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A Wiener algebra for the Fefferman-Phong inequality. (English) Zbl 1122.35163
The object of this paper is to relax the hypothesis needed to have a Fefferman-Phong inequality. The result involves a Wiener-type algebra introduced by Sjöstrand (1994), containing \(S^0_{0,0}\). Let \(\mathbb{Z}^{2n}\) be the standard lattice in \(\mathbb{R}_x^{2n}\) and let \(1= \sum_{j\in\mathbb{Z}^{2n}}\chi_0(X-j)\), \(\chi_0 \in{\mathcal C}_0^\infty (\mathbb{R}^{2n})\), be a partition of unity. We note \(\chi_j(x)= \psi_0(x-j)\). Let \(a\in{\mathcal S}'(\mathbb{R}^{2n})\). We shall say that \(a\in{\mathcal A}\) whenever \(\omega_a\in L^1(\mathbb{R}^{2n})\), with \(\omega_a(\Xi)= \sup_{j\in \mathbb{Z}^{2n}}|{\mathcal F}(\chi_ja)(\Xi)|\). \(\mathcal A\) is a Banach algebra for the multiplication with the norm \(\|a\|_{\mathcal A}=\|\omega_a \|_{L^1(\mathbb{R}^{2n})}\). We have \(S^x_{0,0}\subset{\mathcal A}\subset {\mathcal C}^0(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^{2n})\). The main result is the following:
There exists a constant \(C\) (depending only on the dimension \(n)\) such that for all nonnegative functions \(a\) defined on \(\mathbb{R}^{2n}\) satisfying \(a^{(4)}\in{\mathcal A}\), the Weyl operator \(a^w\) is semi-bounded from below and, more precisely, satisfies \[ a^w+C\| a^{(4)}\|_{\mathcal A}\geq 0. \] As a byproduct, the number of derivatives of \(a\) needed to control the constant in the Fefferman-Phong inequality is \(4+2n+\varepsilon\).

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
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