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Article Dans Une Revue Mathematische Zeitschrift Année : 2024

q-Variationen des Hörmander-Funktionalkalküls und Schrödinger- und Wellen-Maximalabschätzungen

q-variational Hörmander functional calculus and Schrödinger and wave maximal estimates

q-variations du calcul fonctionnel de Hörmander et estimations maximales de Schrödinger et des ondes

Résumé

This article is the continuation of the work [DK] where we had proved maximal estimates ||supt>0|m(tA)f|||Lp(Ω,Y)C||f||Lp(Ω,Y) for sectorial operators A acting on Lp(Ω,Y) (Y being a UMD lattice) and admitting a H\"ormander functional calculus (a strengthening of the holomorphic H calculus to symbols m differentiable on (0,) in a quantified manner), and m:(0,)\C being a H\"ormander class symbol with certain decay at . In the present article, we show that under the same conditions as above, the scalar function tm(tA)f(x,ω) is of finite q-variation with q>2, a.e. (x,ω). This extends recent works by [BMSW,HHL,HoMa1,HoMa,JSW,LMX] who have considered among others m(tA)=etA the semigroup generated by A. As a consequence, we extend estimates for spherical means in euclidean space from [JSW] to the case of UMD lattice-valued spaces. A second main result yields a maximal estimate ||supt>0|m(tA)ft|||Lp(Ω,Y)C\norm||ft||Lp(Ω,Y(Λβ)) for the same A and similar conditions on m as above but with ft depending itself on t such that tft(x,ω) belongs to a Sobolev space Λβ over (\R+,dtt). We apply this to show a maximal estimate of the Schr\"odinger (case A=Δ) or wave (case A=Δ) solution propagator texp(itA)f. Then we deduce from it variants of Carleson's problem of pointwise convergence [Car] exp(itA)f(x,ω)f(x,ω) a. e. (x,ω)(t0+) for A a Fourier multiplier operator or a differential operator on an open domain Ω\Rd with boundary conditions.
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Dates et versions

hal-04526930 , version 1 (29-03-2024)

Identifiants

Citer

Luc Deleaval, Christoph Kriegler. q-variational Hörmander functional calculus and Schrödinger and wave maximal estimates. Mathematische Zeitschrift, 2024, 307 (2), ⟨10.1007/s00209-024-03488-7⟩. ⟨hal-04526930⟩
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