Morrey-Type Spaces on Gauss Measure Spaces and Boundedness of Singular Integrals

In this paper, the authors introduce Morrey-type spaces on the locally doubling metric measure spaces, which means that the underlying measure enjoys the doubling and the reverse doubling properties only on a class of admissible balls, and then obtain the boundedness of the local Hardy–Littlewood maximal operator and the local fractional integral operator on such Morrey-type spaces. These Morrey-type spaces on the Gauss measure space are further proved to be naturally adapted to singular integrals associated with the Ornstein–Uhlenbeck operator. To be precise, by means of the locally doubling property and the geometric properties of the Gauss measure, the authors establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein–Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space.     

Measure Density and Extension of Besov and Triebel–Lizorkin Functions

We show that a domain is an extension domain for a Haj?asz–Besov or for a Haj?asz–Triebel–Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case \(0<p<1\). The necessity of the measure density condition is derived from embedding theorems; in the case of Haj?asz–Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj?asz–Besov spaces are intermediate spaces between \(L^p\) and Haj?asz–Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces \(B^s_{p,q}\), \(0<s<1\), \(0<p<\infty \), \(0<q\le \infty \), defined via the \(L^p\)-modulus of smoothness of a function.     

Morrey空间上Marcinkiewicz积分与R

本文建立了 Marcinkiewicz 积分M与具离散系数的正则有界平均振荡空间RBMO（μ）生成的交换子M_b在非齐性度量测度空间上的有界性. 在控制函数λ满足∈-弱反双倍条件的假设下, 当p∈（1,∞）时,证明了M_b在L^p（μ）上是有界的. 另外,还得到了M_b在 Morrey 空间上的有界性.     

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process ?? on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of ??. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener?CIt? chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris?CFKG-inequalities for monotone functions of ??.     

分数次积分交换子在非齐Morrey空间上的紧性

The aim of this paper is to establish the necessary and sufficient conditions for the compactness of fractional integral commutator[b,I_γ]which is generated by fractional integral I_γand function b∈Lip_β(μ)on Morrey space over non-homogeneous metric measure space,which satisfies the geometrically doubling and upper doubling conditions in the sense of Hytonen.Under assumption that the dominating functionλsatisfies weak reverse doubling condition,the author proves that the commutator[b,I_γ]is compact from Morrey space M_q^p(μ)into Morrey space M_t^s(μ)if and only if b∈Lip_β(μ).     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form r ⁿ for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the L ^p spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (? ⁿ ,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L ¹, boundedness from L ^∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.     

Multilinear commutators of fractional integrals over Morrey spaces with non-doubling measures

In this paper, the authors study the boundedness of multilinear fractional integrals on the product Morrey space with non-doubling measure, and investigate the Morrey boundedness properties of the multilinear commutators generated by multilinear fractional integral operators with a tuple of RBMO functions.     

Several equivalent characterizations of fractional Hajłasz-Morrey-Sobolev spaces

In this article, the authors establish several equivalent characterizations of fractional Haj?asz-Morrey-Sobolev spaces on spaces of homogeneous type in the sense of Coifman and Weiss.     

非齐型空间上分数型Marcinkiewicz积分算子的加权估计

令（X，d，μ）为满足所谓上倍双倍条件和几何双倍条件的度量测度空间.设M_β，ρ，q为（X，d，μ）上的分数型Marcinkiewicz积分算子.在本文中，作者证明了若β ∈[0，∞），ρ ∈（0，∞），q ∈（1，∞）且M_β，ρ，q在L²（μ）上有界，则M_β，ρ，q是从加权Lebesgue空间L^p（w）到加权弱Lebesgue空间L^p，∞（w）上有界和从加权Morrey空间L^p，κ，η（ω）到加权弱Morrey空间WL^p，κ，η（ω）上有界.     

Moser iteration for (quasi)minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.     

Regularity of Sets with Quasiminimal Boundary Surfaces in Metric Spaces

This paper studies regularity of perimeter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure-theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory to study rectifiability issues related to quasiminimal sets in the strong A _∞-weighted Euclidean case.     

A concrete estimate for the weak Poincar�� inequality on loop space

The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with a Hilbert space structure and the Ornstein?CUhlenbeck operator d*d. We give a concrete estimate for the weak Poincaré inequality, assuming positivity of the Ricci curvature of the underlying manifold. The order of the rate function is s ^??? for any ?? >?0.     

Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound ??(B(x,r))??Cr ^d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.     

Morrey spaces and fractional integral operators

The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities.     

非齐度量测度空间上Morrey-Herz空间上的广义分数次积分算子及其交换子

设(χ,d,μ)是一个同时满足上双倍条件和几何双倍条件的非齐度量测度空间,对于引进的一类非齐度量测度空间上的Morrey-Herz空间,利用非齐度量测度空间的特征,证明了广义分数次积分算子及其交换子在非齐度量测度空间上MorreyHerz空间的有界性.     

Weighted Norm Inequalities on Graphs

Let (??,??) be an infinite graph endowed with a reversible Markov kernel p and let P be the corresponding operator. We also consider the associated discrete gradient ?. We assume that ?? is doubling, a uniform lower bound for p(x,y) when p(x,y)>0, and gaussian upper estimates for the iterates of p. Under these conditions (and in some cases assuming further some Poincaré inequality) we study the comparability of (I?P)^1/2 f and ?f in Lebesgue spaces with Muckenhoupt weights. Also, we establish weighted norm inequalities for a Littlewood?CPaley?CStein square function, its formal adjoint, and commutators of the Riesz transform with bounded mean oscillation functions.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

Weighted boundedness of some integral operators on weighted <Emphasis Type="Italic">λ</Emphasis>- central Morrey space

In this paper, the authors prove the weighted boundedness of singular integral and fractional integral with a rough kernel on the weighted λ-central Morrey space. Moreover, the weighted estimate for commutators of singular integral with a rough kernel on the weighted λ-central Morrey space is also given.