In this article, the authors characterize pointwise multipliers for localized MorreyCampanato spaces, associated with some admissible functions on RD-spaces, which include localized BMO spaces as a special case. The results obtained are applied to Schrdinger operators and some Laguerre operators.
We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.
Let￠ ： Nn x [0,∞） -4 [0, ∞） be a function such that ￠（x,.） is an Orlicz function and ￠（.,t） ∈ Aloc∞（Nn） （the class of local weights introduced by Rychkov）. In this paper, the authors introduce a local Musielak-Orlicz Hardy space h￠（Nn） by the local grand maximal function, and a local BMO-type space bmo￠（Nn） which is further proved to be the dual space of h￠（Nn）. As an application, the authors prove that the class of pointwise multipliers for the local BMO-type space bmo￠（Nn）, characterized by Nakai and Yabuta, is just the dual of LI（Rn） ＋ hФ0 （Rn）, where Ф is an increasing function on （0, co） satisfying some additional growth conditions and Ф0 a Musielak-Orlicz function induced by Ф. Characterizations of h￠（Rn）, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic char- acterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h￠（Rn）, from which, the authors further deduce some criterions for the boundedness on h￠（Rn） of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on h￠（Rn）.
This article is devoted to presenting a recapitulative introduction for the theory of Besov-type and Triebel-Lizorkin-type spaces developed in recent years.
Abstract In this paper, the authors characterize the inhomogeneous Triebel-Lizorkin spaces Fs,w p,q （Rn） with local weight w by using the Lusin-area functions for the full ranges of the indices, and then establish their atomic decompositions for s ∈ R, p ∈ （0, 1] and q ∈ [p, ∞）. The novelty is that the weight w here satisfies the classical Muckenhoupt condition only on balls with their radii in （0, 1]. Finite atomic decompositions for smooth functions in Fs,w p,q（Rn） are also obtained, which further implies that a （sub）linear operator that maps smooth atoms of Fs,w p,q（Rn） uniformly into a bounded set of a （quasi-）Banach space is extended to a bounded operator on the whole Fs,w p,q（Rn） As an application, the baundedness of the local Riesz operator on the space Fs,w p,q（Rn） is obtained.
Let L be a one-to-one operator of type w having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k C N. In this paper, the authors introduce the Hardy space HPL（Rn） with p ∈（0, 1] associated with L in terms of square functions defined via {e-t2kL}t〉O and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schr6dinger type operator L2 ：= （-△）k ＋ Vk, where A is the Laplacian and 0≤V C Llkoc（Rn）. Moreover, as an application, for i E {1, 2}, the authors prove that the associated Riesz transform Vk（Li-1/2） p n HP（Rn） for @ （n/（n ＋ k）, 1] and establish the Riesz transform characterizations is bounded from HLI（IR ） to p of HPL1（]Rn） for p C （rn/（n ＋ kr）, 1] if {e-tL1 }t〉o satisfies the Lr - L2 k-off-diagonal estimates with r C （1, 2]. These results when k ：= I and L ：= L1 are known.
