Abstract
We construct extensions of Varopolous type for functions f∈BMO(E), for any uniformly rectifiable set E of codimension one. More precisely, let Ω⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let σ≔Hn⌊∂Ω denote the surface measure on ∂Ω. We show that if f∈BMO(∂Ω,dσ) with compact support on ∂Ω, then there exists a smooth function V in Ω such that |∇V(Y)|dY is a Carleson measure with Carleson norm controlled by the BMO norm of f, and such that V converges in some non-tangential sense to f almost everywhere with respect to σ. Our results should be compared to recent geometric characterizations of Lp-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all f∈Cc(∂Ω), a harmonic extension u, with |∇u(Y)|2dist(Y,∂Ω)dY a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition.
Highlights
We complement recent results related to geometric characterizations of solvability of Dirichlet problems, by showing that an extension property for BMO functions, first proved by Varopoulos in the half-space [41,42], remains true even in settings where harmonic extension of BMO boundary data (i.e., BMO-solvability of the Dirichlet problem) may fail: we show in the present paper that the Varopoulos extension property holds always for UR sets of codimension 1
Azzam, the first author, Martell, Mourgoglou and Tolsa [4] have presented a geometric characterization of quantitative scale-invariant absolute continuity of harmonic measure with respect to the surface measure
The following conditions are equivalent: (1) ∂Ω is UR and Ω satisfies the weak local John condition, (2) harmonic measure belongs to the class weak-A∞ with respect to the surface measure σ := Hn ∂Ω on ∂Ω, (3) the Dirichlet problem is Lp-solvable for some p < ∞, (4) the Dirichlet problem is BMO-solvable
Summary
Ω ⊂ Rn+1 will always be an open set with non-empty d-dimensional ADR boundary ∂Ω (see Definition 2.11). We say that Ω satisfies the weak local John condition if there exist constants λ ∈ (0, 1), θ ∈ (0, 1] and R ≥ 2 such that for every X there exists a Borel set F ⊂ ΔX := B(X, Rδ(X)) ∩ ∂Ω such that σ(F ) ≥ θσ(ΔX ) and for every y ∈ F there is a λ-carrot path connecting y to X. We say that Ω satisfies the corkscrew condition if there exists a uniform constant c such that for every surface ball Δ := Δ(x, r) with x ∈ ∂Ω and 0 < r < diam(∂Ω) there exists a point XΔ ∈ Ω such that B(XΔ, cr) ⊂ B(x, r) ∩ Ω, Definition 2.13 (UR). The constants in the Harnack chain, corkscrew, and ADR conditions are referred to collectively as the chord-arc constants
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