Morrey Spaces Research Articles

Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

Weighted Estimates for Vector-Valued Intrinsic Square Functions and Commutators in the Morrey-Type Spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with \(BMO(\mathbb {R}^{n})\) functions are discussed. We first show the weighted strong-type and weak-type estimates of vector-valued intrinsic square functions in the Morrey-type spaces. Then, we obtain weighted strong-type estimates of vector-valued analogues of commutators in Morrey-type spaces. In the endpoint case, we establish the weighted weak \(L\log L\)-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey-type spaces. In particular, we can obtain strong-type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.

Approximation by Matrix Transforms in Morrey Spaces

Approximation by Matrix Transforms in Morrey Spaces

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderon--Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.

Nontriviality of Riesz–Morrey spaces

In this article, the authors completely answer an open question, presented in Tao et al. [A bridge connecting Lebesgue and Morrey spaces via Riesz norms. Banach J Math Anal. 2021;15(1):20], via showing that the Riesz–Morrey space is truly a new space larger than a particular Lebesgue space with a critical index. Indeed, this Lebesgue space is just the real interpolation space of the Riesz–Morrey space for suitable indices. Moreover, the authors further show that the aforementioned inclusion is also proper, namely, this embedding is sharp in some sense, via constructing two nontrivial spare functions, respectively, on and any given cube of with finite side length. The latter constructed function is inspired by the striking function constructed by Dafni et al. [The space : nontriviality and duality. J Funct Anal. 2018;275:577–603]. All the proofs of these results strongly depend on some exquisite geometrical analysis on cubes of . As an application, the relationship between Riesz–Morrey spaces and Lebesgue spaces is completely clarified on all indices.

Proper Inclusion Between Vanishing Morrey Spaces and Morrey Spaces

In this paper, we give an explicit function which belongs to the Morrey spaces but not in the vanishing Morrey spaces. Therefore, we obtain that the Morrey spaces contain the vanishing Morrey spaces properly.

Characterizations of Morrey type spaces

AbstractFor a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $ , we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$ , denoted by $\mathcal {D}^s_K$ . Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$ , we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Erdélyi–Kober fractional integral operators on ball Banach function spaces

We establish the boundedness of the Erdélyi–Kober fractional integral operators on ball Banach function spaces. In particular, it gives the boundedness of the Erdélyi–Kober fractional integral operators on amalgam spaces and Morrey spaces.

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

Disjoint dynamics of weighted translations on solid spaces

In this paper, applying the new concept of aperiodic functions, we characterize disjoint hypercyclic and disjoint topologically mixing weighted translation operators on general Banach lattices. The obtained results in addition to previous modes of Lebesgue and Orlicz spaces, also cover new classes such as some closed subspaces of Morrey spaces.

Hardy–Sobolev inequality for Sobolev functions in central Herz–Morrey spaces

AbstractOur aim in this paper is to establish Hardy–Sobolev inequality for Sobolev functions in central Herz–Morrey spaces. As an application, we are concerned with Sobolev‐type integral representation for a C1‐function on which vanishes on the unit ball.

An anisotropic regularity condition for the 3D incompressible Navier–Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier–Stokes equations on R3×(−1,1) is regular on R3×(−1,0] if ∂3u belongs to M2p/(2p−3),α((−1,0);Lp(R3)) for any α>1 and p∈(3/2,∞), which is a logarithmic-type variation of a Morrey space in time. For each α>1 this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces Lq((−1,0);Lp(R3)) that are subcritical, that is for which 2/q+3/p<2.

Weighted estimates for θ-type Calderón-Zygmund operator and its commutator on metric measure spaces

Let be a non-homogeneous metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this setting, the author proves that the θ-type Calderón-Zygmund operator is bounded on the weighted Lebesgue space and the weighted Morrey space. Furthermore, the boundedness of the commutator generated by and on weighted weak Lebesgue space and on weighted weak Morrey space is also obtained.

Boundedness of composition operators on Morrey spaces and weak Morrey spaces

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

Muckenhoupt-Type Conditions on Weighted Morrey Spaces

We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderón operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy–Littlewood maximal operator, simplifying the previous characterization of Nakamura–Sawano–Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local \(A_p\) condition. We can extrapolate from Lebesgue \(A_p\)-weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.

Weak Estimates for the Maximal and Riesz Potential Operators in Central Herz–Morrey Spaces on the Unit Ball

Morrey spaces are the powerful tool for the study of partial differential equations. Recently, weak Morrey spaces and weak Herz spaces are known to be useful for the study of Navier–Stokes equations. In this paper we introduce weak central Herz–Morrey spaces W\mathcal H^{p(\cdot),q,\omega}(\mathbf B) and establish the weak estimate for the maximal and Riesz potential operators in the central Herz–Morrey space \mathcal H^{p(\cdot),q,\omega}(\mathbf B) . We also treat generalized Riesz potential operators. Further we obtain the strong estimate for Sobolev functions.

A Gradient Estimate Related Fractional Maximal Operators for a $p$-Laplace Problem in Morrey Spaces

In the present paper, we deal with the global regularity estimates for the $p$-Laplace equations with data in divergence form \[ \operatorname{div}(|\nabla u|^{p-2} \nabla u) = \operatorname{div}(|F|^{p-2} F) \quad \textrm{in $\Omega$}, \] in Morrey spaces with natural data $F \in L^p(\Omega;\mathbb{R}^n)$ and nonhomogeneous boundary data belongs to $W^{1,p}(\Omega)$. Motivated by the work of [M.-P. Tran, T.-N. Nguyen, New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations 268 (2020), no. 4, 1427–1462], this paper extends that of global Lorentz–Morrey gradient estimates in which the `good-$\lambda$' technique was undertaken for a class of more general equations, and further, global regularity of weak solutions will be given in terms of fractional maximal operators.

A regularity criterion for 3D NSE in ‘dynamically restricted’ local Morrey spaces

ABSTRACT It is shown that a local-in-time strong solution u to the 3D Navier–Stokes equations remains regular on an interval provided a smallness -condition on u in a dynamically restricted local Morrey space is stipulated; more precisely, where η is a dynamic dissipation scale consistent with the turbulence phenomenology and α and p are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at T under the framework of -sparseness classes introduced in Bradshaw et al. (An algebraic reduction of the ‘Scaling Gap’ in the Navier–Stokes regularity problem. Arch Ration Mech Anal. 2019;231(3):1983–2005).

Sharp Extrapolation Theorems for Local Morrey Spaces

Sharp Extrapolation Theorems for Local Morrey Spaces

Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

In this paper, we establish the necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$ and the fractional maximal commutators $$M_{b,\alpha }$$ in Orlicz $$L^{\Phi }(X)$$ and generalized Orlicz–Morrey spaces $$\mathcal {M}^{\Phi ,\varphi }(X)$$ on spaces of homogeneous type $$X=(X,d,\mu )$$ in the sense of Coifman-Weiss.