Stein-Weiss inequality revisit on Heisenberg group
Chuhan Sun, Zipeng Wang
TL;DR
This work extends Stein-Weiss-type two-weight inequalities to fractional integration on the Heisenberg group with kernels exhibiting Zygmund dilation. By introducing the operators ${\bf S}_{\rhoup}$ and ${\bf I}_{\alphaup\betaup\vartheta}$ and analyzing their weighted $L^p\to L^q$ bounds, the authors derive a complete set of necessary and sufficient conditions on the weights ${\gammaup},{\deltaup}$ and the dilation parameters, including a key homogeneity relation ${\rhoup\over n+1}={1\over p}-{1\over q}+{{\gammaup+\deltaup}\over 2n+2}$ and sharp bounds ${\gammaup}<{2n\over q}$, ${\deltaup}<2n\left({p-1\over p}\right)$. They reformulate the multi-parameter kernel into a convolution with ${\bf V}^{\alphaup\betaup\vartheta}$ and establish a two-weight bound under ${\alphaup+{\betaup}\over n+1}={1\over p}-{1\over q}+{{\gammaup+\deltaup}\over 2n+2}$ with a lower bound on $\vartheta$. The cone decomposition method then provides a robust approach to handle the full range of parameter signs, yielding exponential decay in the dyadic cone index and ensuring the summability required for the two-weight inequality. Overall, the paper delivers a Stein-Weiss inequality for the Heisenberg group, clarifying the precise weight constraints and offering a multi-parameter, cone-based framework for related kernels.
Abstract
We study a family of fractional integral operators defined on Heisenberg group whose kernels satisfy Zygmund dilation. We give a characterization between a two-weight norm inequality and the necessary constraints by considering the weights to be suitable powers. As a result, we obtain a Stein-Weiss inequality on Heisenberg group.