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) …