# A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

• Volume: 90, Issue: 2, page 105-130
• ISSN: 0066-2216

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## Abstract

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Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\stackrel{\to }{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the ${L}^{q}\left(\Omega ,d\mu \right)$ norm of |∇u| is dominated by the ${L}^{p}\left(\Omega ,dv\right)$ norms of $div\stackrel{\to }{f}$ and $|\stackrel{\to }{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

## How to cite

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Caroline Sweezy. "A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem." Annales Polonici Mathematici 90.2 (2007): 105-130. <http://eudml.org/doc/280936>.

@article{CarolineSweezy2007,
abstract = {Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu = div \vec\{f\}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^\{q\}(Ω,dμ)$ norm of |∇u| is dominated by the $L^\{p\}(Ω,dv)$ norms of $div \vec\{f\}$ and $|\vec\{f\}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.},
author = {Caroline Sweezy},
journal = {Annales Polonici Mathematici},
keywords = {elliptic equations; Lipschitz domains; Littlewood-Paley type inequalities},
language = {eng},
number = {2},
pages = {105-130},
title = {A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem},
url = {http://eudml.org/doc/280936},
volume = {90},
year = {2007},
}

TY - JOUR
AU - Caroline Sweezy
TI - A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem
JO - Annales Polonici Mathematici
PY - 2007
VL - 90
IS - 2
SP - 105
EP - 130
AB - Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu = div \vec{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^{q}(Ω,dμ)$ norm of |∇u| is dominated by the $L^{p}(Ω,dv)$ norms of $div \vec{f}$ and $|\vec{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.
LA - eng
KW - elliptic equations; Lipschitz domains; Littlewood-Paley type inequalities
UR - http://eudml.org/doc/280936
ER -

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