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Solutions avec estimations de l’équation des ondes. (Solutions with estimates of the wave equation). (French) Zbl 0595.35019
Let \(\Omega =R^{n+1}+i\Gamma \subset C^{n+1}\), where \(\Gamma\) is the spherical cone defined by \(y_ 0y_ 1>y^ 2_ 2+...+y^ 2_ n\) and \(y_ 0>0\) in \(R^{n+1}\). The wave operator in \(\Omega\) is \[ \square =4\partial^ 2/\partial z_ 0 \partial z_ 1-\partial^ 2/\partial z^ 2_ 2-...-\partial^ 2/\partial z^ 2_ n. \] When \(n=0\), \(\Omega\) is the upper complex half-plane \(\pi\) and \(\square =d/dz\); when \(n=1\), \(\Omega =\pi \times \pi\) and \(\square =\partial^ 2/\partial z_ o \partial z_ 1\). It is known that for each \(g\in H(\Omega)\) (the space of holomorphic functions in \(\Omega)\) there exists an \(f\in H(\Omega)\) such that \(\square f=g.\)
Let \(A^ p=H(\Omega)\cap L^ p(\Omega)\), \(0<p\leq +\infty\), be the Bergman class in \(\Omega\). The author proves the existence of a linear operator \(T_ p: A^ p\to A^ p\) such that \(\square T_ p=identity\) on \(A^ p\) and \(\| T_ pg\| \leq C\| g\|_{L^ p(\Omega)}\) for all \(g\in A^ p\); the first norm depends on p; it is an \(L^ q\)- norm, a Bloch norm or a Lipschitz norm. The operator \(T_ p\) is defined in terms of the Bergman kernel. The proof makes use of the atomic decomposition given by R. R. Coifman and R. Rochberg [Astérisque 77, 67-151 (1980; Zbl 0472.46040)].
Reviewer: P.Jeanquartier

35C15 Integral representations of solutions to PDEs
58J99 Partial differential equations on manifolds; differential operators
Zbl 0472.46040