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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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