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# Fractional non-Archimedean differentiability [Elektronische Ressource] / vorgelegt von Enno Nagel

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Fractional Non-ArchimedeanDiﬀerentiabilityEnno Nagel2011MathematikFractional Non-ArchimedeanDifferentiabilityInaugural-Dissertationzur Erlangung des Doktorgradesder Naturwissenschaften im FachbereichMathematik und Informatikder Mathematisch-Naturwissenschaftlichen Fakultätder Westfälischen Wilhelms-Universität Münstervorgelegt vonENNO NAGELaus Stadtlohn-2011-Dekan: PROFESSOR DR. MATTHIAS LÖWEErster Gutachter: PROFESSOR DR. PETER SCHNEIDERZweiter PROFESSOR DR. CHRISTOPHER DENINGERTag der mündlichen Prüfung: 01.02.2011Tag der Promotion:AbstractLetr≥ 0 be real number letK be a complete non-Archimedeanly non-trivially valued ﬁeld.dIn the ﬁrst chapter, we give the deﬁnition of a functionf : X→ E on a domainX⊆ Krwith values in aK-Banach spaceE to ber-times differentiable orC at a pointa∈X. Thenrwe endow the K-vector space of all suchC -functions with a locally convex topology andexamine properties of theirs such as completeness, density of (locally) polynomial functions,closure under composition and, for the dual, under convolution.For functions on open domains in one variable, we show this deﬁnition to equal a handier de-rscription through the convergence speedo(1/|h| ) of the rest term of the Taylor-polynomial atdx +h expanded aroundx up to degreebrc. Moreover on the special domainX =Z we showpr dtheC -functionsf :Z →K to be characterized by its Mahler coefﬁcients (a ) d obeyingn n∈Npr|a ||n| → 0 as|n|→∞, where we put|n| :=n +···+n .

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 Publié par Publié le 01 janvier 2011 Nombre de lectures 14 Langue English Poids de l'ouvrage 1 Mo

Exrait

Fractional Non-Archimedean
Diﬀerentiability
Enno Nagel
2011Mathematik
Fractional Non-Archimedean
Differentiability
Inaugural-Dissertation
der Naturwissenschaften im Fachbereich
Mathematik und Informatik
der Mathematisch-Naturwissenschaftlichen Fakultät
der Westfälischen Wilhelms-Universität Münster
vorgelegt von
ENNO NAGEL
-2011-Dekan: PROFESSOR DR. MATTHIAS LÖWE
Erster Gutachter: PROFESSOR DR. PETER SCHNEIDER
Zweiter PROFESSOR DR. CHRISTOPHER DENINGER
Tag der mündlichen Prüfung: 01.02.2011
Tag der Promotion:Abstract
Letr≥ 0 be real number letK be a complete non-Archimedeanly non-trivially valued ﬁeld.
dIn the ﬁrst chapter, we give the deﬁnition of a functionf : X→ E on a domainX⊆ K
rwith values in aK-Banach spaceE to ber-times differentiable orC at a pointa∈X. Then
rwe endow the K-vector space of all suchC -functions with a locally convex topology and
examine properties of theirs such as completeness, density of (locally) polynomial functions,
closure under composition and, for the dual, under convolution.
For functions on open domains in one variable, we show this deﬁnition to equal a handier de-
r
scription through the convergence speedo(1/|h| ) of the rest term of the Taylor-polynomial at
dx +h expanded aroundx up to degreebrc. Moreover on the special domainX =Z we showp
r dtheC -functionsf :Z →K to be characterized by its Mahler coefﬁcients (a ) d obeyingn n∈Np
r
|a ||n| → 0 as|n|→∞, where we put|n| :=n +···+n . Then as a corollary, a characteri-n 1 d
r dzation ofC -functionsf :X→K on openX⊆Q by partial Taylor-polynomials is obtained.p
We turn to the second chapter: LetG be a connected reductive group over a local ﬁeldF andP
a minimal parabolic subgroup. LetK be a complete non-Archimedeanly non-trivally valued
∗ﬁeld of characteristic 0 with valuation ringo. Letθ :P→K be an unramiﬁed character and
G
denote byI(θ) = Ind θ the smooth principal series. LetU be an algebraic representation ofP
G (and ifU is nontrivial, also assumeK⊇F andG to split). ThenV =I(θ)⊗ U is a locallyK
balgebraicG-representation, and we letV be the universalK-Banach space with
aG-invariant norm whereintoV maps continuously with respect to its ﬁnest locally convex
topology. We will then show that the universal unitary lattice L⊆ V , given by the preimage
P
bof the unit ball inV , is of the formL = L withW denoting the Weyl group ofG andw∈W w
¯eachL being a cyclico[P ]-module which is free as ano-module.wAcknowledgements
First of all, I would like to thank very much all my dear former and current colleagues of my
work group, and in particular my advisor Peter Schneider.
I also greatly proﬁted from exchange with Stany de Smedt, Helge Glöckner, Wim Schikhof
and Benjamin Schraen. I am moreover indebted to Marco de Ieso for careful correction read-
ing. It is a pleasure to thank everyone of them.
Most importantly though, I want to thank my family for its support.Contents
Fractional non-Archimedean calculus 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 Locally convexK-vector spaces . . . . . . . . . . . . . . . . . . . . 6
ρ1.2 C -functions forρ∈ [0, 1[ . . . . . . . . . . . . . . . . . . . . . . . 7
ρDeﬁnition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 7
ρProperties of the space ofC -functions . . . . . . . . . . . . . . . . . 9
ρThe locally convex topology onC -functions . . . . . . . . . . . . . 11
ρComponentwise criteria for beingC . . . . . . . . . . . . . . . . . . 15
ρρρ d1.3 C -functions forρρρ∈ [0, 1[ . . . . . . . . . . . . . . . . . . . . . . . 18
ρρρDeﬁnition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 19
ρProperties of the space ofC -functions . . . . . . . . . . . . . . . . . 20
1+ρ1.4 C -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Fractional differentiability in one variable . . . . . . . . . . . . . . . . . . . 32
r2.1 C -functions forr∈R . . . . . . . . . . . . . . . . . . . . . . . . 32≥0
rDeﬁnition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 32
rProperties ofC . . . . . . . . . . . . . . . . . . . . . . . 33
rThe locally convexK-algebra ofC -functions . . . . . . . . . . . . . 35
Description through iterated difference quotients . . . . . . . . . . . 38
2.2 Characterization through Taylor polynomials . . . . . . . . . . . . . 39
rThe Taylor polynomial ofC -functions . . . . . . . . . . . . . . . . 39
rCharacterizingC -functions through Taylor polynomials on general
domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
rSufﬁciency of the Taylor polynomial expansion on B -sets forC -ν
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
rAnother characterization ofC -functions on compact sets and an ap-
plication . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Orthogonal bases onZ . . . . . . . . . . . . . . . . . . . . . . . . 57p
ρThe van der Put base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . 58p
ρThe Mahler base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 59p
rThe Base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 65p
3 Fractional differentiability in many variables . . . . . . . . . . . . . . . . . . 71
r3.1 C -functions forr∈R . . . . . . . . . . . . . . . . . . . . . . . . 71≥0
rDeﬁnition ofC -functions . . . . . . . . . . . . . . . . . . . . . . . 71
rProperties ofC . . . . . . . . . . . . . . . . . . . . . . . 74rThe locally convexK-algebra ofC -functions . . . . . . . . . . . . . 77
rLocally analytic functions inC (X,K) on an open domain . . . . . . 82
rComposition properties ofC -functions . . . . . . . . . . . . . . . . 85
rDensity of (locally) polynomial functions inC (X,K) . . . . . . . . 90
r3.2 Orthogonal bases ofC -functions on a compact domain . . . . . . . . 98
Interlude: Orthogonal bases ofK-Banach spaces . . . . . . . . . . . 98
rThe initialK-Banach algebraC (X,K) of thought topological tensor
rrr dproductsC (X,K) forrrr∈N . . . . . . . . . . . . . . . 100=r
r dThe Mahler base ofC (Z ,K) . . . . . . . . . . . . . . . . . . . . . 106p
r d3.3 Description ofC (X,K) for openX⊆Q through Taylor polynomials 108p
r r3.4 The spaceD (X,K) of distributions onC (X,E) for a compact group
X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
rExample of an inducedC -representation . . . . . . . . . . . . . . . 124
rC -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
The intertwined open cells in the universal unitary lattice of an unramiﬁed
algebraic principal series 133
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
0 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
0.1 The groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
0.2 The representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
0.3 The universal unitary completion of a locally algebraic representation 137
1 The unramiﬁed dominant principal series as a representation ofP . . . . . . 140
2 The universal unitary lattice of the P -representation on an open cell and a
norm of differentiable functions . . . . . . . . . . . . . . . . . . . . . . . . 144
Interlude: The dominant submonoid acting on the afﬁne root factors . . . . . 144
2.1 The necessity criterion . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.2 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Example: The smooth case of small order image . . . . . . . . . . . 155 The split case . . . . . . . . . . . . . . . . . . . . 156
Interlude: Locally polynomial differentiable functions . . . . . . . . . . . . . 157
2.3 The locally algebraic case . . . . . . . . . . . . . . . . . . . . . . . 164
3 The universal unitary lattice of an unramiﬁed dominant principal series . . . . 168
3.1 The universal unitary lattice of the underlyingP -representation . . . 168
3.2 The Jacquet module . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.3 Gluing the universal unitary lattice from the intertwined open cells . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Fractional non-Archimedean calculus
Introduction
Start with two normed ﬁnite dimensional vector spacesV andW over a valued ﬁeldK. Let
f : U→ W some map deﬁned on an open subsetU⊆ V . Thenf is called differentiable or
1C in the pointa∈U if there exists a linear mapD :V→W such that for everyε> 0 therea
is a neighborhoodU 3a inU withε
kf(x +h)−f(x)−D ·hk≤εkhk for allx +h,x∈U .a ε
To iterate this differentiability notion, we need a choice of coordinates on the function’s do-
dmain. We therefore assumeV = K and lete ,...,e be its canonical basis vectors. Then1 d
d∗ ]1[given any two points x +h,x∈ U with h∈ K , we can deﬁne A := f (x +h,h)∈
Hom (V,W ) by the partial difference quotientsK-vctsp.
A(h ·e ) =f(x+h·e +···+h ·e )−f(x+h·e +···+h ·e ) fork = 1,...,d.k k 1 1 k k 1 1 k−1 k−1
]1[ ]1[ [1] [1]Then this mapf :U → Hom (V,W ) extends to a continuous functionf :U →K-vctsp.
[1] 1Hom (V,W ) withU =U×U if and only iff isC at every point ofa. (See RemarkK-vctsp.
1.35.) This function’s domain lies again in the K-vector space V ×V inheriting a natural
choice of coordinates, its range is in a natural way again a K-vector space, and so we can
deﬁnef to be twice continuously differentiable if
]2[ [1] ]1[ [1] ]1[ df = (f ) : (X ) → Hom (Hom (K ,W ),W )K-vctsp. K-vctsp.
[2] [2] [1] [1]extends to a continuous functionf on all ofX =X ×X , and so on. This construction
can also be carried out to yield a notion of pointwise differentiability.
As our goal is a deﬁnition of r-fold differentiability for r∈ R , we introduce the notion≥0
ρ ρof aC -point forρ∈ [0, 1[ as follows: The mappingf isC in the pointa∈ U if for every
ε> 0 there is a neighborhoodU 3a inU withε
ρ
kf(x +h)−f(x)k≤εkhk for allx +h,x∈U .ε
rNow writer = ν +ρ∈R withν∈N andρ∈ [0, 1[. Then forf to be aC -function, we≥0
ρdemand itsν-th iterated difference quotient not merely to extend continuously, butC -wise at
all critical limit points.
rThen to arrive at our Deﬁnition 3.1 of aC -point, we notice that a mapping symmetric in
1two coordinates is partially differentiable in both coordinates if and if only if it is so in
[1]one of them. E.g. if V = K is one-dimensional, we can alternatively write f (x,y) =
[f(x)−f(y)]/(x−y) for its ﬁrst difference quotient. This is a symmetric mapping in both
[1]coordinates. If we deﬁne a mapping to be twice differentiable if ﬁrstlyf exists onU×U and
then is again differentiable, we are hence brought down to checking partial differentiability in
[1]f ’s ﬁrst coordinate, reducing an exponential growth of parameters to a linear one. This ob-
servation underlies the deﬁnition of iterated differentiability in the sense of [Schikhof, 1984],
which we also employ here for our partial difference quotients.
We ﬁrst show this deﬁnition to satisfy a number of properties naively to be expected:
d- Given a locally cartesian subsetX⊆ K (see Section 3.1) and aK-Banach spaceE, the
r rK-vector space of all suchC -functionsC (X,E) can naturally be endowed with a locally
convex topology, which is then complete and also a locally convexK-algebra if the rangeE
is so.
- As a large class of explicit examples, we also ﬁnd all locally analytic functions to ber-times
differentiable for anyr≥ 0. Then we show all locally polynomial of total degree
at most ν and consequently all polynomial functions to constitute dense subspaces on a
rcompact domainX. By this density, we can viewD(X,K) = limD (X,K) as the ﬁltered
−→
K-vector space of allK-linear forms deﬁned on all arbitrarily often differentiable functions
∞ rC (X,E) extending continuously ontoC (X,E) for some r≥ 0. When X is moreover
∞a group withC -multiplication, we can endowD(X,K) with a convolution product and
prove it to be ﬁlteredK-algebra.
- Informed by the above interpretation of theν-th difference quotient as a map with values in
rK-linear homomorphisms, we will also see thatC -functions are closed under composition
rif r≥ 1. We note that thereby and since aC -function is deﬁned pointwise, it is a local
rnotion, so that put together we arrive at a reasonable notion of aC -manifold.
rIn its most naive way presented above, the notion of aC -function is hardly handy, and the
ﬁrst reduction by the symmetry of these difference quotients can be taken further. We want to
give a guideline on the order in which we obtain these simpliﬁcation results:
- On domainsX⊆K in one variable with locally sufﬁciently many points - such as open ones
- the symmetry properties of these iterated difference quotients are strong enough to reduce
the question of iterated differentiability to a more convenient Taylor polynomial criterion in
whichr-fold differentiability can be checked by only one additional variable.
- LetK⊇Q as a valued ﬁeld with valuation ringo. There is a distinguished (see Subsec-p
0 dtion 2.3) orthogonal basis of the continuous K-valued functionsC (Z ,K) relating to thep
∗ ddomain’s topological group structure, the so called Mahler polynomials{ : i∈ N}.
i
Denoting by c (N,K) all zero sequences inK, this means that we have an isomorphism of0
∼ 0 0 dK-Banach spaces c (N,K)→C (Z ,K) withf∈C (Z ,K) corresponding to its Mahler0 p p
r dcoefﬁcients (a ) d. We want to describe the topologicalK-vector subspaceC (Z ,K),→n n∈N p
2