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Weighted inequalities for gradients on non-smooth domains

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Sweezy C., Wilson J.

Instytut Matematyczny Polskiej Akademii Nauk

We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, $u|_{∂Ω} = f$, with f belonging to a reasonable test class, then $(∫_{Ω} |∇u|^{q} dμ) ^{1/q} ≤ (∫_{∂Ω} |f|^{p} dν)^{1/p}$, where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ₊^{d+1}$. As in that case we attack the problem by means of Littlewood-Paley theory. However, the lack of translation invariance forces us to use a general result of Wilson, which must then be translated into the setting of homogeneous spaces. We also consider what can be proved when a strictly elliptic divergence form operator replaces the Laplacian.

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1 Sweezy C.

1 Wilson J.

1 2010

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Weighted inequalities for gradients on non-smooth domains