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A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.
Let -div be a second order elliptic operator with real,
symmetric, bounded measurable coefficients on or on a bounded Lipschitz
domain subject to Dirichlet boundary condition. For any fixed , a necessary and
sufficient condition is obtained for the boundedness of the Riesz transform on the space. As an application, for , we
establish the boundedness of Riesz transforms on Lipschitz domains for operators
with coefficients. The range of is sharp. The closely related boundedness of
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