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Functional calculus for a class of complex elliptic operators in dimension one (and applications to some complex elliptic equations in dimension two). (Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux).) (French) Zbl 0819.35028

Summary: In this paper, we study the functional properties of the square root of differential operators of the form \(b(x)D(a(x)D)\) where \(a(x)\) and \(b(x)\) are bounded, measurable and accretive functions, and \(D=-i{d/dx}\). We prove that \(T^{1/2}D^{-1}\) is a Calderón-Zygmund operator which depends analytically on the pair \((a,b)\). We prove that the semigroup operator \(\exp(-tT^{1/2})\) is bounded on all \(L^ p({\mathbb{R}})\) with sharp pointwise estimates on its kernel. This allows to develop a theory of Hardy spaces associated with \(T\). In a second part we prove existence and uniqueness results of the Dirichlet (and Neumann and regularity) problem for the elliptic equation \(\partial_ t^ 2u-Tu=0\) with data in \(L^ p({\mathbb{R}})\), \(1<p<+\infty\) and we obtain a weak maximum principle (\(p=+\infty\)). We also prove a weak Harnack principle for the gradient of weak solutions of some two dimensional complex elliptic equations.

MSC:

35J15 Second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
42B30 \(H^p\)-spaces
47A20 Dilations, extensions, compressions of linear operators
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