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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

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