On the equivalence of weighted inequalities for a class of operators

Abstract

A characterization is obtained on weight function u so that T : L + p → L + q,u is bounded for 1 < p < ∞ and 0 < q < ∞, where T are integral operators and related maximal operators, and for 0 < p, q < ∞, where T are geometric mean operators and related geometric maximal operators. The equivalence of such weighted inequalities for these operators are established.

Kernel operators on the upper half-space: boundedness andcompactness criteria

“…A similar problem for Hardy-type transforms on star-shaped regions was investigated in [27]. It should be emphasized that the results of [20] were generalized in [16] for kernel operators defined on star-shaped regions.…”