Potential Operators on Cones of Nonincreasing Functions

Abstract EN

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dt and (ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτ on the cone of nonincreasing functions are derived.

In the case of ℐα1,α2, we assume that the right-hand side weight is of product type.

The same problem for other mixed-type double potential operators is also studied.

Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.