We investigate geometric curvature energies on closed curves involving integral versions of the Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding theorems, which may be viewed as counterparts to respective results on one-dimensional sets in the context of harmonic analysis. Mathematics Subject Classification (2010): 28A75 (primary); 53A04,...

In this paper, we consider the problem of boundedness of Hausdorff operator on weighted central Morrey spaces. In particular, we obtain sharp bounds for Hausdorff operators on power weighted central Morrey spaces. Analogous results for the commutators of Hausdorff operators when the symbol functions belong to weighted central-BMO spaces are obtained as well.

We establish a decomposition of Besov-Morrey spaces in terms of smooth “wavelets” obtained from a Littlewood-Paley partition of unity, or more generally molecules concentrated on dyadic cubes. We show that an expansion in atoms supported on dyadic cubes holds. We study atoms in Morrey spaces and prove a Littlewood-Paley theorem. Our results extend those of M. Frazier and B. Jawerth for Besov sp...

This paper focuses on the bounds of weighted multilinear Hardy operators on the product Herz spaces and the product Morrey-Herz spaces, respectively. We present a sufficient condition on the weight function that guarantees weighted multilinear Hardy operators to be bounded on the product Herz spaces. And the condition is necessary under certain assumptions. Finally, we extend the obtained resul...

In this paper, we give the necessary and sufficient conditions for boundedness of Hardy-Ces?ro operators on some weighted function spaces such as central Morrey, local non-local Herz Morrey-Herz type with variable exponent.

We obtain global bounds in Lorentz-Morrey spaces for gradients of solutions to a class of quasilinear elliptic equations with low integrability data. The results are then applied to obtain sharp existence results in the framework of Morrey spaces for Riccati type equations with a gradient source term having growths below the natural exponent of the operator involved. A special feature of our re...

Abstract. We apply wavelets to study the generalized local Morrey-Campanato spaces Mφ,p(R) and their preduals. As applications, we characterize the multipliers on Mφ,p(R) and the stability of these spaces under the perturbation of Calderón-Zygmund operators. Our results indicate that there exist some Mφ,p(R) without unconditional basis. This fact shows that Mφ,p(R) have some different character...

We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A’priori estimates in Sobolev and Sobolev– Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BMO and H...

In this paper we consider the Spanne type boundedness of sublinear operators and prove the Adams type boundedness theorems for these operators and also give BMO (bounded mean oscillation space) estimates for their commutators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are formulated in terms of Zygmund type integral inequalities. Based on the properties of the...

We will prove that under certain conditions on the parameters operators T+f=max(f,0) and Tf=|f| are bounded mappings Triebel–Lizorkin–Morrey Besov–Morrey spaces. Moreover we show some of mentioned before also necessary. Furthermore in many cases spaces do not have Fubini property.