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Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators. (English) Zbl 1163.35346
From the introduction: Let \(A\) be an \((n\times n)\)-matrix of complex \(L^\infty\)-coefficients, defined on \(\mathbb R^n\), with \(\|A\|_\infty\leq\Lambda\), and satisfying the ellipticity (or “accretivity”) condition (1) \(\lambda|\xi^2|\leq \text{Re}\langle A\xi,\xi\rangle\leq \Lambda|\xi|^2\), for \(\xi\in\mathbb C^n\) and for some \(\lambda,\Lambda\) such that \(0<\lambda\leq\Lambda<\infty\). Here \(\langle\cdot,\cdot\rangle\) denotes and the usual inner product in \(\mathbb C^n\), so that \(\langle A\xi,\xi\rangle\equiv \sum_{i,j} A_{ij}(x)\xi_j\cdot\overline{\xi}_i\). We define a divergence-form operator (2) \(Lu\equiv-\text{div} (A(x)\nabla u)\), which we interpret in the usual weak sense via a sesquilinear form.
The accretivity condition (1) enables one to define an accretive square root \(\sqrt{L}\equiv L^{1/2}\), and a fundamental question is to determine when one can solve the “square-root problem”, i.e., to establish the estimate
\[ \big\|\sqrt{L}f\big\|_{L^2(\mathbb R^n)}\leq C\|\nabla f\|_{L^2(\mathbb R^n)}, \tag{3} \]
A long-standing open problem, essentially posed by Kato (but refined by McIntosh), is the following:
Question 1. Let \(A_z\), \(z\in\mathbb C\), denote a family of accretive matrices as above, which in addition are holomorphic in \(z\), and self-adjoint for real \(z\). Let
\[ L_z\equiv-\operatorname{div} A_z(x)\nabla. \]
Is \(L_z^{1/2}\) holomorphic in \(z\), in a neighborhood of \(z=0\)?
In fact, Kato actually formulated this question for a more general class of abstract accretive operators. A counterexample to the abstract problem was found by McIntosh. However, it has been pointed out that, in posing the problem, Kato had been motivated by the special case of elliptic differential operators, and by the applicability of a positive result, in that special case, to the perturbation theory for hyperbolic evolution equations. A positive answer to the question posed above can be restated as
Conjecture 1.4. The estimate (3) holds in a complex neighborhood in \(L^\infty\) of any self-adjoint matrix \(A\) satisfying (1); i.e., (3) holds for the operator \(\widetilde{L}\) (as in (2)) associated to any complex-valued matrix \(\widetilde{A}\), whenever \(\|\widetilde{A}-A\|_\infty\leq \varepsilon_0\), with \(\varepsilon_0\) depending only on \(n\), \(\lambda\) and \(\Lambda\).
In the present paper, we present the solution to Conjecture 1.4, in all dimensions, at least in the case that \(A\) is real, symmetric. Our main result is:
Theorem 1.6. Let \(n\geq 1\). Suppose that \(A\) is a real, symmetric \((n\times n)\)-matrix of \(L^\infty\)-coefficients satisfying (1). Then there exists \(\varepsilon_0\equiv \varepsilon_0(n,\lambda,\Lambda)\) such that for any complex-valued \((n\times n)\)-matrix \(\widetilde{A}\), with \(\|A-\widetilde{A}\|_\infty\leq \varepsilon_0\), the operator
\[ \widetilde{L}\equiv- \text{div}(\widetilde{A}(x)\nabla) \]
satisfies (3), with a constant \(C\) which depends only on \(n,\lambda,\Lambda\). Moreover,
\[ \Big\|\sqrt{\widetilde{L}}f- \sqrt{L}f\Big\|_{L^2(\mathbb R^n)}\leq C(n,\lambda,\Lambda)\big\|A-\widetilde{A}\big\|_\infty \|\nabla f\|_{L^2(\mathbb R^n)}. \]

MSC:
35J15 Second-order elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
47B44 Linear accretive operators, dissipative operators, etc.
47F05 General theory of partial differential operators
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