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\(L^p\) bounds for Riesz transforms and square roots associated to second order elliptic operators. (English) Zbl 1074.35031
Let us consider the following differential operator: \[ Lu = -\operatorname{div} (A(x) \nabla u(x))\,, \] where \(A(x)\) is a matrix with complex entries verifying the uniform ellipticity condition, \[ \exists \;0<\lambda \leq \Lambda <\infty: \lambda | \xi| ^2 \leq \operatorname{Re} A \xi \cdot \overline{\xi}, \quad | A \xi \cdot \bar{\eta}| \leq \Lambda | \xi | | \eta |, \] for all \(\xi\), \(\eta\) in \(\mathbb{C}^n\). The authors show that the Riesz transform \(\nabla L^{-1/2}\) is a weak type \((p_n,p_n)\) operator where \(p_n = \frac{2n}{n+2}\). Their result contains as a particular case the square root problem of Kato that is the case \(p=2\).

MSC:
35J15 Second-order elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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