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Bounds of Riesz transforms on $$L^p$$ spaces for second order elliptic operators. (English) Zbl 1068.47058
The second-order elliptic operator of divergence form ${\mathcal L}= -\text{div}(A(x)\nabla)\quad\text{on }\Omega= \mathbb{R}^n,$ is considered on a bounded open set of $$\mathbb{R}^n$$. In the case of bounded domains, a Dirichlet condition $$u= 0$$ is imposed on $$\partial\Omega$$. Assuming that the differential operator satisfies certain conditions regarding its coefficients, where the Riesz transform $$\nabla({\mathcal L})^{-1/2}$$ is bounded on $$L^p(\Omega)$$ for $$1< p< 2+\varepsilon$$, the boundedness of Riesz transforms is established on Lipschitz domains for an optimal range of $$p$$.

##### MSC:
 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 35J15 Second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
##### Keywords:
Riesz transform; elliptic operator; Lipschitz domain
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