Morrey-type Spaces Research Articles

Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces

Abstract In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

Necessary and sufficient conditions about the fractional-type maximal operator in local mixed Morrey-type spaces

In this paper, we investigate sufficient and necessary conditions about the fractional-type maximal operator in local mixed Morrey-type spaces.

Compactness of Commutators for Riesz Potential on Local Morrey-type spaces

The paper considers Morrey-type local spaces from LMpθw The main work is the proof of the commutator compactness theorem for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. In the proof of the commutator compactness theorem for the Riesz potential, we essentially use the boundedness condition for the commutator for the Riesz potential [b,Iα] in local Morrey-type spaces LMpθw, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LMpθw. In the course of proving the commutator compactness theorem for the Riesz potential, we prove lemmas for the commutator ball for the Riesz potential [b,Iα]. Similar results were obtained for global Morrey-type spaces GMpθw and for generalized Morrey spaces Mpw.

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

Boundedness for Littlewood-Paley functions on weighted Morrey-type spaces related to certain nonnegative potentials

Some estimates of operators on Local mixed Morrey-type spaces

Some estimates of operators on Local mixed Morrey-type spaces

Sharp bounds for multilinear fractional integral operators on Morrey type spaces: the endpoint cases

In this paper, the Strichartz’s result of the exponential integrability of fractional integral operators is improved. Also, we establish the endpoint boundedness of the multilinear fractional integrals acting on the multi-Morrey spaces. The conclusions relax the restriction that $$p_{i}\ne 1$$ for all $$i=1,\ldots ,m$$ and extend some known results.

Sharp constants for multilinear Hausdorff operators on local Morrey-type spaces

Abstract Necessary and sufficient conditions for the boundedness of multilinear Hausdorff operators on local Morrey-type spaces are obtained and sharp constants are also given. As applications, we generalize our results to some concrete operators derived from Hausdorff operators, such as the Hardy operator and its dual operator, the weighted Hardy–Littlewood average, among others.

Composition operators between Morrey type spaces

In this paper, we characterize the boundedness of composition operator Cφ from Morrey space H²K₁ to H²K₂ on the unit complex disk.

Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces Gℳωp,θ,φ(ℝn)

Two-weight inequalities for Riesz potential and its commutators on weighted global Morrey-type spaces Gℳωp,θ,φ(ℝn)

Generalized fractional integrals in the vanishing generalized weighted local and global Morrey spaces

In this paper, we prove the boundedness of generalized fractional integral operators I? in the vanishing generalized weighted Morrey-type spaces, such as vanishing generalized weighted local Morrey spaces and vanishing generalized weighted global Morrey spaces by using weighted Lp estimates over balls. In more detail, we obtain the Spanne-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted local Morrey spaces with wq ? A1+ q/p' for 1 < p < q < ?, and from the vanishing generalized weighted local Morrey spaces to the vanishing generalized weighted weak local Morrey spaces with w A1,q for p = 1, 1 < q < ?. We also prove the Adams-type boundedness of the generalized fractional integral operators I? in the vanishing generalized weighted global Morrey spaces with w Ap,q for 1 < p < q < ? and from the vanishing generalized weighted global Morrey spaces to the vanishing generalized weighted weak global Morrey spaces with w A1,q for p = 1, 1 < q < ?. The our all weight functions belong to Muckenhoupt-Weeden classes Ap,q.

Generalized fractional integral operators on variable exponent Morrey type spaces over metric measure spaces

We prove the boundedness of generalized fractional integral operators I_{\rho,\tau} on variable exponent Morrey type spaces \mathcal{L}^{p(\cdot),\omega,\theta}(X) over non-doubling metric measure spaces X , where both \rho(x,r) and \omega(x,r) depend on x \in X . Our result extends the recent results by the authors.

On uniqueness of mild solutions for Boussinesq equations in Morrey-type spaces

We prove the unconditional uniqueness of mild solutions for the Boussinesq equations in critical weak-Morrey spaces for dimension n⩾3. Our approach relies on duality and block spaces, as well as suitable heat and bilinear estimates in our setting. In comparison with previous results, we provide a new uniqueness class of mild solutions for the system.

ON BOUNDEDNESS OF THE GENERALIZED RIESZ POTENTIAL IN LOCAL MORREY-TYPE SPACES

For all admissible values of the numerical parameters sharp sufficient conditions on the functional parameters are obtained ensuring the boundedness of the generalized Riesz potential from one general local Morrey-type space to another one, which, for a certain range of the numerical parameters, coincide with the necessary ones.

EMBEDDING OF GRAND CENTRAL MORREY-TYPE SPACES INTO LOCAL GRAND WEIGHTED LEBESGUE SPACES

We find conditions for the embedding of grand Morrey-type spaces into the so-called local grand Lebesgue spaces.

ON THE BOUNDEDNESS OF THE RIESZ POTENTIAL AND ITS COMMUTATOR’S IN THE GLOBAL MORREY TYPE SPACES WITH VARIABLE EXPONENTS

The paper considers the global Morrey-type spaces GMp(.),θ(.),w(.)(Ω) with variable exponents p(.), θ(.), where Ω ⊂ Rn is an unbounded domain. The questions of boundedness of the Riesz potential and its commutator in these spaces are investigated. We give the conditions for variable exponents (p1(.), p2(.)), (θ1(.), θ2(.)) and on the functions (w1(.), w2(.)) under which the Riesz potential I α , will be bounded from GMp1(.),θ1(.),w1(.)(Ω) to GMp2(.),θ2(.),w2(.)(Ω). The same conditions are obtained for the boundedness of the commutator of the Riesz potential in these spaces. In the case when the exponents p, θ constant numbers, the questions of boundedness of the Riesz potential and its commutator in global Morrey spaces were previously studied by other authors. There are also well-known results on the boundedness of the Riesz potential in global Morrey-type spaces with variable exponents, when the domain Ω ⊂ Rn is bounded.

On the Boundedness of Integral Operators in Morrey-Type Spaces with Variable Exponents

On the Boundedness of Integral Operators in Morrey-Type Spaces with Variable Exponents

Nonhomogeneous central Morrey-type spaces in Lp(⋅) and weak estimates for the maximal and Riesz potential operators

Our aim in this paper is to discuss the weak estimate for the maximal and potential operators in the nonhomogeneous central Morrey-type space Mp(⋅),q,ν(Rn), which consists of all measurable functions f on Rn satisfying ‖f‖Mp(⋅),q,ν(Rn)=(∫1∞(r−ν‖f‖ Lp(⋅)(B(0,r)))qdr r)1∕q<∞, where ν>0, 0<q<∞ and p(⋅) is a variable exponent satisfying the log-Hölder condition at infinity. When q=∞, we apply a necessary modification.

Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline cases

AbstractWe define the grand Lebesgue space corresponding to the case and similar grand spaces for Morrey and Morrey type spaces, also for , on open sets in . We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases for Lebesgue spaces and for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.

The local characterizations of the singularity formation for the MHD equations

This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some ϵ-regularity criteria in spaces for suitable weak solutions, and then together with an embedding theorem from space into a Morrey type space we characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if is a singular point, then for any r>0 it holds that where is a positive constant independent on ν and p.

The boundedness of fractional integral operators in local and global mixed Morrey-type spaces

In this paper, we introduce the local and global mixed Morrey-type spaces and show some properties. Besides, we investigate the boundedness of the fractional integral operators \(I_\alpha \) in these spaces. Firstly, we investigate sufficient and necessary conditions in mixed-norm Lebesgue spaces and necessary conditions for boundedness in the local mixed Morrey-type spaces. Then, we prove the boundedness of \(I_{\alpha }\) by Hardy operators’ boundedness in weighted Lebesgue spaces. Furthermore, we obtain the boundedness of \(I_\alpha \) in global mixed Morrey-type spaces and some corollaries.