## $$L^p$$ 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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