A Wiener algebra for the Fefferman-Phong inequality

Nicolas Lerner; Yoshinori Morimoto

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The verification of the Nirenberg-Treves conjecture

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In a series of recent papers, Nils Dencker proves that condition $(ψ)$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac{3}{2} + ϵ$ derivatives for all positive $ϵ$ ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of $\frac{3}{2}$ derivatives. ...

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This paper begins with a broad survey of the state of the art in matters of solvability for differential and pseudo-differential equations. Then we proceed with a Hilbertian lemma which we use to prove a new solvability result.