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

Let μ be a positive locally finite Borel measure on Rⁿ that is doubling, and define the homogeneous W^s(μ) -Sobolev norm squared ∥f∥Ws(μ)2 of a function f∈Lloc2(μ) by ∫Rn∫Rn(f(x)-f(y)|x-y|s)2dμ(x)dμ(y)|B(x+y2,|x-y|2)|μ,and denote by W^s(μ) the corresponding Hilbert space completion (when μ is Lebesgue measure, this is the familiar Sobolev space on Rⁿ). We prove in particular that for 0 ≤ α< n, and σ and ω doubling measures on Rⁿ, there is a positive constant θ such for 0 < s< θ, any smooth α-fractional convolution singular integral T^α with homogeneous kernel that is nonvanishing in some coordinate direction, is bounded from W^s(σ) to W^s(ω) if and only if the classical fractional Muckenhoupt condition on the measure pair holds, A2α≡supQ∈Qn|Q|ω|Q|σ|Q|2(1-αn)<∞,as well as the Sobolev 1-testing and 1^∗-testing conditions for the operator T^α, ∥Tσα1I∥Ws(ω)≤TTα(σ,ω)|I|σℓ(I)-s,I∈Qn,∥Tωα,∗1I∥Ws(σ)∗≤TTα,∗(ω,σ)|I|ωℓ(I)s,I∈Qn,taken over the family of indicator test functions {1I}I∈Pn. Here Qⁿ is the collection of all cubes with sides parallel to the coordinate axes, and W^s(μ) ^∗ denotes the dual of W^s(μ) determined by the usual L²(μ) bilinear pairing, which we identify with a dyadic Sobolev space Wdyad-s(μ) of negative order. The sufficiency assertion persists for more general singular integral operators T^α.