Let X be a doubling metric measure space. In this paper, the authors introduce the notions of Property (M̃) and Property (P ) of X , prove that Property (M̃) implies Property (P ) and give some equivalent characterizations of Property (M̃) and Property (P ). If X has Property (P ), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the se...

10 This theorem generalizes a result proved by 0. Lokki, "Uber das Randwertproblem der analytischen Funktionen," Ann. Acad. Sci. Fennicae, Ser. A-I, 144, 1-8, 1952. 11 Under the present assumptions, the result for harmonic functions corresponding to Theorem 3 was conjectured by W. Kaplan in a conversation with one of the present authors; a proof of it was obtained jointly, and the result appear...

Let A1 and A2 be expansive dilations, respectively, on R n and R. Let ~ A ≡ (A1, A2) and Ap( ~ A) be the class of product Muckenhoupt weights on R × R for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ Ap( ~ A), the authors characterize the weighted Lebesgue space L w (R × R) via the anisotropic Lusin-area function associated with ~ A. When p ∈ (0, 1], w ∈ A∞( ~ A), the authors introduce the weighted anis...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,topological space), in order to obtain Egoroff and Lusin type theorems forset multifunctions in the Vietoris hypertopology. Some mathematicalapplications are established and several physical implications of thema...

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and (q−(L), q+(L)) be the maximal interval of exponents q ∈ [1, ∞] such that the gradient semigroup { √ t∇e}t>0 is bounded on L(R). In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces H L(R) for all p ∈ (0, q+(...

The aim of this paper is to establish the vector-valued inequalities for Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g∗μ-functions, and their commutators on the Herz-Morrey spaces with variable exponentMK̇ p,q(·)(R n). By applying the properties of Lp(·)(Rn) spaces and the vector-valued inequalities for Littlewood-Paley operators and their...