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.     

Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces

The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato. The first author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. The second author is supported by Fūjyukai foundation and the 21st century COE program at Graduate School of Mathematical Sciences, the University of Tokyo.     

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)     

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)     

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.     

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.     

Local means,wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces.     

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Singular integral operator,Hardy–Morrey space estimates for multilinear operators and Navier–Stokes equations

After establishing the molecule characterization of the Hardy–Morrey space, we prove the boundedness of the singular integral operator and the Riesz potential. We also obtain the Hardy–Morrey space estimates for multilinear operators satisfying certain vanishing moments. As an application, we study the existence and the uniqueness of the solutions to the Navier–Stokes equations for the initial data in the Hardy–Morrey space ????(p?n) for q as small as possible. Here, the Hardy–Morrey space estimates for multilinear operators are important tools. Copyright © 2010 John Wiley & Sons, Ltd.     

Boundedness of certain commutators on Triebel–Lizorkin spaces

In this paper, two types of commutators are considered, and the boundedness of these operators on Triebel–Lizorkin spaces are discussed.     

Some specific unboundedness property in smoothness Morrey spaces. The non-existence of growth envelopes in the subcritical case

We study smoothness spaces of Morrey type on Rn and characterise in detail those situations when such spaces of type A_(p,q)~(s,r)(R~n) or A_(u,p,q)~s(R~n) are not embedded into L_(∞)(R~n).We can show that in the so-called sub-critical,proper Morrey case their growth envelope function is always infinite which is a much stronger assertion.The same applies for the Morrey spaces M_(u,p)(R~m) with p u.This is the first result in this direction and essentially contributes to a better understanding of the structure of the above spaces.     

Complex interpolation of Herz‐type Triebel–Lizorkin spaces

We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderón product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces . Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between (or ) spaces and Herz spaces.     

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.     

Riesz Potentials in Besov and Triebel–Lizorkin Spaces over Spaces of Homogeneous Type

By using the discrete Calderón reproducing formulae, the author first establishes the boundedness of the Riesz-potential-type operator in homogeneous Besov and Triebel–Lizorkin spaces over spaces of homogeneous type. Then, by use of the T1 theorems for these spaces, the author proves that this operator of Riesz potential type can be used as the lifting operator of these spaces.     

The duality between Morrey spaces and its pre-dual spaces characterized by heat kernel

Let L be an elliptic operator, we give arguments about the duality estimate between Morrey spaces and its pre-dual spaces characterized by the heat kernel associated to L that is given in [6],[8].     

Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents

In this paper, Morrey type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Then equivalent quasi-norms of these new spaces in terms of Peetre?s maximal functions are obtained. Finally, applying those equivalent quasi-norms, the authors obtain the atomic, molecular and wavelet decompositions of these new spaces.     

Properties of analytic Morrey spaces and applications

In this note, we aim to study analytic Morrey spaces . We first give the canonical factorization for . Then by applying p‐Carleson measure, we prove an atomic decomposition theorem of . As an application of the decomposition theorem, the interpolation problem of is solved. Finally, we show the boundedness and compactness of Toeplitz operators on .     

加权Morrey空间上分数次极大算子的双权不等式

Ye与Wang研究了Hardy-Littlewood极算子在加权Morrey空间的双权不等式.该文将Ye与Wang的结果拓展到分数次极大算子,此外也得到了Ap型的充分条件.     

Weighted Hardy operators and commutators on Morrey spaces

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ ⁿ )) on Morrey spaces.