Vakhtang Kokilashvili

Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates

Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.

The Boundedness of Sublinear Operators in Weighted Morrey Spaces Defined on Spaces of Homogeneous Type

The boundedness of sublinear integral operators in weighted Morrey spaces defined on spaces of homogeneous type is established under the Muckenhoupt conditions on weights. These operators involve Hardy-Littlewood and fractional maximal operators, Calderon-Zygmund operators, potential operators, etc. The boundedness problem for commutators of sublinear operators is also studied. Applications to estimates for hypoelliptic operators in weighted Morrey spaces defined on nilpotent Lie groups are also given.

The Riemann and Dirichlet problems with data from the grand Lebesgue spaces

In Section 1, we present a solution of the following boundary value problem: find an analytic function Φon the plane cut along a closed piecewisesmooth curve Γ which is represented by a Cauchy type integral with a density from the Grand Lebesgue Space ( L^{p)},theta(Gamma)(1 < p < infty, 0 < theta < infty) ) and whose boundary values satisfy the conjugacy condition n n

Sublinear operators in generalized weighted Morrey spaces

Phi^{+}(t)=G(t)Phi^{-}(t)+g(t),quad t in Gamma

Boundedness of sublinear operators in weighted grand Morrey spaces

n nHere G and g are functions defined on Γ such that G is a piecewise continuous function, ( G(t)neq 0 ) and ( g in L^{{p}),theta}(Gamma) ) The conditions for the problem to be solvable are established and the solutions are constructed in explicit form.

Applications to Singular Integral Equations

In this chapter we introduce grand Lebesgue spaces on open sets Ω of infinite measure in ( mathbb{R}^n ), controlling the integrability of (vert{f}(x)vert^{p-varepsilon}) at infinity by means of a weight (depending also on e); in general, such spaces are different for different ways to introduce dependence of the weight on e.

Variable Exponent Hölder Spaces

Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmonic analysis, such as, e.g., maximal functions, singular and fractional integrals, Bochner–Riesz means, and so on.

One-sided Operators

The paper is devoted to weighted inequalities for positive kernel operators in variable exponent amalgam spaces. In particular, a characterization of a weight v governing the boundedness/compactness of the weighted kernel operators Kv and Kv, defined on R+ and ℝ, respectively, under the log-Hölder continuity condition on exponents of spaces is established. These operators involve, for example, weighted variable parameter fractional integrals. The results are new even for constant exponent amalgam spaces.MSC:46E30, 47B34.