Parabolic oblique derivative problem in generalized Morrey spaces

We study the regularity of the solutions of the oblique derivative problem for linear uniformly parabolic equations with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized Morrey space than the strong solution belongs to the corresponding generalized Sobolev-Morrey space

Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized local Morrey spaces

WOS: 000350676000002In this paper, we will study the boundedness of a large class of sublinear operators with rough kernel T-Omega on the generalized local Morrey spaces LM rho,phi{x0}, for s' 1 are homogeneous of degree zero. In the case when b is an element of LCp,lambda{x0} is a local Campanato spaces, 1 <p < infinity, and T-Omega,T-b be is a sublinear commutator operator, we find the sufficient conditions on the pair (phi(1),phi(2)) which ensures the boundedness of the operator T-Omega,T-b, from one generalized local Morrey space LMp,phi 1{x0} to another LMp,phi 2{x0}. In all cases the conditions for the boundedness of T-Omega are given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)), which do not make any assumptions on the monotonicity of phi(1), phi(2) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudo-differential operators, Littlewood-Paley operators, Marcinkiewicz operators, and Bochner-Riesz operators.Science Development Foundation under the President of the Republic of AzerbaijanScience Development Foundation (SDF) - Azerbaijan [EIF-2013-9(15)-46/10/1]; Ahi Evran University Scientific Research ProjectsAhi Evran University [PYO.FEN.4001.13.012]The authors would like to express their gratitude to the referees for their very valuable comments and suggestions. The research of V Guliyev was partially supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-2013-9(15)-46/10/1 and by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4001.13.012)

Generalized fractional maximal operator on generalized local morrey spaces

In this paper, we study the boundedness of generalized fractional maximal operator M-rho on generalized local Morrey spaces LMP,phi{x0} and generalized Morrey spaces M-p,M-phi, including weak estimates. Firstly, we prove the Spanne type boundedness of M-rho from the space LMp,phi 1{x0} to another LMq,phi 2{x0} 1 < p < q < infinity and from LM1,phi 1{x0} to the weak space WLMq,phi 2{x0} for p = 1 and 1 < q < infinity. Secondly, we prove the Adams type boundedness of M-rho from the space M-p,M-phi 1/p to another M-q,M-phi 1/q for 1 < p < q < infinity and from to the weak M-1,M-phi, space M-q,M-phi 1/q for p = 1 and 1 < q < infinity. In all cases the conditions for the boundedness of M-rho are given in terms of supremal-type integral inequalities on (phi(1), phi(2), rho) and (phi, rho), which do not assume any assumption on monotonicity of phi(1) (x, r), phi(2)(x, r) and phi(x, r) in r

Two weighted inequalities for fractional integrals on Laguerre hypergroup

Let K = [0,infinity) x R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group, vertical bar.vertical bar its homogeneous norm and Q its homogeneous dimension. In this paper we prove the two weighted inequality for fractional integrals I-beta on K. The obtained result is an analog of the Heinig result [Heinig HP. Weighted norm inequalities for classes of operators. Indiana Univ Math J. 1984;33(4):573-582] for fractional integrals on Laguerre hypergroup. Furthermore, the Stein-Weiss inequality for I-beta is proved as an application of this result.Presidium of Azerbaijan National Academy of SciencesThe research of the second author was supported by the grant of Presidium of Azerbaijan National Academy of Sciences 2015