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On $$L^p$$ estimates for square roots of second order elliptic operators on $$\mathbb R^n$$. (English) Zbl 1107.42003
Consider $$A$$ an $$n\times n$$ matrix of $$L^\infty({\mathbb R}^n)$$ functions satisfying the accretivity condition $\lambda | \xi |^2 \leq \operatorname{Re}\, A\xi \cdot\overline{\xi}, \; | A \xi\cdot \overline{\zeta} | \leq \Lambda | \xi| \, |\zeta|, \quad \xi,\zeta\in {\mathbb C}$ for some $$\lambda$$, $$\Lambda$$ such that $$0<\lambda\leq \Lambda <\infty$$. Then $L f=-\text{div } (A\nabla f)$ can be seen as a maximal accretive operator in $$L^2({\mathbb R}^n)$$, and the square root $$L^{1/2}$$ is defined in the sense of maximal accretive operators. The Kato conjecture, that $\| L^{1/2} f\|_{L^2({\mathbb R}^n)}\sim \| \nabla f \|_{L^2({\mathbb R}^n)}, \quad n\geq 1$ has a positive answer (see R. R Coifman, A. McIntosh and Y. Meyer [Ann. Math. (2) 116, 361–387 (1982; Zbl 0497.42012)] for $$n=1$$, S. Hofmann and A. McIntosh [Publ. Mat., Barc. 2002, Spec. Vol., 143–160 (2002; Zbl 1020.47031)] for $$n=2$$, and P. Auscher, S.  Hofmann, M. Lacey, Michael, A. McIntosh and Ph. Tchamitchian [Ann. Math. (2) 156, No. 2, 633–654 (2002; Zbl 1128.35316)] for general $$n$$).
In the paper under review the norms of $$L^{1/2} f$$ and $$\nabla f$$ in $$L^p$$ are compared, for some $$p\neq 2$$, as a part of the programme initialized by the present author and Ph. Tchamitchian in [Square root problem for divergence operators and related topics (Astérisque 249, Société Mathématique de France, Paris) (1998; Zbl 0909.35001)]. In fact one proves that $\| L^{1/2} f\|_{L^p({\mathbb R}^n)}\sim \| \nabla f \|_{L^p({\mathbb R}^n)}$ whenever $\sup \big(1, \frac{2n}{n+4} -\epsilon\big) <p< \frac{2n}{n-2} +\epsilon.$ The method involves proving weak estimates by using properties of the semigroup $$\text{e}^{-tL}$$ and relies on a Calderón-Zygmund decomposition for locally integrable functions with $$L^p$$ gradients. The method is also generalized to high-order operators.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory 35J15 Second-order elliptic equations 35J45 Systems of elliptic equations, general (MSC2000) 47B44 Linear accretive operators, dissipative operators, etc. 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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