# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Auscher, P., Regularity theorems and heat kernels for elliptic operators.J. London Math. Soc. (2), 54 (1996), 284–296. · Zbl 0863.35020  Auscher, P. &Tchamitchian, P.,Square Root Problem for Divergence Operators and Related Topics. Astérisque, 249. Soc. Math. France, Paris, 1998. · Zbl 0909.35001  Christ, M.,Lectures on Singular Integral Operators. CBMS Regional Conf. Ser. in Math., 77. Conf. Board Math. Sci., Washington, DC, 1990. · Zbl 0745.42008  Coifman, R. R., Deng, D. G. &Meyer, Y., Domains de la racine carrée de certains opérateurs différentiels accrétifs.Ann. Inst. Fourier (Grenoble), 33 (1983), 123–134.  Coifman, R. R., McIntosh, A. &Meyer, Y., L’intégrale de Cauchy définit un opérateur borné surL 2 pour les courbes lipschitziennes.Ann. of Math. (2), 116 (1982), 361–387. · Zbl 0497.42012  Coifman, R. R. &Meyer, Y., Nonlinear harmonic analysis, operator theory and P.D.E., inBeijing Lectures in Harmonic Analysis (Beijing, 1984), pp. 3–45. Ann. of Math. Stud., 112. Princeton Univ. Press, Princeton, NJ, 1986.  Dahlberg, B. E. J. &Kenig, C. E.,Harmonic Analysis and Partial Differential Equations. Department of Mathematics, Chalmers University of Technology and the University of Göteborg, Göteborg, 1985.  Dorronsoro, J. R., A characterization of potential spaces.Proc. Amer. Math. Soc., 95 (1985), 21–31. · Zbl 0577.46035  Fabes, E. B., Jerison, D. S. &Kenig, C. E., Multilinear square functions and partial differential equations.Amer. J. Math., 107 (1985), 1325–1368. · Zbl 0655.35007  – Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure.Ann. of Math. (2), 119 (1984), 121–141. · Zbl 0551.35024  Hofmann, S. &Lewis, J. L.,The Dirichlet Problem for Parabolic Operators with Singular Drift Terms. Mem. Amer. Math. Soc., 151 (719). Amer. Math. Soc., Providence, RI, 2001. · Zbl 1149.35048  Hofmann, S. & McIntosh, A., The solution of the Kato problem in two dimensions. Unpublished manuscript. · Zbl 1020.47031  Journé, J.-L., Remarks on Kato’s square-root problem, inMathematical Analysis (El Escorial, 1989).Publ. Math., 35 (1991), 299–321.  Kato, T., Fractional powers of dissipative operators.J. Math. Soc. Japan, 13 (1961), 246–274. · Zbl 0113.10005  – Integration of the equation of evolution in a Banach space.J. Math. Soc. Japan, 5 (1953), 208–234. · Zbl 0052.12601  Kenic, C. &Meyer, Y., Kato’s square roots of accretive operators and Cauchy kernels on Lipschitz curves are the same, inRecent Progress in Fourier Analysis (El Escorial, 1983), pp. 123–143. North-Holland Math. Stud., 111. North-Holland Amsterdam, 1985.  Lacey, M., Personal communication.  Lewis, J. L. &Murray, M. The Method of Layer Potentials for the Heat Equation in Time-Varying Domains. Mem. Amer. Math. Soc., 114 (545). Amer. Math. Soc., Providence, RI, 1995. · Zbl 0826.35041  McIntosh, A., Square roots of operators and applications to hyperbolic PDEs, inMini-conference on Operator Theory and Partial Differential Equations (Camberra, 1983), pp. 124–136. Proc. Centre Math. Anal. Austral. Nat. Univ., 5. Austral. Nat. Univ., Canberra, 1984.  – On the comparability ofA 1/2 andA *1/2.Proc. Amer. Math. Soc., 32 (1972), 430–434. · Zbl 0248.47020  – On representing closed accretive sesquilinear forms as (A 1/2 u,A *1/2 v) inNonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. III (Paris, 1980/1981), pp. 252–267. Res. Notes in Math., 70, Pitman, Boston, MA-London, 1982.  – The square root problem for elliptic operators: a survey, inFunctional-Analytic Methods for Partial Differential Equations (Tokyo, 1989), pp. 122–140. Lecture Notes in Math., 1450. Springer-Verlag, Berlin, 1990.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.