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 t↦m(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)=e−tA 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 t↦ft(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 t↦exp(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)(t→0+)
for A a Fourier multiplier operator or a differential operator on an open domain Ω⊆\Rd with boundary conditions.
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