# Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy; J. Michael Wilson

- 2010

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topCaroline Sweezy, and J. Michael Wilson. Weighted inequalities for gradients on non-smooth domains. 2010. <http://eudml.org/doc/285995>.

@book{CarolineSweezy2010,

abstract = {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.},

author = {Caroline Sweezy, J. Michael Wilson},

keywords = {elliptic equations; boundary value problems; weighted norm inequalities; Littlewood-Paley theory; Lipschitz domains},

language = {eng},

title = {Weighted inequalities for gradients on non-smooth domains},

url = {http://eudml.org/doc/285995},

year = {2010},

}

TY - BOOK

AU - Caroline Sweezy

AU - J. Michael Wilson

TI - Weighted inequalities for gradients on non-smooth domains

PY - 2010

AB - 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.

LA - eng

KW - elliptic equations; boundary value problems; weighted norm inequalities; Littlewood-Paley theory; Lipschitz domains

UR - http://eudml.org/doc/285995

ER -

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