Maximal functions, functional calculus, and generalized Triebel-Lizorkin spaces for sectorial operators [Elektronische Ressource] / von Alexander Ullmann

Alexander Ullmann

Sujets

• Mathematics

Informations

• Publié par: karlsruher_institut_fur_technologie
• Publié le: 01 janvier 2010
• Nombre de lectures: 13
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Maximal functions, functional calculus,
and generalized Triebel-Lizorkin spaces
for sectorial operators
Zur Erlangung des akademischen Grades eines
DOKTORS DER NATURWISSENSCHAFTEN
von der Fakultät für Mathematik des
Karlsruher Instituts für Technologie
genehmigte
DISSERTATION
von
Alexander Ullmann
aus Bad Segeberg
Tag der mündlichen Prüfung: 22. Dezember 2010
Referent: HDoz. Dr. Peer Christian Kunstmann
Korreferent: Prof. Dr. Lutz WeisContents
Contents iii
Introduction 1
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Notations and Preliminaries 9
1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
11.2 The functional calculus for sectorial operators and the H -calculus . . . . . . . . 10
11.3 R-sectorial operators and the operator-valued H -calculus . . . . . . . . . . . . 12
1.4 UMD spaces and the operator-valued Mikhlin Theorem . . . . . . . . . . . . . . . 15
1.5 Interpolation of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Interpolation couples and interpolation functors . . . . . . . . . . . . . . . 17
1.5.2 Real interpolation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.3 Complex interpolation spaces and multilinear Stein interpolation . . . . . 20
1.6 Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6.1 Definition and elementary properties . . . . . . . . . . . . . . . . . . . . . 22
1.6.2 Duality in Banach function spaces . . . . . . . . . . . . . . . . . . . . . . 29
1.6.3 p-convexity and q-concavity . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.4 Some approximation results . . . . . . . . . . . . . . . . . . . . . . . . . . 35
11.6.5 H -calculus in q-concave Banach function spaces . . . . . . . . . . . . . . 37
1.7 Classical function spaces: Besov- and Triebel-Lizorkin spaces . . . . . . . . . . . 37
2 Maximal functions for sectorial operators 43
12.1 The H -maximal function for sectorial operators . . . . . . . . . . . . . . . . . . 430
12.2 Examples of operators with a bounded H -maximal function . . . . . . . . . . . 480
2.3 Equivalence of maximal estimates for f2E( ) . . . . . . . . . . . . . . . . . . . 52
2.4 Interpolation of maximal functions . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5 Maximal functions for tensor-extensions of A in vector-valued Banach function
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 R -boundedness andR -sectorial operators 67s s
3.1 R -boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67s
3.2 R -sectorial operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82s
iiiContents iv
3.2.1 Definition and elementary properties ofR -sectorial operators . . . . . . . 82s
3.2.2 Equivalence of s-power function norms . . . . . . . . . . . . . . . . . . . . 87
13.2.3 R -bounded H -calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 94s
3.3 The associated s-intermediate spaces . . . . . . . . . . . . . . . . . . . . . . . . . 98
_3.3.1 Definition end elementary properties of the spaces X and X . . . . . 98
s;A s;A
3.3.2 The s-spaces as intermediate spaces and interpolation . . . . . . . . . . . 111
3.3.3 The part of A in the s-intermediate spaces . . . . . . . . . . . . . . . . . . 116
3.4 Comparison and perturbation forR -sectorial operators . . . . . . . . . . . . . . 118s
p3.5 Weighted estimates andR -boundedness in L . . . . . . . . . . . . . . . . . . . 139s
13.5.1 Weighted estimates andR -bounded H -calculus . . . . . . . . . . . . . 139s
3.5.2 Applications to differential operators . . . . . . . . . . . . . . . . . . . . . 143
3.5.3 Proof of Theorem 3.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.6 The s-intermediate spaces for differential operators and Triebel-Lizorkin spaces . 158
d3.6.1 Elliptic differential operators in non-divergence form onR . . . . . . . . 158
nd d3.6.2tial op of 2 order in divergence form onR . . . . 174
Bibliography 181Introduction 1
Introduction
In 1986, Alan McIntosh introduced in the fundamental paper Operators which have an H1
1functional calculus ([McI86]) his notion of a boundedH -calculus for sectorial operators: LetA
be a sectorial operator in a complex Banach spaces X, i.e. the set of resolventsf R ( ;A )j2
Cn g is bounded for some !2 (0;), where :=fz2Cnf0gjj arg(z)j < !g denotes the! !
open sector symmetric around the positive axis with half opening angle !. Then based on the
ideas of the Dunford functional calculus one defines
Z
1
’(A) := ’()R( ;A )d 2L(X);
2i
where is the canonical parametrization of the oriented boundary of a suitable sector, and’ is of
1classH , i.e. a bounded holomorphic function on a larger open sector that decays polynomially0
1to 0 as z tends to 0 or1. Then ’7! ’(A) defines a functional calculus on H , which can0
be naturally extended to the larger algebra of holomorphic functions with at most polynomial
growth at 0 and1, where in this case the resulting operators are in general unbounded. In
1particular, f(A) is defined for all bounded holomorphic functions f 2 H . Now one central
question is the following:
1Is f(A) bounded for any f2H , and does an estimatekf(A)k.kfk hold?1
1In this case, A is said to have a bounded H -calculus. McIntosh was able to give various
1characterizations of the boundedness of theH -calculus in the case that the underlying space is
a Hilbert space. One of these is given in terms of so-called square functions and can be rewritten
1in the following form: A sectorial operatorA in a Hilbert spaceX has a boundedH -calculus if
1and only if the following norm equivalence holds for one (and then for all) ’2H with’ = 0:0
Z 1=21 dt2kxk k’(tA)xk for x2X: (1)X X
t0
Thisconditionwasmotivatedbywellknownconceptsofsquarefunctionsfromharmonicanalysis.
Indeed, the methods McIntosh used were operator theoretic, but many of them are motivated
by harmonic analysis. McIntosh himself says the following in his paper [McI86]:
The material in this paper has two heritages: One is operator theory [...]; the other is harmonic
analysis [...],
and this thesis follows the same tradition.
pThe condition (1) has been generalized to other classes of spaces, in a first step to spacesX =L ,
pp2 (1; +1), where it takes the following form: A sectorial operator A in the space X =L has
1a boundedH -calculus if and only if the following norm equivalence holds for one (and then for
1all) ’2H with ’ = 0:0
Z 1=21 dt2 kxk j’(tA)xj for x2X: (2)X t0 X
66Introduction 2
2This has first been treated in [CDMY96]. Note that for X = L , the norm expressions in (1)
and (2) coincide by Fubini. Again, the idea for (2) is based on methods from classical harmonic
panalysis in L , in particular Littlewood-Paley theory. Let us mention that this concept of char-
1acterizing the boundedness of the H -calculus has finally been transferred to general Banach
spaces by Nigel Kalton and Lutz Weis in [KW-1], cf. also [KW04] and [KKW06], where the
square-function norms in (2) are replaced by more general square functions in terms of so called
Rademacher-norms and in terms of -norms. Furthermore, square function estimates are used
in various fields of analysis, e.g. questions of admissibility of certain operators for control sys-
tems have been treated in [LeM03] using square function norms of the form (1), and the related
concept ofR-admissibility is treated in [LeM04] in terms of the square function norms in (2).
Moreover, [KW-1] and the survey [LeM07] give a nice overview of different characterizations and
applications for square functions and square function estimates.
In this thesis, we will concentrate on a certain class of Banach function spaces instead of general
Banach spaces, so in particular, we have an additional lattice structure, and expressions as in (2)
pare still well defined. We note that this class of spaces covers the spaces L , where p2 [1; +1),
p qbut also certain kinds of Lorentz-, Orlicz- and mixedL L -spaces. The central challenge we meet
in this work is to change the power 2 in (2) to a power s2 [1; +1]. This leads to the following
expressions:
Z 1=s1 dts kxk := j’(tA)xj if s< +1, andkxk := supj’(tA)xj : (3)s;A;’ 1;A;’ Xt t>00 X
Although starting from the same idea, i.e. generalizing the square function norms (2), we will
use these two expressions for two different ideas:
In the first part of this thesis, we will study the terms sup j’(tA)xj. These are well known int>0
classical situations and are referred to as maximal functions. In this context, the question arises
naturally, if an estimate of the form

supj’(tA)xj .kxk for x2X (4)XX
t>0
holds. Here we will work more generally in vector-valued Banach function spaces X(E) (e.g.
pvector-valued Lebesgue spacesL ( ;E)), whereE is a Banach space. Given a sectorial operator
A in X(E) we ask for the validity of a maximal estimate

supj’(zA)xj .kxk for x2X(E): (5)E X(E)X
z2
One important issue in this context is the Banach principle, which states that if the estimate (5)
holds, then the set of all x2X(E) such that
(’(zA)x) converges pointwise a.e. if 3z! 0z2
p dis closed in X(E). If e.g. A = is the Laplacian and X = L