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For a class of sublinear operators, we find conditions on the variable exponent Morrey-type space L p (⋅) , q , ω (⋅ , ⋅) (R n) ensuring the boundedness in this space. A priori assumptions on this class are that the operators are bounded in L p (⋅) (R n) and satisfy some size condition. This class includes in particular the maximal operator, singular operators with the standard kernel, and the Hardy operators. We also prove embedding of variable exponent Morrey-type spaces into weighted L p (⋅) -spaces. [ABSTRACT FROM AUTHOR]

Applying the new class of multiple weights A p → 0 , we establish some weighted norm inequalities for certain classes of multilinear operators in the Morrey-type spaces. In addition, the new multiple weighted norm inequalities for multilinear commutators of T with the new BMO functions BMOθ are also obtained. [ABSTRACT FROM AUTHOR]

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form L u : = ∑ i , j = 1 n a ij (x) u x i x j = ∇ · f , for almost all x ∈ Ω where the coefficients a ij belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces LM ~ { x 0 } p , φ . Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results. [ABSTRACT FROM AUTHOR]

In this paper, we generalize constructions of Morrey spaces by using as a basic space a general rearrangement invariant space E(Rn) instead of Lp(Rn) and a general ideal space F(R+) for outer norm instead of L∞(R+) or Lq,w(R+). Embeddings, nontriviality conditions, and inclusion into a general concept of ideal spaces and generalized rearrangement invariant spaces are discussed. [ABSTRACT FROM AUTHOR]

We introduce a class of Morrey-type spaces M p , q , Ω λ , which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. This theorem is then applied to obtaining an analogue of O'Neil's inequality for convolutions and to proving the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operators. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators I α $I_{\alpha}$ in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators [ b , I α ] $[b,I_{\alpha}]$ formed by b and I α $I_{\alpha}$ are established. Also we study related problems about two-weight, weak type inequalities for I α $I_{\alpha}$ and [ b , I α ] $[b,I_{\alpha}]$ in the Morrey-type spaces and give partial results.

We develop and apply a decomposition theory for generic local Morrey-type spaces. Our result is nonsmooth decomposition, which follows from the fact that local Morrey-type spaces are isomorphic to Hardy local Morrey-type spaces in the generic case. As an application of our results, we consider the Hardy operator. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators $I_{\alpha}$ in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators $[b,I_{\alpha}]$ formed by b and $I_{\alpha}$ are established. Also we study related problems about two-weight, weak type inequalities for $I_{\alpha}$ and $[b,I_{\alpha}]$ in the Morrey-type spaces and give partial results. [ABSTRACT FROM AUTHOR]

In this paper, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator $\mathcal{M}_{\Omega,\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\Omega,\alpha}$ with rough kernels on Morrey spaces and modified Morrey spaces, respectively. This extends some recent results of Guliyev, Hasnov and Zeren; the necessary and sufficient conditions for the boundedness of $M_{\alpha}$ and $I_{\alpha}$ on modified spaces are considered. [ABSTRACT FROM AUTHOR]