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, , with f belonging to a reasonable test class, then , where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on . 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.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm471-0-1, author = {Caroline Sweezy and J. Michael Wilson}, title = {Weighted inequalities for gradients on non-smooth domains}, series = {GDML\_Books}, year = {2010}, zbl = {1210.35060}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm471-0-1} }
Caroline Sweezy; J. Michael Wilson. Weighted inequalities for gradients on non-smooth domains. GDML_Books (2010), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm471-0-1/