Trudinger's exponential integrability for Riesz potentials of functions in generalized grand Morrey spaces

Our aim in this paper is to discuss Trudinger's exponential integrability for Riesz potentials of functions in generalized grand Morrey spaces. Our result will imply the boundedness of the Riesz potential operator from a grand Morrey space to a Morrey space.     

On Stein's extension operator preserving Sobolev–Morrey spaces

We prove that Stein's extension operator preserves Sobolev–Morrey spaces, that is spaces of functions with weak derivatives in Morrey spaces. The analysis concerns classical and generalized Morrey spaces on bounded and unbounded domains with Lipschitz boundaries in the n‐dimensional Euclidean space.     

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space.     

Inclusion properties of generalized Morrey spaces

This paper discusses the structure of Morrey spaces, weak Morrey spaces, generalized Morrey spaces, and generalized weak Morrey spaces. Some necessary and sufficient conditions for the inclusion property of these spaces are obtained through a norm estimate for the characteristic functions of balls.     

Doob's inequality,Burkholder-Gundy inequality and martingale transforms on martingale Morrey spaces

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob's inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob's inequality on martingale block spaces, we obtain the Davis' decompositions for martingale Morrey spaces.     

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.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces on domains

The purpose of this paper is to develop a theory of the Besov‐Morrey spaces and the Triebel‐Lizorkin‐Morrey spaces on domains in R ⁿ. We consider the pointwise multiplier operator, the trace operator, the extension operator and the diffeomorphism operator. Not only to domains in R ⁿ we extend our definition of function spaces to compact oriented Riemannian manifolds. Among the properties above, the result for the trace operator is in particular interesting, which reflects the property of the parameters p, q in the Morrey space ??^p_q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Sobolev embeddings into spaces of Campanato, Morrey, and Hölder type

Necessary and sufficient conditions on a rearrangement-invariant Banach function space X(Q) on a cube Q in , n?2, are given for the corresponding Sobolev space W¹X(Q) to be continuously embedded into (generalized) Campanato, Morrey, or Hölder spaces. The optimal such r.i. spaces X(Q) are found. As a by-product, sharp inclusion relations are proved among Campanato, Morrey, and Hölder type spaces.     

Sparse non-smooth atomic decomposition of Morrey spaces

Recently, a non-smooth atomic decomposition in Morrey spaces was obtained by the author, Iida and Tanaka. This note refines the above existing result using the notion of sparse families. As an application, we prove the equivalence between fractional integral operators and fractional maximal operators under the Morrey norm.     

Maximal and fractional integral operators on generalized Morrey spaces over metric measure spaces

We establish the boundedness and weak boundedness of the maximal operator and generalized fractional integral operators on generalized Morrey spaces over metric measure spaces without the assumption of the growth condition on μ. The results are generalization and improvement of some known results. We also give the vector‐valued boundedness. Moreover we prove the independence of the choice of the parameter in the definition of generalized Morrey spaces by using the geometrically doubling condition in the sense of Hytönen.     

Multipliers and Morrey Spaces

We study the pointwise multipliers from one Morrey space to another Morrey space. We give a necessary and sufficient condition to grant that the space of those multipliers is a Morrey space as well.     

An oscillating operator related to wave equations in the block spaces

In this article, we study certain oscillating multipliers related to Cauchy problem for the wave equations on the Euclidean space and on the torus. We obtain that, at the end point, these operators are bounded from the L _p spaces to certain block spaces.     

Sobolev's inequality for Riesz potentials of functions in Morrey spaces of integral form

Morrey spaces have become a good tool for the study of existence and regularity of solutions of partial differential equations. Our aim in this paper is to give Sobolev's inequality for Riesz potentials of functions in Morrey spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

Fourier–Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier–Jacobi expansions in Morrey spaces.     

Commutators of fractional integral operators on Vanishing-Morrey spaces

In this note we prove a sufficient condition for commutators of fractional integral operators to belong to Vanishing Morrey spaces VL ^p,λ. In doing this we use an extension on Morrey spaces of an inequality by Fefferman and Stein concerning the sharp maximal function and the fractional maximal function and related Morrey inequalities.      

Absolutely summing multipliers on abstract Hardy spaces

We study summing multipliers from Banach spaces of analytic functions on the unit disc of the complex plane to the complex Banach sequence lattices. The domain spaces are abstract variants of the classical Hardy spaces generated by the complex symmetric spaces. Applying interpolation methods, we prove the Hausdorff Young and Hardy-Littlewood type theorems. We show applications of these results to study summing multipliers from the Hardy-Orlicz spaces to the Orlicz sequence lattices. The obtained results extend the well-known results for the Hp spaces.     

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.