## 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

### Citations:

Zbl 0497.42012; Zbl 1020.47031; Zbl 0909.35001; Zbl 1128.35316
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