Espaces fonctionnels associés au calcul de Weyl-Hörmander

Abstract

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Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces

“…For such pseudodifferential calculi, adapted Sobolev spaces were introduced in [BC94]. Here, we use such Sobolev spaces; the semi-classical setting we follow allows us however to introduce such spaces without relying on the more intricate analysis of [BC94]. These calculi allow us to perform a microlocal analysis of the conjugated parabolic operator with a characterisation of the behavior of the roots of the principal symbol.…”

“…Since we face parabolic operators here, such refined calculi are needed to compare the action of the time derivative and the second-order space derivatives. For such pseudodifferential calculi, adapted Sobolev spaces were introduced in [BC94]. Here, we use such Sobolev spaces; the semi-classical setting we follow allows us however to introduce such spaces without relying on the more intricate analysis of [BC94].…”

On the Well-Posedness for a Class of Pseudo-Differential Parabolic Equations

“…Here we adopt the Beals's definition for simplicity in the presentation of the basic theory. Comprehensive treatments on Sobolev spaces in this setting can be found in [3,20]. Definition 2.9.…”

“…Indeed, that is a consequence of the existence (cf. [3]), for every g-weight M of a one parameter group (for the operation #) of symbols…”

On the Generalization of Hörmander's Inequality

“…As a consequence of the above facts, χt λ ∈ S(Λ −m d −k λ , g), and arguing as in 2.4.9 of [9] it is easily seen that (b + λ k )#(χt λ ) = χ + s λ , with s λ ∈ S(d, g). For λ = 0 the above "parametrix" construction shows that B is self-adjoint as an unbounded operator in L 2 (R n ) with domain the Bony-Chemin's Sobolev space H(Λ m d k , g) (see [2] for details). Moreover, we see from (A.9) and (A.10) that, if we choose N, M such that m−N/2 ≤ 0 and −2N +k+M ≥ 0, B+λ k Id has a right inverse for |λ| large enough, thus it is surjective.…”