Sufficiency for positive integer power weights

Establish sufficient conditions for the positive-exponent power-weighted Stein–Weiss inequality in weighted L^1 when the weight exponent a is positive and an integer, including the cases not covered by the stated vanishing-moment and polyharmonic-moment theorems.

Background

The paper proves necessary moment conditions for the power-weighted Stein–Weiss inequality, using ordinary vanishing moments in the generic case and polyharmonic moment conditions when the Riesz-potential order $\lambda$ is an even integer. The stated sufficiency results, however, are restricted to non-integer positive exponents $a$.

The authors explicitly note that their necessity proposition also applies when $a$ is an integer, but they do not state a corresponding sufficiency theorem in that setting. Thus, determining sufficient conditions for positive integer weight exponents remains unresolved within the paper.