Two weight Sobolev norm inequalities for smooth Calderón–Zygmund operators and doubling weights

Abstract

Let $\mu$ be a positive locally finite Borel measure on ${\mathbb {R}}^{n}$ that is doubling, and define the homogeneous $W^{s}\left( \mu \right)$-Sobolev norm squared $\left\| f\right\| _{W^{s}\left( \mu \right) }^{2}$ of a function $f\in L_{{\textrm{loc}}}^{2}\left( \mu \right)$ by

$$\begin{aligned} \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\left( \frac{f\left( x\right) -f\left( y\right) }{\left| x-y\right| ^{s}}\right) ^{2}\frac{d\mu \left( x\right) d\mu \left( y\right) }{\left| B\left( \frac{x+y}{2}, \frac{\left| x-y\right| }{2}\right) \right| _{\mu }}, \end{aligned}$$

and denote by $W^{s}\left( \mu \right)$ the corresponding Hilbert space completion (when $\mu$ is Lebesgue measure, this is the familiar Sobolev space on ${\mathbb {R}}^{n}$). We prove in particular that for $0\le \alpha <n$, and $\sigma$ and $\omega$ doubling measures on ${\mathbb {R}}^{n}$, there is a positive constant $\theta$ such for $0<s<\theta$, any smooth $\alpha$-fractional convolution singular integral $T^{\alpha }$ with homogeneous kernel that is nonvanishing in some coordinate direction, is bounded from $W^{s}\left( \sigma \right)$ to $W^{s}\left( \omega \right)$ if and only if the classical fractional Muckenhoupt condition on the measure pair holds,

$$\begin{aligned} A_{2}^{\alpha }\equiv \sup _{Q\in {\mathcal {Q}}^{n}}\frac{\left| Q\right| _{\omega }\left| Q\right| _{\sigma }}{\left| Q\right| ^{2\left( 1-\frac{\alpha }{n}\right) }}<\infty , \end{aligned}$$

as well as the Sobolev ${\textbf{1}}$-testing and ${\textbf{1}}^{*}$-testing conditions for the operator $T^{\alpha }$,

$$\begin{aligned} \left\| T_{\sigma }^{\alpha }{\textbf{1}}_{I}\right\| _{W^{s}\left( \omega \right) }\le & {} {\mathfrak {T}}_{T^{\alpha }}\left( \sigma ,\omega \right) \sqrt{\left| I\right| _{\sigma }}\ell \left( I\right) ^{-s},\quad I\in {\mathcal {Q}}^{n}, \\ \left\| T_{\omega }^{\alpha ,*}{\textbf{1}}_{I}\right\| _{W^{s}\left( \sigma \right) ^{*}}\le & {} {\mathfrak {T}}_{T^{\alpha ,*}}\left( \omega ,\sigma \right) \sqrt{\left| I\right| _{\omega }} \ell \left( I\right) ^{s},\quad I\in {\mathcal {Q}}^{n}, \end{aligned}$$

taken over the family of indicator test functions $\left\{ {\textbf{1}} _{I}\right\} _{I\in {\mathcal {P}}^{n}}$. Here ${\mathcal {Q}}^{n}$ is the collection of all cubes with sides parallel to the coordinate axes, and $W^{s}\left( \mu \right) ^{*}$ denotes the dual of $W^{s}\left( \mu \right)$ determined by the usual $L^{2}\left( \mu \right)$ bilinear pairing, which we identify with a dyadic Sobolev space $W_{{\textrm{dyad}} }^{-s}\left( \mu \right)$ of negative order. The sufficiency assertion persists for more general singular integral operators $T^{\alpha }$.