In this paper, we introduce grand weighted Herz–Morrey spaces with a variable exponent and prove the boundedness of fractional integrals on these spaces.

We show that b ∈ BMO( n) if and only if the commutator [b, Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from generalized Morrey spaces Lp,φ( n) to Lq,φ q/p ( n), where φ is non-decreasing, and 1 < p < ∞, 0 < α < n and 1/q = 1/p− α/n.

In this article a new method for moving from local to global results in variable exponent function spaces is presented. Several applications of the method are also given: Sobolev and trace embeddings; variable Riesz potential estimates; and maximal function inequalities in Morrey spaces are derived for unbounded domains.

We derive uniform weighted L 2 and Morrey-Campanato type estimates for Helmholtz Equations in a medium with a variable index which is not necessarily constant at innnity. Our technique is based on a multiplier method with appropriate weights which generalize those of Morawetz for the wave equation. We also extend our method to the wave equation.

In this paper the boundedness for a large class of multisublinear operators is established on product generalized Morrey spaces with non-doubling measures. As special cases, the corresponding results for multilinear Calderón-Zygmund operators, multilinear fractional integrals and multi-sublinear maximal operators will be obtained.

The problem of the boundedness of the fractional maximal operator Mα, 0 ≤ α < n in local Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters.

We prove the existence of conformally parametrized minimizers for parametric two-dimensional variational problems subject to partially free boundary conditions. We establish regularity of class H loc ∩C1,α, 0 < α < 1, up to the free boundary under the assumption that there exists a perfect dominance function in the sense of C.B. Morrey. Mathematics Subject Classification (2000): 49J45, 49Q10, 4...

In this article, we study the regularity criteria for the 3D NavierStokes equations involving derivatives of the partial components of the velocity. It is proved that if ∇he u belongs to Triebel-Lizorkin space, ∇u3 or u3 belongs to Morrey-Campanato space, then the solution remains smooth on [0, T ].