Non‐smooth atomic decomposition for generalized Orlicz‐Morrey spaces

In the present paper, we consider the non‐smooth atomic decomposition of generalized Orlicz‐Morrey spaces. The result will be sharper than the existing results. As an application, we consider the boundedness of the bilinear operator, which is called the Olsen inequality nowadays. To obtain a sharp norm estimate, we first investigate their predual space, which is even new, and we make full advantage of the vector‐valued inequality for the Hardy‐Littlewood maximal operator.     

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)     

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.     

Generalized weighted Morrey spaces and classical operators

We introduce the notion of generalized weighted Morrey spaces and investigate the boundedness of some operators in these spaces, such as the Hardy–Littlewood maximal operator, generalized fractional maximal operators, generalized fractional integral operators, and singular integral operators. We also study their boundedness in the vector‐valued setting.     

Decomposition for Morrey type Besov–Triebel spaces

In this paper, the author establishes the decomposition of Morrey type Besov–Triebel spaces in terms of atoms and molecules concentrated on dyadic cubes, which have the same smoothness and cancellation properties as those of the classical Besov–Triebel spaces. The results extend those of M. Frazier, B. Jawerth for Besov–Triebel spaces and those of A. L. Mazzucato for Besov–Morrey spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Morrey spaces and trace operator

The theory of generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, which Mizuhara and Nakai proposed, are equipped with a parameter and a function. The trace property is one of the main focuses of the present paper, which will clarify the role of the parameter of generalized Morrey spaces. The quarkonial decomposition is obtained as an application of the atomic decomposition. In the end, the relation between the function spaces dealt in the present paper and the foregoing researches is discussed.     

Besov‐Morrey spaces and Triebel‐Lizorkin‐Morrey spaces for nondoubling measures

We define Morrey type Besov‐Triebel spaces with the underlying measure non‐doubling. After defining the function spaces, we investigate boundedness property of some class of the singular integral operators (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Characterization of multiplier spaces by wavelets and logarithmic Morrey spaces

In this paper, we consider multipliers from Sobolev spaces to Lebesgue spaces. We establish some wavelet characterization of multiplier spaces without using capacity. Further, we give a sharp logarithmic Morrey space condition for multipliers which lessens Fefferman’s Morrey space condition to the logarithm level and generalizes Lemarié’s counter-example to non-integer cases and expresses his results in a more precise way.     

Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces

We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz–Morrey and weak Orlicz–Morrey spaces. To do this, we prove the weak–weak type modular inequality of the Hardy–Littlewood maximal operator with respect to the Young function. Orlicz–Morrey spaces contain $L^p$ spaces ($1\le p\le \infty$), Orlicz spaces, and generalized Morrey spaces as special cases. Hence, we get necessary and sufficient conditions on these function spaces as corollaries.     

Anisotropic Lizorkin–Triebel spaces with mixed norms — traces on smooth boundaries

This article deals with trace operators on anisotropic Lizorkin–Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self‐contained exposition of Lizorkin–Triebel spaces on manifolds and extend these results to mixed‐norm Lizorkin–Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right‐inverse of the trace is given explicitly and proved to be continuous in the scale of mixed‐norm Lizorkin–Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.     

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.     

Gradient estimates for parabolic equations in generalized weighted Morrey spaces

We consider the Cauchy–Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain.The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others.We obtain Calderón–Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.     

On the coincidence of certain approaches to smoothness spaces related to Morrey spaces

In this paper, we compare the recent approach of Hans Triebel to introduce smoothness spaces related to Morrey‐Campanato spaces with Besov type and Triebel‐Lizorkin type spaces. These two scales have been introduced some years ago and represent a further variant to measure smoothness by using Morrey spaces.     

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.     

The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces

The boundedness of maximal Bochner-Riesz operator Bδ* and that of maximal commutator Bδb,* generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.     

Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

In this paper, we study the boundedness of the fractional integral operator I _α on Carnot group G in the generalized Morrey spaces M _{p, φ} (G). We shall give a characterization for the strong and weak type boundedness of I _α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces

In this paper we are concerned with Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces.     

Weak Morrey spaces with applications

The main object of this investigation is to study weak Morrey spaces. Block spaces, which are preduals of weak Morrey spaces, are characterized. Besides, the Fatou property of block spaces is proved. Finally, as an application, we study the boundedness of singular integral operators in weak Morrey spaces.