spaces on X . The main novelty of this article is that all these results get rid of their dependence on the reverse doubling property of μ and also the triangle inequality of d , by fully using the geometrical property of X expressed via its equipped quasi-metric d , dyadic reference points, dyadic cubes, and wavelets.

Quantitative Weighted Estimates for Some Singular Integrals Related to Critical Functions

Let $$(X, d, \mu )$$ be a space of homogeneous type with a metric d and a doubling measure $$\mu $$ . Assume that $$\rho $$ is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note that the assumptions on the kernels of these singular integrals do not have any regularity conditions. Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators.

Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

In this paper, we establish the necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$ and the fractional maximal commutators $$M_{b,\alpha }$$ in Orlicz $$L^{\Phi }(X)$$ and generalized Orlicz–Morrey spaces $$\mathcal {M}^{\Phi ,\varphi }(X)$$ on spaces of homogeneous type $$X=(X,d,\mu )$$ in the sense of Coifman-Weiss.

Weak type (p,p) bounds for Schrödinger groups via generalized Gaussian estimates

Let L be a non-negative self-adjoint operator acting on L2(X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e−tL satisfies the generalized Gaussian (p0,p0′)-estimates of order m for some 1≤p0<2. It is known that the operator (I+L)−seitL is bounded on Lp(X) for s≥n|1/2−1/p| and p∈(p0,p0′) (see for example, [5,7,9,10,13,26]). In this paper we study the endpoint case p=p0 and show that for s0=n|12−1p0|, the operator (I+L)−s0eitL is of weak type (p0,p0), that is, there is a constant C>0, independent of t and f so thatμ({x:|(I+L)−s0eitLf(x)|>α})≤C(1+|t|)n(1−p02)(‖f‖p0α)p0,t∈R for α>0 when μ(X)=∞, and α>(‖f‖p0/μ(X))p0 when μ(X)<∞. Our results can be applied to Schrödinger operators with rough potentials and higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.

A note on the trace inequality for Riesz potentials

Abstract We establish a necessary and sufficient condition on a non-negative locally integrable function v guaranteeing the (trace) inequality ∥ I α ⁢ f ∥ L v p ⁢ ( ℝ n ) ≤ C ⁢ ∥ f ∥ L p , 1 ⁢ ( ℝ n ) \lVert I_{\alpha}f\rVert_{L^{p}_{v}(\mathbb{R}^{n})}\leq C\lVert f\rVert_{L^{p% ,1}(\mathbb{R}^{n})} for the Riesz potential I α {I_{\alpha}} , where L p , 1 ⁢ ( ℝ n ) {L^{p,1}(\mathbb{R}^{n})} is the Lorentz space. The same problem is studied for potentials defined on spaces of homogeneous type.

Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure $$\mu $$ satisfies the doubling property only. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure $$\mu .$$ As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel–Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher–Hytonen on spaces of homogeneous type in the sense of Coifman and Weiss.

Variable Besov Spaces Associated with Heat Kernels

Let $$({\mathcal {X}},\rho , \mu )$$ be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ are such that both $$1/p(\cdot )$$ and $$1/q(\cdot )$$ satisfy the globally log-Holder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ is a bounded function satisfying the locally log-Holder continuous condition. In this article, the authors introduce the variable Besov space $$B_{p(\cdot ),q(\cdot )}^{s(\cdot ),L}({\mathcal {X}})$$ , associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Holder continuity, and the Markov property, which is new even on the sphere and the ball of $$\mathbb R^d$$ . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that $$\mu $$ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on $$\mathbb R^d$$ , this space coincides with the existing variable Besov space.

Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type

We study the Lorentz boundedness properties of Calderón-type commutators and relate those properties to the spaces of bounded mean oscillation and vanishing mean oscillation. In application, we obtain Lorentz boundedness and compactness characterization of integral commutators with singular kernel of Calderón–Zygmund type on strictly pseudoconvex domains in ℂn, or convex domain of finite type in ℂn.

Some inequalities for homogeneous $B_n$-potential type integrals on $H^{p}_{\Delta_{\nu}}$ Hardy spaces

We prove the norm inequalities for potential operators and fractional integrals related to generalized shift operator defined on spaces of homogeneous type. We show that these operators are bounded from $H^{p}_{\Delta_{\nu}}$ to $H^{q}_{\Delta_{\nu}}$, for $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{Q}$, provided $0&amp;lt;\alpha&amp;lt;\frac{1}{2}$, and $\alpha&amp;lt;\beta\leq 1$ and $\frac{Q}{Q+\beta}&amp;lt;p\leq\frac{Q}{Q+\alpha}$. By applying atomic-molecular decomposition of $H^{p}_{\Delta_{\nu}}$ Hardy space, we obtain the boundedness of homogeneous fractional type integrals which extends the Stein-Weiss and Taibleson-Weiss's results for the boundedness of the $B_n$-Riesz potential operator on $H^{p}_{\Delta_{\nu}}$ Hardy space.

On Pointwise ell ^r-Sparse Domination in a Space of Homogeneous Type

We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual ell ^1-sum in the sparse operator is replaced by an ell ^r-sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the A_2-theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

Spectral multipliers without semigroup framework and application to random walks

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates à la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results in some situation. Finally, we consider the random walk on the integer lattice Zn and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on Rn.

Boundedness of paraproducts on spaces of homogeneous type II

This article is the second part of two works by the same authors on the same topic. Let be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors introduce the notion of paraproducts on and obtain their boundedness from into for any given satisfying , where , for any given , denotes the Hardy space on . Moreover, the authors also establish the endpoint boundedness of these paraproducts on when p or q is ∞ or 1 with or replaced by , or , and hence give a complete picture on the boundedness of paraproducts on . The main novelty of this article is the avoidance of dependence on the reverse doubling assumption of the considered measure μ, which is achieved by fully using the geometrical properties of expressed via its equipped quasi-metric d, dyadic reference points and dyadic cubes.

Calderón–Zygmund singular operators in extrapolation spaces

We study the boundedness of the Hardy–Littlewood maximal operator in abstract extrapolation Banach function lattices and their Köthe dual spaces. The extrapolation spaces are generated by compatible families of Banach function lattices on quasi-metric measure spaces with doubling measure. These results combined with a variant of the integral Coifman–Fefferman inequality imply that every Calderón–Zygmund singular operator is bounded in considered extrapolation spaces. We apply these results to extrapolation spaces determined by compatible families of Calderón–Lozanovskii spaces, in particular to compatible families of Orlicz spaces that are interpolation of weighted Lp-spaces (1<p<∞) with Ap weights defined on spaces of homogeneous type.

Boundedness of paraproducts on spaces of homogeneous type I

This article is the first part of two works by the same authors on the same topic. Let be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish some basic estimates and a modified inhomogeneous discrete Calderón reproducing formula, which play essential roles in the proof of the boundedness of paraproducts in the second part of this work, and are also of independent interest. To escape the reverse doubling property of the equipped measure μ, the authors make full use of the geometrical properties of expressed by its equipped quasi-metric d, dyadic reference points and dyadic cubes.

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let ( $${\cal X}$$ , d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderon-Zygmund operator and $$\vec b: = ({b_1}, \ldots ,{b_m})$$ be a finite family of $$\widetilde{{\rm{RBMO}}}(\mu)$$ functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator $${{\rm{T}}_{\Pi \vec b}}$$ generated by T and $$\vec b$$ are obtained.

Two-weighted inequalities for maximal operators related to Schrödinger differential operator

Abstract We introduce classes of pairs of weights closely related to Schrödinger operators, which allow us to get two-weight boundedness results for the Schrödinger fractional integral and its commutators. The techniques applied in the proofs strongly rely on one hand, boundedness results in the setting of finite measure spaces of homogeneous type and, on the other hand, Fefferman–Stein-type inequalities that connect maximal operators naturally associated to Schrödinger operators.