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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 \(p\).

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.)
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