Sharp integral estimates for the fractional maximal function and interpolation

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between L ₁ and the Morrey space . The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.     

乘子算子的Toeplitz型算子的M~2型Sharp极大函数估计和有界性

该文对与乘子算子相关的Toeplitz型算子证明了其sharp极大函数估计,做为应用,得到了该算子在Lebesgue空间和Morrey空间上的有界性.     

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.     

与一类奇异积分算子相关的Toeplitz型算子的有界性

对与具有一般核的分数次奇异积分算子相关的Toeplitz型算子,本文证明了其sharp极大函数不等式,作为应用,得到了该算子在Lebesgue空间,Morrey空间和Triebel-Lizorkin空间的有界性.     

Fractional Integrals on Variable Hardy–Morrey Spaces

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy–Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy–Morrey spaces, which extends the Taibleson–Weiss’s results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered.     

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)     

Sharp Maximal Inequalities and Its Application to?Some Bilinear Estimates

In this note we establish the sharp maximal inequalities for Herz spaces and Morrey spaces by use of good ??-inequality. As an application, we obtain estimates of some bilinear forms which include usual product of functions and the nonlinear term of Euler and Navier-Stokes equations on Herz spaces and Morrey spaces.     

Variable Lebesgue norm estimates for BMO functions

In this paper, we are going to characterize the space BMO(? ⁿ ) through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space BMO(? ⁿ ) by using various function spaces. For example, Ho obtained a characterization of BMO(? ⁿ ) with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.     

Some Characterizations of Lipschitz Spaces via Commutators on Generalized Orlicz–Morrey Spaces

In this paper, we give some new characterizations of the Lipschitz spaces via the boundedness of commutators associated with the fractional maximal operator, Riesz potential and Calderón–Zygmund operator on generalized Orlicz–Morrey spaces.     

The Fractional Maximal Operator and Marcinkiewicz Integrals Associated with Schr?dinger Operators on Morrey Spaces with Variable Exponent

In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schr ?dinger operator on Morrey spaces with variable exponent.     

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.     

Orlicz–Morrey Spaces and Fractional Operators

By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz–Morrey spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz–Morrey spaces are established. The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also verified. The result obtained here sharpens the one in our earlier papers.     

A characterization for Adams-type boundedness of the fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz–Morrey 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.     

Boundedness of maximal function related to the Monge-Ampère equation

In this paper, we establish the boundedness of maximal function on Morrey spaces related to the Monge-Ampère equation.     

λ-Central BMO estimates for multilinear commutators of fractional integrals

The author establishes λ-central BMO estimates for commutators of multilinear fractional integral operators on central Morrey spaces.     

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 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.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.