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Spectral multiplier theorems of Hörmander type via generalized Gaussian estimates [Elektronische Ressource] / Matthias Uhl. Betreuer: P. C. Kunstmann

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133 pages
Spectral multiplier theoremsof H ormander type viageneralized Gaussian estimatesZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakultat fur Mathematik desKarlsruher Instituts fur Technologie (KIT)genehmigteDISSERTATIONvonMatthias Uhlaus Haslach im KinzigtalTag der mun dlichen Prufung: 29. Juni 2011Referent: Priv.-Doz. Dr. Peer Christian KunstmannKorreferent: Prof. Dr. Lutz WeisContents1 Introduction 12 Preliminaries 92.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Spaces of homogeneous type . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Generalized Gaussian estimates and Davies-Ga ney estimates . . . . . . . . 122.4 The H ormander condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Spectral multipliers on Lebesgue spaces 253.1 A H ormander type multiplier theorem for operators satisfying generalizedGaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Variation for operators with non-empty point spectrum . . . . . . . . . . . 374 Hardy spaces 434.1 Hardy spaces via square functions . . . . . . . . . . . . . . . . . . . . . . . 454.2 Tent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Hardy spaces via molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 Relationship between Hardy spaces and Lebesgue spaces . . . . . . . . . . .
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Spectral multiplier theorems
of H ormander type via
generalized Gaussian estimates
Zur Erlangung des akademischen Grades eines
DOKTORS DER NATURWISSENSCHAFTEN
von der Fakultat fur Mathematik des
Karlsruher Instituts fur Technologie (KIT)
genehmigte
DISSERTATION
von
Matthias Uhl
aus Haslach im Kinzigtal
Tag der mun dlichen Prufung: 29. Juni 2011
Referent: Priv.-Doz. Dr. Peer Christian Kunstmann
Korreferent: Prof. Dr. Lutz WeisContents
1 Introduction 1
2 Preliminaries 9
2.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Spaces of homogeneous type . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Generalized Gaussian estimates and Davies-Ga ney estimates . . . . . . . . 12
2.4 The H ormander condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Spectral multipliers on Lebesgue spaces 25
3.1 A H ormander type multiplier theorem for operators satisfying generalized
Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Variation for operators with non-empty point spectrum . . . . . . . . . . . 37
4 Hardy spaces 43
4.1 Hardy spaces via square functions . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Tent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Hardy spaces via molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Relationship between Hardy spaces and Lebesgue spaces . . . . . . . . . . . 62
15 Spectral multipliers on the Hardy space H (X) 77L
15.1 A criterion for boundedness of spectral multipliers on H (X) . . . . . . . . 77
L
15.2 A H ormander type multiplier theorem on H (X) . . . . . . . . . . . . . . . 88L
p p6 Boundedness of spectral multipliers on H (X) and L (X) 95
L
7 Applications 101
7.1 Maxwell operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2 Stokes operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3 Lame operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Bibliography 123
iiiContents
iv1 Introduction
In this thesis we investigate spectral multiplier theorems on Lebesgue and Hardy spaces,
where our focus is on the required regularity order.
2Consider a non-negative, self-adjoint operator L on the Hilbert space L (X), where X is
an arbitrary measure space. If E denotes the resolution of the identity associated to L,L
then L can be represented in the form
Z ∞
L= dE ()L
0
(see e.g. [Rud73, Theorem 13.33]). The spectral theorem asserts that the operator
Z

F(L):= F()dE ()L
0
2is well dened and acts as a bounded linear operator on L (X) whenever F: [0,∞)→C is
a bounded Borel function. Spectral multiplier theorems provide regularity assumptions on
p 2F which ensure that the operator F(L) extends from L (X)∩L (X) to a bounded linear
poperator on L (X) for all p ranging in some symmetric interval containing 2.
DIn 1960, L. H ormander addressed this question for the Laplacian L= onR during
Dhis studies on the boundedness of Fourier multipliers onR . His famous Fourier multiplier
theorem ([H or60 , Theorem 2.5]) reads as follows:
If ∈N with >D/2 and F: [0,∞)→C is a bounded function such that F is -times
continuously dierentiable on (0,∞) and the inequality
Z 1/22RX 1 2k (k) sup R F () d <∞ (1.1)
RR>0 R/2k=0
p Dholds, then F( ) is a bounded linear operator on L (R ) for every p∈(1,∞).
Note that F( ) corresponds to a Fourier multiplier operator and that the spectrum
of is the set [0,∞) on which the function F is de ned. This observation justi es
the terminology “spectral multiplier” which will also be used for other operators than the
Laplacian.
1