Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations

The real interpolation method is considered and it is proved that for general local Morrey-type spaces, in the case in which they have the same integrability parameter, the interpolation spaces are again general local Morrey-type spaces with appropriately chosen parameters. This result is a particular case of the interpolation theorem for much more general spaces defined with the help of an operator acting from some function space to the cone of nonnegative nondecreasing functions on (0, ∞). It is also shown how the classical interpolation theorems due to Stein-Weiss, Peetre, Calderón, Gilbert, Lizorkin, Freitag and some of their new variants can be derived from this theorem.     

Dual spaces of local Morrey-type spaces

In this paper we show that associated spaces and dual spaces of the local Morrey-type spaces are so called complementary local Morrey-type spaces. Our method is based on an application of multidimensional reverse Hardy inequalities.     

Decompositions of local Morrey-type spaces

Positivity - We develop and apply a decomposition theory for generic local Morrey-type spaces. Our result is nonsmooth decomposition, which follows from the fact that local Morrey-type spaces are...     

An analog of Young’s inequality for convolutions of functions for general Morrey-type spaces

An analog of the classical Young’s inequality for convolutions of functions is proved in the case of general global Morrey-type spaces. The form of this analog is different from Young’s inequality for convolutions in the case of Lebesgue spaces. A separate analysis is performed for the case of periodic functions.     

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.     

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator _Mα$M_{\alpha }$, 0<α<n$0<\alpha <n$, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted _Lp$L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.     

Successive approximation by spaces of local functions

One considers interpolation and approximation by local functions on a nonuniform net and stable algorithms for their successive construction. One introduces the concept of H-stability of a family of interpolations and one determines sufficient conditions for H-stability. One constructs nonhomogeneous spaces of local functions which realize an a priori approximation of a given order; the basis functions of these spaces have in a certain sense a minimal support and possess interpolation properties.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 1l1, pp. 31–51, 1981.     

Interpolation for martingale Hardy spaces over weighted measure spaces

Several interpolation theorems on martingale Hardy spaces over weighted measure spaces are given. Our proofs are based on the atomic decomposition of martingale Hardy spaces over weighted measure spaces. As applications of interpolation theorems, some inequalities of martingale transform operator are obtained.     

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.     

Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces

This paper is devoted to the interpolation principle between spaces of weak type. We characterise interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such operators and their behavior on Lorentz gamma spaces. A particular emphasis is placed on elementary and comprehensive proofs.     

Wavelets and local Triebel–Lizorkin spaces with the Lorentz index

In this paper, we apply wavelets to consider local norm function spaces with the Lorentz index. Triebel–Lizorkin–Lorentz spaces are based on the real interpolation of the Triebel–Lizorkin spaces. Triebel–Lizorkin–Morrey spaces are based on local norm of the Triebel–Lizorkin spaces. We give a unified depict of spaces that include these two kinds of spaces. Each index of the five index spaces represents a property of functions. We prove the wavelet characterization of the Triebel–Lizorkin–Lorentz–Morrey spaces and use such characterization to study some basic properties of these spaces.     

Use of abstract Hardy spaces, real interpolation and applications to bilinear operators

This paper can be considered as the sequel of Bernicot and Zhao (J Func Anal 255:1761–1796, 2008), where the authors have proposed an abstract construction of Hardy spaces H ¹. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderón-Zygmund operators in a far more abstract framework as in the Euclidean case.     

Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.     

Weak descartes systems in generalized spline spaces

A class of generalized spline spaces is introduced for which a basis of functions with local support is constructed by using a recursion relation. It is shown that this basis forms a weak Descartes system. Moreover, an interpolation property is given.     

Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper, we focus on the case of Hilbert spaces and introduce, in particular, a version of the Hardy space. Then we define Blaschke factors and Blaschke products and consider an interpolation problem. In the second part of the paper, we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carathéodory functions.     

Interpolation properties of scales of Banach spaces

It is shown that any interpolation scales joining weight spaces L _p or similar spaces have many remarkable properties. Not only are such scales intrinsically interpolation scales, but an analog of the Arazy-Cwikel theorem describing interpolation spaces between the spaces from the scale is valid.     

Interpolation theorem on Lorentz spaces over weighted measure spaces

In 1997 Ferreyra proved that it is impossible to extend the Stein-Weiss theorem in the context of Lorentz spaces. In this paper we obtain an interpolation theorem on Lorentz spaces over weighted measure spaces.

     

Real interpolation between martingale hardy and BMO spaces

In this paper, we consider the real interpolation with a function parameter between martingale Hardy and BMO spaces. An interpolation theorem for martingale Hardy and BMO spaces is formulated. As an application, real interpolation between martingale Lorentz and BMO spaces is given.