Global and Local Regularity of Fourier Integral Operators on Weighted and Unweighted Spaces

86 Pages·2014·0.636 MB·English

by  Ferreira, David Dos Santos, Staubach, Wolfgang

MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1074 (first of 5 numbers) • May 2014
Global and Local Regularity of
Fourier Integral Operators on
Weighted and Unweighted Spaces
David Dos Santos Ferreira
Wolfgang Staubach
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 229 • Number 1074 (first of 5 numbers) • May 2014
Global and Local Regularity of
Fourier Integral Operators on
Weighted and Unweighted Spaces
David Dos Santos Ferreira
Wolfgang Staubach
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Ferreira,DavidDosSantos,1975-
GlobalandlocalregularityofFourierintegraloperatorsonweightedandunweightedspaces/
DavidDosSantosFerreira,WolfgangStaubach.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 229,
number1074)
“May2014,volume229,number1074(firstof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-9119-3(alk. paper)
1. Fourier integral operators. 2. Mathematical analysis. I. Staubach, Wolfgang, 1970-
II.AmericanMathematicalSociety. III.Title.
QA403.5.F47 2014
515(cid:2).723–dc23 2013051215
DOI:http://dx.doi.org/10.1090/memo/1074
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Contents
Introduction vii
Chapter 1. Prolegomena 1
1.1. Definitions, notations and preliminaries 1
1.2. Tools in proving Lp boundedness 3
1.3. Links between nonsmoothness and global boundedness 13
Chapter 2. Global Boundedness of Fourier Integral Operators 17
2.1. Global L1 boundedness of rough Fourier integral operators 17
2.2. Local and global L2 boundedness of Fourier integral operators 19
2.3. Global L∞ boundedness of rough Fourier integral operators 34
2.4. Global Lp-Lp and Lp-Lq boundedness of Fourier integral operators 37
Chapter 3. Global and Local Weighted Lp Boundedness of Fourier Integral
Operators 39
3.1. Tools in proving weighted boundedness 39
3.2. Counterexamples in the context of weighted boundedness 42
3.3. Invariant formulation in the local boundedness 47
3.4. WeightedlocalandglobalLp boundednessofFourierintegraloperators 48
Chapter 4. Applications in Harmonic Analysis and Partial Differential
Equations 55
4.1. Estimates in weighted Triebel-Lizorkin spaces 55
4.2. Commutators with BMO functions 56
4.3. Applications to hyperbolic partial differential equations 61
Bibliography 63
iii

Abstract
We investigate the global continuity on Lp spaces with p ∈ [1,∞] of Fourier
integral operators with smooth and rough amplitudes and/or phase functions sub-
ject to certain necessary non-degeneracy conditions. In this context we also prove
the optimal global L2 boundedness result for Fourier integral operators with non-
degenerate phase functions and the most general smooth H¨ormander class ampli-
tudes i.e. those in Sm with (cid:2),δ ∈ [0,1]. We also prove the very first results con-
(cid:2),δ
cerning the continuity of smooth and rough Fourier integral operators on weighted
Lp spaces, Lp with 1 < p < ∞ and w ∈ A , (i.e. the Muckenhoupt weights)
w p
for operators with rough and smooth amplitudes and phase functions satisfying a
suitable rank condition. These results are shown to be optimal for operators with
amplitudesinclassical Ho¨rmanderclassesandcanalsobegivenageometricallyin-
variant formulation. The weighted results are in turn applied to prove, for the first
time, weighted and unweighted estimates for the commutators of Fourier integral
operatorswith functions of bounded mean oscillation BMO, estimates on weighted
Triebel-Lizorkinspaces, andfinallyglobalunweightedandlocalweightedestimates
for the solutions of the Cauchy problem for m-th and second order hyperbolic par-
tial differential equations on Rn. The global estimates in this context, when the
Sobolev spaces are L2 based, are the best possible.
ReceivedbytheeditorNov172011,and,inrevisedform,May31,2012.
ArticleelectronicallypublishedonSeptember24,2013.
DOI:http://dx.doi.org/10.1090/memo/1074
2010 MathematicsSubjectClassification. Primary35S30,42B99.
Keywordsandphrases. Fourierintegraloperators,Weightedestimates,BMOcommutators.
During thepreparationofthismanuscriptthefirstauthorwaspartiallysupportedbyANR
grantEqua-disp.
Duringthepreparationofthismanuscriptthesecondauthorwaspartiallysupportedbythe
EPSRCFirstGrantScheme,referencenumberEP/H051368/1.
(cid:3)c2013 American Mathematical Society
v

Introduction
A Fourier integral operator is an operator that can be written locally in the
form
(cid:2)
(0.1) T u(x)=(2π)−n eiϕ(x,ξ)a(x,ξ)u(cid:3)(ξ)dξ,
a,ϕ
Rn
where a(x,ξ)is the amplitudeand ϕ(x,ξ)is the phase function. Historically, asys-
tematic study of smooth Fourier integral operators with amplitudes in C∞(Rn×
Rn) with |∂α∂βa(x,ξ)| ≤ C (1+|ξ|)m−(cid:2)|α|+δ|β| (i.e. a(x,ξ) ∈ Sm ), and phase
ξ ξ αβ (cid:2),δ
functionsinC∞(Rn×Rn\0)homogenousofdegree1inthefrequencyvariableξand
with non-vanishing determinant of the mixed Hessian matrix (i.e. non-degenerate
phase functions), was initiated in the classical paper of L. Ho¨rmander [H3]. Fur-
thermore, G. Eskin [Esk] (in the case a(x,ξ)∈ S0 ) and H¨ormander [H3] (in the
1,0
case a(x,ξ)∈S0 , 1 <(cid:2)≤1) showed the local L2 boundedness of Fourier inte-
(cid:2),1−(cid:2) 2
gral operators with non-degenerate phase functions. Later on, H¨ormander’s local
L2 resultwasextendedbyR.Beals[RBE]andA.GreenleafandG.Uhlmann[GU]
to the case of amplitudes in S0 .
1,1
2 2
After the pioneering investigations by M. Beals [Be], the optimal results con-
cerning local continuity properties of smooth Fourier integral operators (with non-
degenerate and homogeneous phase functions) in Lp for 1≤p≤∞, were obtained
in the seminal paper of A. Seeger, C. D. Sogge and E.M. Stein [SSS]. This also
paved the way for further investigations by G. Mockenhaupt, Seeger and Sogge in
[MSS1]and[MSS2], seealso[So]and[So1]. Intheseinvestigationsthebounded-
ness,fromLp toLp andfromLp toLq ofsmoothFourierintegraloperators
comp loc comp loc
withnon-degeneratephasefunctionshavebeenestablished, andfurthermoreitwas
shown that the maximal operators associated with certain Fourier integral opera-
tors (and in particular constant and variable coefficient hypersurface averages) are
Lp bounded.
InthecontextofH¨ormandertypeamplitudesandnon-degeneratehomogeneous
phasefunctionswhicharemostfrequentlyusedinapplicationsinpartialdifferential
equations,ithasbeencomparativelysmallamountofactivityconcerningglobalLp
boundedness of Fourier integral operators. Among these, we would like to men-
tion the global L2 boundedness of Fourier integral operators with homogeneous
phases in C∞(Rn ×Rn \0) and amplitudes in the Ho¨rmander class S0 , due to
0,0
D. Fujiwara [Fuji]; the global L2 boundedness of operators with inhomogeneous
phases in C∞(Rn×Rn) and amplitudes in S0 , due to K. Asada and D. Fujiwara
0,0
[AF]; the global Lp boundedness of operators with smooth amplitudes in the so
called SG classes, due to E. Cordero, F. Nicola and L. Rodino in [CNR1]; the
vii
viii INTRODUCTION
boundedness of operators with amplitudes in Sm on the space of compactly sup-
1,0
ported distributions whose Fourier transform is in Lp(Rn) (i.e. the FLp spaces)
due to Cordero, Nicola and Rodino in [CNR2] and Nicola’s refinement of this in-
vestigation in [Nic] (where the roles of the smooth spatial factorizations and affine
fibrations have been emphasised); and finally, S. Coriasco and M. Ruzhansky’s
globalLp boundednessofFourierintegraloperators[CR],withsmoothamplitudes
in a suitable subclass of the Ho¨rmander class S0 , where certain decay of the am-
1,0
plitudes in the spatial variables are assumed. We should also mention that before
the appearance of the paper [CR], M. Ruzhansky and M. Sugimoto had already
proved in [Ruz 2] certain weighted L2 boundedness (with some power weights) as
well as the global unweighted L2 boundedness of Fourier integrals operators with
phasesinC∞(Rn×Rn)thatarenotnecessarilyhomogeneousinthefrequencyvari-
ables, and amplitudes that are in the class S0 . In all the aforementioned results,
0,0
one has assumed certain bounds on the derivatives of the phase functions and also
astrongernon-degeneracyconditionthantheonerequiredinthelocalLpestimates.
Inthispaper,weshalltakealltheseresultsasourpointofdepartureandmake
a systematic study of the global Lp boundedness of Fourier integral operators with
amplitudes in Sm with (cid:2) and δ in [0,1], which cover all the possible ranges of
(cid:2),δ
(cid:2)’s and δ’s. Furthermore we initiate the study of weighted norm inequalities for
FourierintegraloperatorswithweightsintheA classofMuckenhouptanduseour
p
global unweighted Lp results to prove a sharp weighted Lp boundedness theorem
forFourierintegraloperators. Theweightedresultsinturnwillbeusedtoestablish
thevalidityofcertainvector-valuedinequalitiesandmoreimportantlytoprovethe
weighted and unweighted boundedness of commutators of Fourier integral opera-
tors with functions of bounded mean oscillation BMO. Thus, all the results of this
paper are connected and each chapter uses the results of the previous ones. This
has beenreflected in the structure of the paper and the presentation of the results.
As mentioned earlier, in [SSS] the sharp local Lp boundedness of the Fourier
integral operators was established under the assumption of the non-degeneracy of
the phase function. Furthermore, in the same context, it was shown in [Nic] that
if the rank of the spatial Hessian ∂2 ϕ(x,ξ) is bounded from above, then for an
xx
appropriateorderoftheamplitude whichdependsonthatupperbound(andturns
out to be sharp), the corresponding Fourier integral operator is bounded on FLp.
Concerningthespecificconditionsthatareputinthispaperonthephasefunctions,
ithasbeenknownatleastsincetheappearanceofthepapers[Fuji],[AF],[Ruz 2]
and [CR], that one has to assume stronger conditions, than mere non-degeneracy,
onthephasefunctioninordertoobtainglobalLp boundednessresults. Itturnsout
that the assumption on the phase function, referred to in this paper as the strong
non-degeneracy condition, which requires a nonzero lower bound on the modulus
of the determinant of the mixed Hessian of the phase, is actually necessary for the
validityofglobalregularityofFourierintegraloperators,seesection1.2.5. Further-
more, we also introduce the class Φk of homogeneous (of degree 1) phase functions
with a specific control over the derivatives of orders greater than or equal to k,
and assume our phases to be strongly non-degenerate and belong to Φk for some
k. At first glance, these conditions might seem restrictive, but fortunately theyare
INTRODUCTION ix
general enough to accommodate the phase functions arising in the study of hyper-
bolicpartial differential equationsandwill still apply tothe most genericphases in
practical applications.
Concerning our choice of amplitudes, there are some features that set our in-
vestigations apart from those made previously, for example partly motivated by
the investigation of C.E. Kenig and W. Staubach [KS], of the Lp boundedness of
the so called pseudo-pseudodifferential operators, we consider the global and local
Lp boundedness of Fourier integral operators when the amplitude a(x,ξ) belongs
to the class L∞Sm, wherein a(x,ξ) behaves in the spatial variable x like an L∞
(cid:2)
function, and in the frequency variable ξ, the behaviour is that of a symbol in the
H¨ormander class Sm .
(cid:2),0
It is worth mentioning that the conditions defining classes Φk, L∞Sm and the
(cid:2)
assumptionofstrongnon-degeneracymaketheglobalresultsobtainedherenatural
extensions of the local boundedness results of Seeger, Sogge and Stein’s in [SSS].
Apartfromtheobviouslocaltoglobalgeneralizations, thisisbecauseononehand,
ourmethodscanhandle thesingularity ofthephasefunctioninthefrequencyvari-
ables at the origin and therefore the usual assumption that ξ (cid:6)=0 in the support of
theamplitudebecomesobsolete. Ontheotherhand,wedonotrequireanyregular-
ity(andthereforenodecayofderivatives)inthespatialvariablesoftheamplitudes.
Therefore, our amplitudes are close to, and in fact are spatially non-smooth ver-
sionsofthoseintheSeeger-Sogge-Stein’spaper[SSS].Indeed,in[SSS]theauthors
although dealing with spatially smooth amplitudes, assume neither any decay in
the spatial variables nor the vanishing of the amplitude in a neighbourhood of the
singularity of the phase function.
There are several steps involved in the proof of the results of the paper and
there are discussions about various conditions that we impose on the operators as
well as some motivations for the study of rough operators. Moreover, giving exam-
ples and counterexamples when necessary, we have strived to give motivations for
our assumptions in the statements of the theorems. Here we will not mention all
theresults thathave beenproveninthis paper, insteadwe chose tohighlight those
that are the main outcomes of our investigations.
In Chapter 1, we set up the notations and give the definitions of the classes of
amplitudes, phase functions and weights that will be used throughout the paper.
Wealsoincludethetoolsthatweneedinprovingourglobalboundednessresults,in
thischapter. Weclosethechapterwithadiscussionabouttheconnectionsbetween
rough amplitudes and global boundedness of Fourier integral operators.
Chapter 2 is devoted to the investigation of the global boundedness of Fourier
integral operators with smooth or rough phases, and smooth or rough amplitudes.
To achieve our global boundedness results, we split the operators in low and high
frequency parts and show the boundedness of each and one of them separately. In
proving the Lp boundedness of the low frequency portion, see Theorem 1.18, we
utilise Lemma 1.17 which yields a favourable kernel estimate for the low frequency