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.


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
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