Alexander Meskhi

Weighted criteria for generalized fractional maximal functions and potentials in Lebesgue spaces with variable exponent

Necessary and sufficient conditions governing two-weight inequalities with general-type weights for fractional maximal functions and Riesz potentials with variable parameters are established in the Lebesgue spaces with variable exponent. In two-weight inequalities the right-hand side weight to the certain power satisfies the reverse doubling condition. In particular, from the general results we have: generalization of the Sobolev inequality for potentials; criteria governing the trace inequality for fractional maximal functions and potential operators; theorem of Muckenhoupt–Wheeden type (one-weight inequality) for fractional maximal functions defined on a bounded interval when the parameter satisfies the Dini–Lipschitz condition. Sawyer-type two-weight criteria for fractional maximal functions are also derived.

On Some Weighted Inequalities for Fractional Integrals on Nonhomogeneous Spaces

Necessary and sufficient conditions on a measure governing two-weight inequality with the weights of power type for fractional integrals on nonhomogeneous spaces are established. Various applications are given, in particular to potentials with Radon and Hausdorff measures.

Fractional Integrals on the Line

In this chapter, boundedness and compactness problems are investigated for various fractional integrals defined on the real line. Our main objective is to give complete descriptions of those pairs of weight functions for which these fractional integrals generate operators which are bounded or compact from one weighted Banach function space into another. This problem was studied earlier by many authors, for instance, for fractional Riemann-Liouville operators R ± when ± ≥ 1. Here the problem is studied in a more general setting. Transparent, easy to verify criteria are presented for a wider range of fractional integral orders. At the end of the chapter, some applications to nonlinear Volterra-type integral equations are given.

Multipliers of Fourier Transforms

In this chapter weighted Triebel-Lizorkin spaces are defined in a general settiing. The two-weighted criteria for fractional and singular integrals derived in the previous chapters enable us to develop a new approach to the theory of multipliers of Fourier transforms. For (L p, L q) multipliers we establish two-weight estimates involving weight functions which do not necessarily belong to the class A p . It should be noted that the derived conditions are not only sufficient but also necessary for a whole class of multipliers under consideration.

Potentials on R N

In this chapter we develop a new approach to truncated potentials. We introduce the extension of truncated potentials and prove necessary and sufficient conditions for boundedness from L p (R n ) into L v q (R n ), when 1 n/p. A generalization of Sawyer’s result [258] is presented. Then a compactness criterion for this operator is proved, and upper and lower estimates of its distance from the class of compact operators are derived.

Ball Fractional Integrals

A new class of fractional integrals connected with balls in R n was introduced and investigated by B. Rubin in [246] (see also [247]). The special interest in ball fractional integrals (BFI’s) arises from the fact that Riesz potentials I a f over a ball B may be represented by a composition of such integrals. This enables one to derive necessary and sufficient solvability conditions for the equation Iαφ = f in Lebesgue spaces with power weights and to construct the solution in closed form.