Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators

By David Lannes. 2005

90 citations recorded by the source corpus.

We consider here pseudo-differential operators whose symbol σ(x,ξ) is not infinitely smooth with respect to x. Decomposing such symbols into four—sometimes five—components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol σ(x,ξ). We also study commutator estimates involving such operators, and generalize or improve the so-called Kato–Ponce and Calderon–Coifman–Meyer estimates in various ways.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af226e93fe2996230c1d7d2259864958c21d2c9a

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In Applied Mathematics

In Computational Theory and Mathematics

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Citations in the corpus

Asymptotic Models for Internal Waves (2007, unknown): “Using the expression for Qμ2[ε2ζ ] and the commutator estimate, ∣∣[Λs,f ]g∣∣2 C|∇f |Hmax{t0,s−1} |g|Hs−1 , which holds for some constant C which depends upon s > − d2 and t0 > d2 (see Theorem 6 of [27]), we obtain ∥∥[Qμ2[ε2ζ ],Λs]∇μ2 X,zu∥∥ C( 1 H2 , εmax 2 ,μmax 2 , |ζ |Hmax{t0+2,s+1} )(∥∥Λs−1∇μ2 X”
Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity (2009, background): “However, as diverse problems in Analysis and PDEs demand, the case in which the symbol has mild or no regularity in x has received considerable attention, see, for instance, [32–34, 41, 43, 44], and references therein.”
The Cauchy Problem for the Euler–Poisson System and Derivation of the Zakharov–Kuznetsov Equation (2012, unknown): “Classical commutator estimates (see for instance [18]) yield for s > d/2 + 1, ∣α[Λs, n 1 + ǫn ]∇ñ ∣∣ H1 ǫ ≤ C( 1 c0 , |n|Hs+1 ǫα )|ñ|Hs+1 ǫα , |f̃ |H1 ǫα ≤ |f |Hs+1 ǫα + ( |n|Hs+1 ǫα + |v|Hs+1 ǫ ) × ( |v|Hs+1 ǫ + |n|Hs+1 ǫα ) ,”
The Cauchy problem on large time for surface waves Boussinesq systems (2012, background): “Thanks to Theorems 3 and 6 in [19] or by the classical theory of pseudo-differential operator, the lemma is proved.”
On the well-posedness for the Euler-Korteweg model in several space dimensions (2007, background): “2 Let us now state a more accurate result (which is a variation on an exercise in [1], page 179, see also [13] for similar inequalities pertaining to more general pseudo-differential operators).”
A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations (2007, background): “3 and 6 of [11]), and the last one is a classical “quasilinear type” estimate (e.”