Weighted Sobolev inequalities for the Riesz potentials and transforms

Abstract

Thanks to isomorphism results for Laplace's operator in weighted Sobolev spaces, we derive some applications. Various related spaces, intersection of Hardy spaces and weighted Sobolev spaces, are also considered. We show that the Riesz potentials spaces, consisting of all Riesz potentials $I_\alpha$ of order $\alpha$ of $L^p$ functions, with $1<p<\infty$, are equivalent to some appropriate weighted Sobolev spaces. It is known that if $ 1< p\le q<\infty$ and if $w$ and $v^{1-p'}$ satisfy the reverse doubling condition, then $I_\alpha$ satisfy the following continuity property: $$ \Big (\int_{\mathbb R }|I_\alpha f(x)|^q w(x) dx\Big )^{1/q}\le C \Big (\int_{\mathbb R }|f(x)|^p v(x) dx\Big )^{1/p}, \tag*{$(\star)$} $$ if and only if the weights satisfy the $(A_{p,q})$ condition $$ \frac {1}{|Q|^{1-\alpha/ {\rm N}}} \Big (\int_Q w dx\Big )^{1/q}\Big (\int_Q v^{1-p'} dx\Big ) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_{p,q})$} $$ A similar property holds for the Riesz transforms $R_j$ with $j = 1,\ldots , {\scriptstyle N}$: $$ \int_{\mathbb R }|R_j f(x)|^p u(x) dx\le C \int_{\mathbb R }|f(x)|^p u(x) dx, \tag*{$(\star\star)$} $$ if and only if the weight $u$ satisfies the Muckenhoupt $A_p$ condition: $$ (\dfrac{1}{|Q|}\int_Q u dx)^{1/p}(\dfrac{1}{|Q|}\int_Q u^{1-p'} dx) ^{1/p'}\le C \ \ \ \hbox{ for all cubes } Q\subset\mathbb R . \tag*{$(A_p)$} $$ Here, $f$ belongs to a weighted Sobolev space $W$. We show that even if these conditions are not satisfied, the continuity property of $I_\alpha$ and $R_j$ still holds in appropriate subspaces of $W$. By interpolation, a weighted version of the Sobolev imbedding theorem is also proved.