In this paper, we find sufficient conditions on functions ω 1 , ω 2 which ensure the boundedness of Riesz potentials and their commutators with BMO functions from one local complementary generalized Orlicz–Morrey spaces to the spaces . As a consequence of the boundedness of the Riesz potential, we give the boundedness the fractional maximal operator in local complementary generalized Orlicz–Morrey spaces.

We give necessary and sufficient conditions for the boundedness of the anisotropic fractional maximal operator M d α in total anisotropic Morrey spaces L d p,λ,µ (R n ).

We study the regularity theory of linear elliptic systems with discontinuous coefficients in generalized local Morrey spaces. Precisely, we obtain local regularity results for the strong solutions to 2b-order linear elliptic systems dustur where the principal coefficients Aα are assumed to be functions with vanishing mean oscillation (VMO). Keywords: elliptic systems; generalized local Morrey space; vanishing mean oscillation.

We establish a global generalized weighted Sob olev-Morrey W1Mwp ,phi-regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non -smooth domains.

In this paper, we give the definition of local variable Morrey Lorentz spaces which are a new class of functions. Also, we prove the boundedness of the Hardy Littlewood maximal operator M and Calderon Zygmund operators T on these spaces. Finally, we apply these results to the Bochner Riesz operator, identity approximation and the Marcinkiewicz operator on these spaces.

In this study, we obtain the necessary and sufficient conditions for the boundedness of the fractional maximal operator M_α in the local Morrey–Lorentz spaces M_p,q;λ^loc(ℝ^n) . We use sharp rearrangement inequalities while proving our result. We apply this result to the Schrödinger operator -Δ + V on ℝ^n , where the nonnegative potential V belongs to the reverse Hölder class B_∞(ℝ^n) . The local Morrey–Lorentz M_p,r;λ^loc(ℝ^n) → M_q,s;λ^loc(ℝ^n) estimates for the Schrödinger type operators V^γ (-Δ +V)^-β and V^γ∇ (-Δ +V)^-β are obtained.

Abstract In this paper, we prove the generalized Morrey estimates for the gradient of weak solutions to a class of nonlinear elliptic equations in a very general irregular domain. The nonlinearity ${\bf a}(x,\xi)$ is assumed to be measurable in $x$ for almost every $\xi$ and belongs to the small $BMO$ class.

We obtain the boundedness of parabolic fractional integral operators $$T_{\Omega ,\alpha }$$ with variable kernels $$\Omega (\cdot ,\cdot )$$ belonging to $$L^{\infty }({\mathbb {R}^n}) \times L^{s}({\mathbb {S}}^{n-1}), s>n/(n-\alpha )$$ , and their commutators $$[b,T_{\Omega ,\alpha }]$$ with BMO functions in variable exponent generalized Morrey spaces $$M^{p(\cdot ),\varphi }$$ and variable exponent vanishing generalized Morrey spaces $$\textrm{VM}^{p(\cdot ),\varphi }$$ . We find the sufficient conditions on the pair $$(\varphi ,\psi )$$ which ensures the boundedness of the operators $$T_{\Omega ,\alpha }$$ and $$[b,T_{\Omega ,\alpha }]$$ from $$M^{p(\cdot ),\varphi }$$ to $$M^{q(\cdot ),\psi }$$ and from $$\textrm{VM}^{p(\cdot ),\varphi }$$ to $$\textrm{VM}^{q(\cdot ),\psi }$$ .

In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(\cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${\mathcal{\widetilde{M}}}_{p(\cdot),\lambda(\cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.

In this paper, we study the boundedness of multilinear commutators of Calderón–Zygmund operators $$T_{\mathbf {b}}$$ on generalized variable exponent Morrey spaces $$M^{p(\cdot ), \varphi }$$ . Let $$\mathbf {b}=(b_1,\ldots ,b_m)$$ and $$b_i \in BMO$$ for $$i=1,\ldots ,m$$ . Then the sufficient conditions on the pair $$(\varphi _1,\varphi _2)$$ , which ensure the boundedness of the operator $$T_{\mathbf {b}}$$ from $$M^{p(\cdot ), \varphi _1}$$ to $$M^{p(\cdot ), \varphi _2}$$ , are found.

We study the continuity properties of the generalized fractional integral operator Iρ on the generalized local Morrey spaces LMp,φ{x0} and generalized Morrey spaces Mp,φ. We find conditions on the triple (φ1,φ2,ρ) which ensure the Spanne-type boundedness of Iρ from one generalized local Morrey space LMp,φ1{x0} to another LMq,φ2{x0}, 1

In this paper, the necessary and sufficient conditions are found for the boundedness of the Riesz potential I-alpha in the local Morrey- Lorentz spaces M-p,q(loc);lambda (R-n). This result is applied to the boundedness of particular operators such as the fractional maximal operator, fractional Marcinkiewicz operator and fractional powers of some analytic semigroups on the local Morrey-Lorentz spaces M-p,q(loc);lambda (R-n).

In this paper a two-weight boundedness of multidimensional Hardy operator and its dual operator acting from one weighted variable Lebesgue spaces with mixed norm into other weighted variable Lebesgue spaces with mixed norm spaces is proved. In particular, a new type two-weight criterion for multidimensional Hardy operator is obtained.

In this paper we study the potential operator I-Gamma(alpha), 0 < 1 in the modified Morrey space <(L)over tilde>(p,lambda)(Gamma) and the spaces BMO(Gamma) defined on Carleson curves Gamma. We prove that for 1 < p < (1 - lambda)/alpha the potential operator I-Gamma(alpha) is bounded from the modified Morrey space (L) over tilde (p,lambda)(Gamma) to (L) over tilde (q,lambda)(Gamma) if and in the case of infinite curve only if alpha <= 1 - 1/q <= alpha/1-lambda. Furthermore, for the limiting case (1 - lambda)/alpha <= p <= 1/alpha we show that if Gamma is an infinite Carleson curve, then the modified potential operator (L) over tilde (alpha)(Gamma) is bounded from (L) over tilde (p,lambda)(Gamma) to BMO(Gamma), and if Gamma is a finite Carleson curve, then the operator I-Gamma(alpha) is bounded from (L) over tilde (p,lambda)(Gamma) to BMO(Gamma). (C) 2017 Ivane Javakhishvili Tbilisi State University. Published by Elsevier B.V.

In this paperwe prove the boundedness of certain sublinear operators T , a, a. 0, n , generated by fractional integral operators with rough kernels . Ls( Sn- 1), s > 1, from one generalized local Morrey space LM {x0} p,.1 to another LM {x0} q,.2, 1 < p < q < 8, 1p - 1q = a n, and from the space LM {x0} 1,.1 to the weak space WLM {x0} q,.2, 1 < q < 8, 1 - 1q = a n. In the case b belongs to the local Campanato space LC {x0} p2,. and T , b, a is a linear operator, we find the sufficient conditions on the pair (.1,.2) which ensures the boundedness of the commutator operators T , b, a from LM {x0} p1,.1 to LM {x0} q,.2, 1 < p < 8, 1p = 1 p1 + 1 p2, 1q = 1p - a n, 1 q1 = 1 p1 - a n. In all cases the conditions for the boundedness of T , a are given in terms of Zygmund- type integral inequalities on (.1,.2), which do not assume any assumption on monotonicity of.1,.2 in r.

In this paper we proved the boundedness of the Hardy- Littlewood maximal operator M, the Calderon- Zygmund operators T and the maximal Calderon- Zygmund operators T on the local Morrey- Lorentz spaces M-p,q ,lambda(loc) (R-n). Finally, we give some applications of these results.

In this paper we prove a two weighted inequality for Riesz potentials \(I_{\alpha,\gamma} f\) (B-fractional integrals) associated with the Laplace-Bessel differential operator \(\Delta_{B}=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} + \sum_{j=1}^{k} \frac{\gamma _{j}}{x_{j}}\frac{\partial}{\partial x_{j}}\). This result is an analog of Heinig’s result (Indiana Univ. Math. J. 33(4):573-582, 1984) for the B-fractional integral. Further, the Stein-Weiss inequality for B-fractional integrals is proved as an application of this result.

In this paper, we investigate the boundedness of the Hilbert transform H in the local Morrey-Lorentz spaces[GRAPHICS],[GRAPHICS],[GRAPHICS]. We prove that the operator H is bounded in[GRAPHICS]under the condition[GRAPHICS],[GRAPHICS]. In the limiting case[GRAPHICS],[GRAPHICS], we prove that the operator H is bounded from the space[GRAPHICS]to the weak local Morrey-Lorentz space[GRAPHICS]. Also we show that for the limiting case[GRAPHICS],[GRAPHICS], the modified Hilbert transform[GRAPHICS]is bounded from the space[GRAPHICS]to the bounded mean oscillation space.

In this paper we prove the boundedness of the p(x)-admissible sublinear singular operators on generalized Morrey spaces Mp(·),ω(Rn) with variable exponent.

The aim of this paper is to apply the well-known ordinary differential operator to certain multivalent functions which are analytic in the certain domains of the complex plane and then to determine some criteria concerning analytic and geometric properties of the related complex functions.