The product Stein–Weiss theorem
Volume 256 / 2021
Abstract
We show that the Stein–Weiss extension of the classical Hardy–Littlewood–Sobolev inequality to power weights carries over to the $2$-parameter setting with nonproduct power weights and product fractional integrals $$ I_{\alpha ,\beta }^{m,n}f\left ( x,y\right ) =\mathop{\int\int} _{\mathbb {R}^{m}\times \mathbb {R}^{n}}| x-u| ^{\alpha -m}| y-t| ^{\beta -n}f\left ( u,t\right ) \,du\,dt $$ in $\mathbb {R}^{m}\times \mathbb {R}^{n}$.
We also show that almost none of the other standard $1$-parameter results carry over without additional side conditions on the weights: for example, the two-tailed rectangle characteristic $$ \widehat {A}_{p,q}^{\left ( \alpha ,\beta \right ) ,\left ( m,n\right ) }\left ( v,w\right ) =\sup _{I,J}| I| ^{\frac {\alpha }{m}-1}| J| ^{\frac {\beta }{n}-1} \left ( \mathop{\int\int} _{\mathbb {R}^{m}\times \mathbb {R}^{n}}( \widehat {s}_{I\times J}w) ^{q} \right ) ^{\frac {1}{q}}\left ( \mathop{\int\int} _{\mathbb {R}^{m}\times \mathbb {R}^{n}}( \widehat {s}_{I\times J}v^{-1}) ^{p’} \right ) ^{\frac {1}{p’}} $$ fails to control the operator norm of $I_{\alpha ,\beta }:L^{p}\left ( v^{p}\right ) \rightarrow L^{q}\left ( w^{q}\right ) $ in general, even in the presence of testing or $T1$ type conditions.
Finally, we characterize the one-weight inequality in terms of the characteristic, and compute its optimal power.
