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
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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