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

Let μ$$\mu $$ be a positive locally finite Borel measure on Rn$${\mathbb {R}}^{n}$$ that is doubling, and define the homogeneous Wsμ$$W^{s}\left( \mu \right) $$-Sobolev norm squared fWsμ2$$\left\| f\right\| _{W^{s}\left( \mu \right) }^{2}$$ of a function f∈Lloc2μ$$f\in L_{{\textrm{loc}}}^{2}\left( \mu \right) $$ by ∫Rn∫Rnfx-fyx-ys2dμxdμyBx+y2,x-y2μ,$$\begin{aligned} \int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\left( \frac{f\left( x\right) -f\left( y\right) }{\left| x-y\right| ^{s}}\right) ^{2}\frac{d\mu \left( x\right) d\mu \left( y\right) }{\left| B\left( \frac{x+y}{2}, \frac{\left| x-y\right| }{2}\right) \right| _{\mu }}, \end{aligned}$$and denote by Wsμ$$W^{s}\left( \mu \right) $$ the corresponding Hilbert space completion (when μ$$\mu $$ is Lebesgue measure, this is the familiar Sobolev space on Rn$${\mathbb {R}}^{n}$$). We prove in particular that for 0≤α