Pseudo-Differential Operators and Markov Processes Volume 1. Fourier Analysis and Semigroups 9781860942938

Table of contents :
Contents
Preface
Notation
General Notation
Functions and Distributions
Measures and Integrals
Spaces of Functions, Measures and Distributions
Some Families of Functions
Norms, Scalar Products and Seminorms
Notation from Functional Analysis, Operators
Introduction: Pseudo Differential Operators and Markov Processes
Part I. Fourier Analysis and Semigroups
Chapter 1. Introduction
Chapter 2. Essentials from Analysis
2.1 Calculus Results
2.2 Some Topology
2.3 Measure Theory and Integration
2.4 Convexity
2.5 Analytic Functions
2.6 Functions and Distributions
2.7 Some Functional Analysis
2.8 Some Interpolation Theory
Chapter 3. Fourier Analysis and Convolution Semigroups
3.1 The Fourier Transform in S(R^n)
3.2 The Fourier Transform in L^p(R^n), 1

Citation preview

PSEUDO DIFFERENTIAL OPERATORS PROCESSES Fourier Analysis and Semigroups

N.Jacob

Imperial College Press

PSEUDO DIFFERENTIAL OPERATORS MARKOV PROCESSES

PSEUDO DIFFERENTIAL OPERATORS MARKOV PROCESSES Volume I

Fourier Analysis and Semigroups

N.Jacob University of Wales Swansea, UK

ICP

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PSEUDO-DIFFERENTIAL OPERATORS AND MARKOV PROCESSES Volume I: Fourier Analysis and Semigroups Copyright © 2001 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 1-86094-293-8

Printed in Singapore by Utopia Press Pte Ltd

Contents Preface

ix

Notation General Notation Functions and Distributions Measures and Integrals Spaces of Functions, Measures and Distributions Some Families of Functions Norms, Scalar Products and Seminorms Notation from Functional Analysis, Operators

xi xi xii xiii xiv xv xvi xvi

Introduction: Pseudo Differential Operators and Markov Processes

xix

I

Fourier Analysis and Semigroups

1 Introduction 2 Essentials from Analysis 2.1 Calculus Results 2.2 Some Topology 2.3 Measure Theory and Integration 2.4 Convexity 2.5 Analytic Functions 2.6 Functions and Distributions 2.7 Some Functional Analysis

1 3 11 11 20 23 37 40 43 52

Contents

VI

2.8

Some Interpolation Theory

3 Fourier Analysis and Convolution Semigroups 3.1 The Fourier Transform in S(Rn) 3.2 The Fourier Transform in L p (M n ), 1 a u = {~id)au ^u{x) = lim »('+«)-»(*? (-•0

t>o 0 , is a continuous negative definite function and contains all information about ( X t ) t > 0 . Some results for ( X t ) t > 0 are best proven by looking directly at ip, for example t h e process is conservative if and only if "0(0) = 0. But for pathwise considerations it is useful t o take the Levy-Khinchin representation

JR"\{O}\

i + i2/r/

Our starting point is t h e following observation made in [169].

Let

x

(Xt)t>0, P ) be a (nice) Feller process with state space R™. T h e n t h e V — J x£M.n function q(x, 0 : = - lim — ^ t->o

>-

(0.3)

t

completely characterises ( ( X t ) t > 0 , P x ) \

—

• In analogy t o the theory of par/xERn

tial differential operators we will call q(a;, £) t h e symbol of the process ((Xt)t>0,Px) . Let us t r y t o understand (0.3) heuristically from two dif\

—

/x£Mn

ferent starting points. First suppose t h a t ( ( X t ) f > f ) , Px)

is given as a nice Feller process. For

fixed i £ I " we may consider the random variables X t under Px and look at their characteristic functions, i.e. we may consider

At(i.O := WU^-^A

= e-ix^Ex(eiXt^).

(0.4)

Denoting by (T t )t>o the semigroup associated with ( ( X t ) t > 0 , Px) \

Ttu(x)

= E > ( X t ) ) = (27r)-"/ 2 /

eixo is a family of pseudo-differential operators a n d t h e symbol of T t is At(a;, £). By assumption (T t )t>o is a Feller semigroup. Hence we may look at its generator Au=limTtU~~U, t->o t

(0.6)

where the limit is taken in the strong sense in t h e space C 0 0 (M n ;]R). Substituting (0.5) into (0.6) we arrive at Au(x) = - ( 2 T T ) - " / 2 [

eix* Xt(x, £) must be positive definite (and continuous). Supposing for simplicity for a moment t h a t the process is conservative we have Xt(x,0) = 1 and from t h e general theory of negative definite functions it follows that £ i->- —' t^xf'~ ' must be negative definite, which must also hold for t h e limit. Hence, we find t h a t t h e symbol q(x, £) of the process must be a continuous negative definite function with respect t o £. There is another way to understand (0.7) (or (0.3)). As generator of a Feller semigroup A has to satisfy the positive maximum principle, i.e. sup u(x) = u(a;o) > 0 implies Au(a; 0 ) < 0.

(0.8)

In [62] P h . Courrege characterises the operators satisfying t h e positive maximum principle. In particular he proved under t h e reasonable assumption t h a t Cg°(K";R) C D(A), then for u £ Cg°(M n ;R) t h e operator A has t h e representation (0.7), where q(x, £) is a locally bounded function in x which is continuous and negative definite in £. Clearly, t h e generator completely characterises t h e semigroup, and further, the family of characteristic functions (Xt(x,£))t>0 completely characterise the process. Now we are in t h e analogous situation as S. Bochner was in case of Levy processes: T h e study of t h e process is reduced to t h e study of its symbol! As in t h e case of Levy processes it is sometimes more advantageous to use q(a;, £) directly, but sometimes it is better to use its Levy-Khinchin

xxii

Introduction: Pseudo Differential Operators and Markov Processes

decomposition n

n

- q ( z , 0 = c(x) + ^ b j ( x ) ^ - + ^2 j=i

+/

aki(x)£k£i

k,i=i

(l-e-^- I f T ^)^fj#N(,,d y ) .

(0.9)

Now we may state the aim of the monograph: Following Bochner: Construct and study Markov processes by s y s t e m a t i c a l l y m a k i n g u s e of t h e i r s y m b o l s . It turns out t h a t in doing so a lot of analysis is to be developed which is not covered by standard theories of pseudo-differential operators, of function spaces etc. This is due to the fact t h a t the symbols under considerations do in general not belong to classical symbol classes. Thus, although stochastic processes are our aim, we have to develop our analytic tools first. We divide our presentation into three parts: 1. Fourier analysis and semigroups. 2. Generators and their potential theory. 3. Markov processes and applications. For each part we will give a separate introduction.

Part I

Fourier Analysis and Semigroups

Chapter 1

Introduction In this volume we discuss those parts of the theory which are necessary to understand why certain pseudo-differential operators are generators of Feller and sub-Markovian semigroups, and we provide the reader with tools from Fourier analysis and the theory of (one parameter operator) semigroups needed later on to construct semigroups starting with pseudo-differential operators, studying their potential theory and examine the associated stochastic processes. Since we want to reach analysists as well as probabilists starting with advanced graduate students, we felt the necessity to achieve a certain selfcontainedness. For this reason in our first chapter "Essentials from Analysis" we collect many results which we do not believe are standard knowledge of graduate students (in both analysis and probability theory) or probabilists. (When starting with probabilistic considerations we will supply an analogous chapter "Essentials from Probability Theory" for graduate students and analysists.) We do not prove these results but give precise references. Further we do not give notes to this chapter. Readers who feel familiar with these topics should start reading from Chapter 3 and consider Chapter 2 as a reference text. Our third chapter is an introduction to Fourier analysis in R™ with special emphasis to positivity. We start with introducing the Fourier transform on R, u > 0 and v ^ 0 the following formulas, 7 € NQ and I = \j\: 1

cF(lnu) =

Y, 7 1 +...+7 1 =7

c

j

W } I l ^ r ' 7 ^ 0 ; J= l

(2.26)

16

Chapter 2 Essentials from Analysis

7 1 + .-.+7 i =7

J=

1

J2

C vanishes at infinity if for any e > 0 there exists a compact set K C G such t h a t |u(x)| < e, if a; G if c . We define Coo (G) := {u G C(G) | u vanishes at infinity } ,

(2.34)

C™(G) := {u G C m ( G ) | dau G Coo(G)for |a| < m } .

(2.35)

and

Note t h a t C ( £ ) , C f e (£), CU(E), C^E) and C 0 ( £ ) are well defined for any topological Hausdorff space E. T h e spaces C ^ ( G ) , C™(G) and C™(G) are Banach spaces with respect to the norm I M I c r == E \a\ OO.

Sometimes, when it is necessary to emphasise t h a t we will work for a while only with real-valued functions belonging to such a function space, we write u G C m ( G ; R ) , etc. When constructing Markov transition functions we work with the one— point-compactification Goo of an open set G C R n . By definition Goo is given by G U {oo}, oo denoting an ideal point, and the topology Coo on G ^ is the system Ooo = O U O, where O is the usual topology on G and O := {O = Goo \ K | K c G c o m p a c t } . It follows t h a t (Goo, Coo) is a compact space and G C G oo is an open and dense subset. L e m m a 2.1.2 The following

assertions

are equivalent

1. f G C o o ( G ) ; 2. f G C(G) and for every e > 0 the set {|f| > e} is compact; of n m (x — k), is a partition of unity, i.e. J2keZ tp (x — k) = 1 for all x £ R™. It has the property that for m £ No given, there exists a constant M = M(n,m,e) such that \daipk(x)\ < e~^M for all \a\ < m, and there exists a number N = N(e, n) such that the intersection of N + 1 of the supports of the functions ipk is empty. C. For any £ > 0 there exists 4>e £ C°°(R) such that -e < (f>E(x) < 1 + e for all x £ R, 4>e\[o,i] = 1 and whenever x < y it follows that 0 < (j)e{y) — 4>e(x) < y — x. A proof of part A and B is given in [237], Corollary 1.3.2 and Theorem 1.5.2, respectively. A proof of part C is indicated in [102], Problem 1.2.1. In some situations we have to work with lower semicontinuous or upper semicontinuous functions. A function f : G —>• (—00,00], G C R n open, is said to be lower semicontinuous at XQ £ G if f(a;o) < 00, then for every e > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > / ( z o ) — £ for all x £ U{XQ), and if f(a;o) = 00, then for every M > 0 there exists a neighbourhood U{XQ) C G such t h a t i(x) > M for all x £ U(x0). If f is lower semicontinuous at all points x £ G, then it is called lower semicontinuous on G. T h e set of all lower semicontinuous functions on G is denoted by %i.s.c{G). We call a function g : G —>• [—00, 00) upper semicontinuous if —g : G —> (—00, 00] is lower semicontinuous. The set of all upper semicontinuous functions on G is denoted by rHu.s.c.{G). P r o p o s i t i o n 2 . 1 . 4 A. Leti,g £ 'Hi.s.c.{G) and a > 0. Thena-i, f+g andiAg are lower semicontinuous too. B. For any non-empty set % c 'Hi.s.c. {G) the function defined by u(x) := sup {f(a;) | / £ V.} is also lower semicontinuous. C. Let u : G —¥ [0, 00) be a non-negative lower semicontinuous function. Then we have u(x) := sup {i(x)

I f £ C 0 (R") andO < i(x) < u(x) } .

D. Let u £ T-li.s.c.{G). Then there exists a sequence ({V)V£H of functions f„ : G —> R such that f„ < f„+i for all u G N and VL{X) = lim iu{x) — supf„(a:).

continuous

2.1

19

Calculus Results

E. Let u £ V.i.s.c.{G) and K C G a compact set. such that u(x0) = inf u(x). xeK

Then there exists XQ £ K

A proof of Proposition 2.1.4 is given (partly) in [126], p.89, or in [8], § 3. D e f i n i t i o n 2 . 1 . 5 A family {ij : G -» [—00,00] \j £ 1} is called filtering increasing if for j \ , J2 £ I there exists J3 £ I such that ij1 < f,3 and f,-2 < f,-3. T h e o r e m 2.1.6 Let H C 'Hi.s.c.{G) be a filtering increasing family such that f(a;) ;= sup{g(a;) | g £ T-L} is upper semicontinuous. Then f is continuous and f is approximated uniformly on compact sets K C G by the family H, i.e. for every e > 0 and every compact set K C G there exists g£ £ % such that 0 < f(ir) - ge(a;) < s for all x £ K. A proof of Theorem 2.1.6, which is often called Dini's theorem, is given in [8], § 4. Let G C R n be an open set. We call u : G —> C Holder continuous exponent A £ (0,1] if sup K

g

) - y i


£ C X defined for s > 0 and p £ P by UPt£ := {x £ X \p(x) < £ } . Then the set Bo of all finite intersections of elements ofB'0 form a base ofO £ X which turns X into a locally convex topological (Hausdorff) space. Conversely, let X be a locally convex topological vector space and for a convex, balanced and absorbing set U C X define the Minkowski functional Pu(x) : = i n f { A > 0 \x€ AC/}. Then P\j is a seminorm and the family of all these seminorms generates the topology on X in the sense described in the first part of this proposition. A linear operator A : X —» Y mapping a topological vector space X into another topological vector space Y is continuous if and only if it is continuous in 0 £ X. Moreover, when the topologies in X and Y are generated by families Px and Py of separating seminorms, then A is continuous if and only if for any q £ Py there exist finitely many seminorms pj £ Px, 1 < j < I = l(q), such that q(Ax) < c max Pj(x)

(2-43)

holds. In particular, a linear functional u : X —¥ C is continuous if and only if there exist finitely many seminorms pj £ Px, 1 < j < I = 2(u), such that |u(a;)| < c max Pj(x)

(2.44)

holds. Let q be a seminorm on X. We call q continuous with respect to the family Px if (2.43) holds with A = id^- Suppose that Pi and P 2 are

22

Chapter 2 Essentials from Analysis

two separating families of seminomas each turning the vector space X into a locally convex topological vector space. These topologies on X are equivalent if and only if every seminorm in Pi is continuous with respect to Pi and every seminorm of P\ is continuous with respect to Pi. In this situation we call Pi equivalent to P2. For a locally convex topological vector space we denote by X* the space of all continuous linear functionals u : X —¥ C, i.e. X* is the dual space of X. Let X be a locally convex topological vector space. The weak topology on X is the weakest topology on X which makes every u. £ X* continuous. Note that every weak neighbourhood of 0 € X contains a neighbourhood of the type {x e X I \UJ(X)\ < £j for 1 < j < m}

(2.45)

where u, € X* and £j > 0. Next we want to introduce a suitable topology on X*. For this let x € X be given. We can define a linear functional lx on X* by lx(u) := u(x) for all u 0. For the weak-*-topology we have the following compactness result. Theorem 2.2.2 (Banach—Alaoglu) Let X be a topological vector space and U(0) C X a neighbourhood ofOeX. Then the set K:={ueX*

I [u(sc)| < I for allx £ U(0) }

(2.47)

is weak-*-compact. For a separable space X, i.e. there is a countable dense set in X, we have more results. Theorem 2.2.3 A. Let X be a separable topological vector space and let K C X* be a weak-*-compact set. Then there exists a metric on K turning K into

2.3

23

Measure Theory and Integration

a metric space with metric topology equivalent to the weak-*-topology (on K). B. Let U(0) C X be a neighbourhood ofO£X,X being a separable topological vector space. Further let (u„)„gN be a sequence in X* such that |u„(:r)| < 1 for all x £ U(0) and i / £ N . Then there exists a subsequence (u„fc)fc6N of (u„)i,gN and u £ X* such that for all x £ X we have lim ul/k(x) = u(x). k—>oo

R e m a r k 2 . 2 . 4 In general we call a topology metrizable, if there exists a metric generating an equivalent topology. A locally convex topological vector space which is metrizable such that the metric space is complete is called a Frechet space. W h e n working with mappings f : X —>• Y from a topological vector space into another, it is often sufficient to restrict the attention to sequentially continuous mappings, i.e. mappings with the property t h a t xv —>• x in X implies t h a t f(a;„) ->• f(a:) in Y.

2.3

Measure Theory and Integration

We assume the reader to be familiar with the basic notions and results of measure theory and the theory of integration including the notion of properties which do hold almost everywhere (a.e. as abbreviation), the completion of a afield, central theorems like the Caratheodory extension theorem, the dominated and monotone convergence theorems, the theorems of Tonelli and Fubini, the Radon-Nikodym theorem, etc. For these topics and for the following results which we state without proofs and without reference, our standard references are the book [18] of H. Bauer, the beautiful work [217] of P. Malliavin, and the monograph [256] of W. Rudin. Let Q be a set and S C V(£l). We call S a n-system in fl, whenever A,B € S implies A D B e S. A 7r-system is called a d-system in fi, if ft G S, A,B £ S and A C B then B\A £ S, and for any sequence (j4„)„ e N, Au £ S such t h a t A„ C Av+\ it follows t h a t (Ji/gN ^ 6 &. T h e o r e m 2.3.1 ( M o n o t o n e class t h e o r e m ) Suppose that S is Then it follows that d ( 5 ) = a(S).

air-system.

Here d(S) denotes the d-system generated by S and cr(S) is the cr-field generated by S.

24

Chapter 2 Essentials from Analysis

C o r o l l a r y 2.3.2 Let ft be a set and S a •n-system in Q. Further let rl be a vector space of real-valued functions f : fi —»• R such that 1 £ %, XA € H for all A € S, and for any sequence (fv)v€^, f„ G H, such that 0 < iv < f„+i and f = supf„ < oo it follows that f € T-L. Then "H contains all real-valued bounded functions

which are measurable with respect to o~(S).

For a locally compact space G its B orel-o-field is denoted by B(G), but for G = R™ we write B^n\ Note t h a t B^ is already generated by the bounded open sets in R™. T h e Lebesgue measure on R n is denoted by \(n\dx). By definition measures are non-negative, and measures on B(G), G being a locally compact space, are called Borel measures. Let fi be a Borel measure. We call H locally finite if fJ-(K) < oo for any compact set K C G. A Borel measure is inner regular if for any B € B{G) we have / i ( 5 ) = sup {n(K)

| K C B and K c o m p a c t } ,

and \i is outer regular if for any B € B{G) it follows t h a t H(B) = inf {fi(U)

| B C Uand Uopen} .

A Borel measure fj, is called a Radon measure if it is locally finite and inner regular. On fi(") Radon and Borel measures coincide. T h e support supp fi of a Radon measure /x is the complement of the largest open set G such t h a t /i(G) - 0. Let (fi, .4) be an arbitrary measurable space. T h e total mass of a measure H on (Q,,A) is denoted by ||/x|| := //(fi), and .MjJ~(fi) is the set of all bounded measures on (fi,-4), i.e. fi € A4jj"(fi) implies ||/x|| < oo. By Ml(fl) we denote the set of all probability measures, i.e. measures fi with total mass ||/z|| = 1. A sub-probability measure is an element of .Mjj"(fi) such t h a t ||/x|| < 1. For a G 0 the Z>irac measure at a is denoted by e a , but in case fi = R n and a = 0 we often write £ instead of £Q. Now let G be a compact space. A signed measure on (G, B(G)) is a mapping /i : B(G) —> R such t h a t we can write y,{A) = Hi(A) — fJ>2(A) for two measures /xi,/X2 G A^j"(G). Whenever p is a signed measure there exist two measures Hi and y% in M£(G) such t h a t fi = (1° — (i% where /i° and /i° are mutually singular, i.e. there exists a Borel set A0 C G such t h a t

A(A°) = /i?(G) and /^(^°) = 0.

(2.48)

2.3

25

Measure Theory and Integration

Let (fi, A, //) be a measure space. We call AG A a fi-atom if fi(A) > 0 and for any B C A, B e A, either //(£?) — 0 or //(A \ 5 ) = 0 holds. In particular we have / / ( J B ) = 0 or n(B)

=

p(A).

T h e total mass of // is given by ||//|| := //°(G) + //°(G). T h e set of all signed measures is denoted by M.(G). Let G be a locally compact space. A signed Radon measure // is given by two mutually singular Borel measures //J and / J ° . W i t h |//| := /i° + ^2 w e consider the completion of 23(G) with respect to |//| and denote this cr-field by B^G). For A G B^{G) such t h a t |//|(A) < oo we define //(A) = //?(A) — n\(A). T h e bounded signed Radon measures on G are denoted by M.b(G). It follows t h a t (Mb, ||-||) is a Banach space. T h e o r e m 2.3.3 The Banach space (Mb(Rn), ||.||) is the dual space of n Coo(]R ;]R). For /J, G ^Vl6(]Rn) a linear continuous functional on Coo(]R™;R) is defined by UH

/" u(a:)/x(da;) = /" u(a:) ^°(da;) 7R" 7R"

/

u(x) (4(dx).

(2.49)

JU"

This theorem is also related to the various variants of Riesz' representation theorem. A linear functional I on C ( G ; R ) or any of its subspaces, G being a locally compact space, is called a positive functional if l(u) > 0 for all u > 0, u € C(G; R) (or in case where I is defined only on a subspace of C(G; R), u > 0 should be taken only from this subspace). T h e o r e m 2 . 3 . 4 A. Let G be a metrizable locally compact space which is countable at infinity. Then I is a positive linear functional on Co (G; R) if and only if there exists a unique locally finite Borel measure fj, such that Z(u) = / u{x) //(da;)

(2.50)

JG

for all u € Co(G;R). B. In case that G is compact, I is a positive linear functional on C ( G ; R ) if and only if (2.50) holds with a unique measure // £ Ai^(G) and all u e C(G; R ) . C. A linear functional I on Coo(R n ; R) is positive if and only if there exists a unique Borel measure // on R™ such that (2.50) holds for all u G Coo(R n ;R). Moreover, for all u G C 0 0 ( R " ; R ) it follows that JRn u(x) //(da;) < oo. We may also extend the considerations to complex-valued measures. This is possible in two equivalent ways. Either we define a complex-valued measure

26

Chapter 2 Essentials from Analysis

with the help of two pairs (M?, M°) and (^°, fi°) of mutually singular measures by (i(A) = fi{A) - &{A) + i(£(A) -

fi(A))

(2.51)

whenever (2.51) makes sense, or we introduce complex-valued measures as the dual space of Co(G), now we consider of course complex-valued functions. T h e total mass of M given by (2.51) is now defined by ||/i|| := Y^=i M?(^)> a n d IMI is the measure $^,- =1 M?- We denote the space of all complex-valued measures by MC(G); M.f(G) is the set of all bounded complex-valued measures. Let ( f i i , - 4 i , / / i ) and (0,2, .4.2, M2) be two measure spaces. T h e product measure of MI and M2 is denoted by Mi ® M2- We introduce the mapping Ak : Rnk ->• R n , (xi,... ,Xk) i-+ xi + ... + Xk and give D e f i n i t i o n 2.3.5 Let Mj G At+(R™), 1 < j < k, be measures. The image of Mi ® • • • ® Mfc under Ak is called the convolution of these measures and is denoted by Mi * . ..*/zjb := Ak(fJ*i ...® Mfc)-

(2.52)

Obviously we have Mi * . . . *Mfc G A^j"(R n ) and ||MI * • • • *Mfc|| = ||MI|| • . . . • ||/ifc||. T h e convolution is associative and commutative, i.e. (Mi * fj,2) * M3 = Mi * (A*2 * M3) and Mi * M2 = M2 * Mi

(2.53)

for all Mj ^ A^j"(R"). For any non-negative measurable function on Rn we have /

i(x) (Mi *M2)(dz) =

JR"

/

/

f(a; + y)Mi(da;)M2(dy)

,/R" ./R"

= 1 1

i(x + y)^(dy)tx1(dx)

(2.54)

which yields in particular (Mi*M2)(5)=/

Mi(5-y)M2(d2/)= /

M2(S-z)Mi(dz)

(2.55)

n

for all i? G B( \ Using (2.51) we may extend the convolution to .Mf,(R n ) as well as to M%(Rn), and (2.53) remains valid for this extension. Moreover we have Mi * (M2+M3) = Mi * M 2 + M i *M3

(2.56)

2.3

27

Measure Theory and Integration

and «(Mi * M2) = {afJ-i) * M2 = Mi * ( 0, € R or € C depending on whether Hj € M^(Rn), respectively. In each case we have fj,*e

(2-57) Mb{Rn)

= £*H = H

or

Mf(Rn),

(2.58)

Thus it follows t h a t _Mb(R n ) and Mf(Rn) are algebras with respect to the vector space operations and the convolution as product. Moreover, £ is a unit in these algebras. For integrable functions f, g : R n ->• C or R, iX^ and gA^n) are elements in M%(M.n) or .Mi,(R n ), respectively. Thus we can consider the convolution iX^ * gA< n \ and we find f A ^ * gAo, Vt S M^(G), we may define ^lim-fo+,-77,) dt

S_K)

S

(2.63)

2.3

29

Measure Theory and Integration

in these topologies. For example for the vague convergence (2.63) does mean t h a t for a l l u G C 0 (G) lim - ( / u(x) rjt+s(dx) «->o s \JG

-

/ u(x) JG

r]t{dx)

exists and leads to a measure fi, defined by / u(x) fi(dx) = lim - ( / JG

u ( i ) Vt+s{dx) -

s->0 S \JG

[ u(x) r,t(dx)) Jo

(2.64) J

for all u G C 0 ( G ) . Let (Cl, A) be a measurable space and f, g : Cl —)• R be measurable functions. We write (f A g)(x) := min(i(x),g(x))

and (f V g)(x) := max(f(x),

g(x)).

(2.65)

Both functions f Ag and f Vg are measurable too. Thus we may define f+ := f VO as the positive part of f and f ~ := — (f A 0) as the negative part of f. We always have t h a t f+, f_ > 0 and f + (a:) • i~(x) = 0 for x € 0 . Moreover it follows t h a t f=f+-f"

and | f | = f + + r .

(2.66)

Note however t h a t f+ and f - do not have disjoint supports since they coincide on the set {x £ fi | f(a;) = 0 } . In case t h a t fl = G is a topological space and A = B{G) is the Borel cr-field, the set of all measurable functions f: G —>• C is denoted by B(G), and Bf,{G) is the set of all bounded functions f S B(G). When we want to emphasise t h a t we have to work with real-valued functions we write f G B(G;W) or f G Bb(G;R), respectively. For any measure space (CI, A,n) the spaces Lp(Cl,n), 1 < p < oo, are the usual Lebesgue spaces (of equivalence classes) of measurable functions f : £~i —>C with finite norm

Hf|| L P :=^JfW| P Md^))

P

,l L P (R") is a Ziraear operator. Moreover we have IIWHLP

< IHIL?

(2.79)

and lim || J e ( u ) - u|| L P = 0.

(2.80)

£->0

In case p = 2 we also have (JE(u),v)0 = (u,Js(v))0 for all u, v G L 2 ( R n ) , i.e. J £ is a bounded selfadjoint Moreover, for u G Co(R n ) we have lim/

jv(x - y)u(y) dy = u(x) =

*-e- (JTJ)TJ>O converges vaguely to EQ and (^(x

operator on L 2 ( R n ) .

u(y) ex(dy),

(2.81)

— .))v>o converges vaguely to ex.

Note t h a t the approximation result holds also true for u € Coo(R n ). Moreover, the special type of j e is not necessary if only the convergence is of interest. For any € L 1 ( R n ) such t h a t / R „ (x) dx = 1 let us define e(x) = e~n ( | ) . Then it follows t h a t ||u * E - u|| L P -> 0 as e ->• 0 for all u G L p ( R n ) , 1 < p < oo. Once again the result remains true for u G C 0 0 ( R n ) . A proof for this is given in [289]. Now let u, v G L P ( R " ) , 1 < p < oo, be functions such t h a t u > 0 a.e. and 0 < v < 1 a.e., respectively. It follows t h a t J £ (u) > 0 and 0 < J £ (v) < 1 hold. Furthermore, since |((0 V u{x)) A 1) - ((0 V v(x)) A 1)| < |u(x) - v(a;)| a.e.

(2.82)

holds for all u,v G L p ( R n ) , we find t h a t a sequence ( u „ ) „ e N , uv G Cg°(R n ), converging in L p ( R n ) to u G L P ( R " ) , has the property t h a t ((0 V u„) A 1 ) „ 6 N converges in L p ( R n ) to (0 V u) A 1 and ( 0 V u „ ) M £ C 0 ( R n ) . T h e next theorem characterises the conditionally compact subsets in L P (G). T h e o r e m 2.3.18 A subset K C L P ( G ) , G c l " a Borel set, 1 < p < oo, is conditionally compact if and only if the following three conditions hold

33

2.3 Measure Theory and Integration 1. sup ||u||LP < c; 2. lim sup ||u(. + h) - u ( . ) | | L P = 0; |h|->0u€if

3. lim sup

G\BR(0)U

= 0.

X

LP

iVoie rAai for a bounded Borel set G C R™ t/ie Zas£ condition is empty. Let g : G —• R be a strictly positive measurable function and consider the measure fx := gA(")|G on G. The space L P (G, /x) consists of all measurable functions u such that u • g G L P (G), or equivalently, for any v G L P (G) the function ^ • v belongs to L P (G, fi). Moreover, we have ||u|| LP , G •. = ||ug|| LP , G j. Thus a set K C L P (G, fj,) is conditionally compact if and only if the set Kg := {v = • Ju(x — y) n{dy) defines an element in L 1 (R n ). More generally, we may introduce the notion of a kernel. Definition 2.3.19 Let (SI, A) and (SI', A') be two measurable spaces. We call K : SI x A' ->• [0, oo] a kernel from (SI, A) to (SV, A') if x i-> K(x, A') is for all A' € A' a measurable function and A' H-> K(X, A') is for all x G SI a measure. A kernel K is called a Markovian kernel ifK(x,Sl') — 1 for all x G SI, and it is called a sub-Markovian kernel ifK(x, SI') < 1 for all x G SI. Given a kernel K from (SI, A) to (SI', A'). We may define an operator K op on all non-negative measurable functions u : SI' —> K by (K op u)(x) := J u(x')K(x,dx').

(2.83)

It follows that K op u is a non-negative measurable function on SI. Suppose that Kop is an operator from all non-negative measurable functions u : ST.' —• R to the space of all measurable functions v : SI —> R having the representation (2.83). Then the right hand side in (2.83) is called the kernel representation of K op .

34

Chapter 2 Essentials from Analysis

T h e o r e m 2 . 3 . 2 0 Suppose thatK.op is an operator from all non-negative measurable functions u : Cl' —>• R to all measurable functions v : Cl —> R. Assume further that Kop has the property that u > 0 implies K o p u > 0, and that for every sequence (u„)j,gN of non-negative measurable functions \xv : f2' —> R such that 0 < u„ < u „ + i it follows that supKopU^ = K o p ( s u p u t / ) . Then there exists a kernel K /rom (fi,»4) io (Cl',A') Kopu(z) = /

such that Kop has the kernel

representation

u{x')K(x,dx')

for all measurable functions

(2.84)

u : $7' —>• R, u > 0.

Clearly, we may extend K o p in the usual way to suitable integrable functions. In particular, if all the measures (K(a;, .)) x en are bounded, K o p is well defined on the space of all bounded, real-valued measurable functions u : Q' —¥ R and K o p u is a bounded measurable function on fi. Suppose t h a t N : G" x B(G) -> [0,oo], G,G' C R n Borel sets, has the property t h a t for every x € G' the mapping N(x,.) : 13(G) —• [0, oo] is a measure. Still it is possible to define the operator N op u(a;) := /

u(y)N(x,dy).

JG

P r o p o s i t i o n 2 . 3 . 2 1 Suppose that for all u € C Q ° ( G ) the

mapping

a; M- / u(y) N(z, dy) is continuous. Then x \-t N(a;, A) is measurable for every A S B(G), N(:r, dy) is a kernel.

i.e.

This proposition is proved by approximating the functions x A , ^ € B(G), pointwise by suitable sequences of functions belonging to Co°(G). Suppose t h a t Ki is a kernel from (£li,Ai) to ( ^ 2 , ^ 2 ) and K2 is a kernel from (^2,^.2) to (^3,^.3). Suppose further t h a t the measures (Ki(a;i, . ) ) I i e n i as well as (K 2 (a;2, -))x 2 en2 a r e bounded. Then the operators K„ p , K„ p and K ^ o K ^ are defined by K^p := ( K i ) o p and K ^ := ( K 2 ) o p , respectively, acting on bounded measurable functions, and K^ p o K ^ is just the usual composition of operators. For a bounded, measurable function u : $^3 —>• R we find Kopu(x1):=(K2opoKfp)u(x1)=

[ [ Jn2 Jn3

u(x3)K2(x2,dx3)K1(x1,dx2).

2.3

35

Measure Theory and Integration

On the other hand, by Theorem 2.3.20 the operator Kop has a kernel representation with a kernel K3i(a:i, dx^) which yields K31(Xl,A3)=

f

K2(x2,A3)K1(x,dx2)

(2.85)

for all x\ S fli and A3 e .43. In particular, when a family (Kt)t>o of kernels from (R™, Bn) into itself is given, we may consider the kernels K,it{x,A)=

[

Ks(y,A)Kt(x,dy).

(2.86)

Often we have to use the following two results. L e m m a 2 . 3 . 2 2 Let E be a metric space and (£l,A,n) be a measure space. Further suppose that u : ExQ, —» R is a function with the following properties: u> i-t \i(x,u>) is for all x G E integrable with respect to fi, x i-> U(X,OJ) is continuous at XQ £ E for all ui £ ft, and there exists a function h £ L 1 (r2,/n) such that |f(a;,o;)| < h(u>) for all x £ E and w £ fi. Then the function x i->JQ{(x,u>) fi(dui) is continuous at XQ. L e m m a 2 . 3 . 2 3 Let I c R be a non-empty interval and (Cl,A,fJ,) a measure space. Assume that u : I x fi —>• R has the following properties: u(x,.) £ L 1 (fl,/i) for all x € 1, for all ui £ Q, the function x H> U(X, UI) is differentiate on I, and there exists a function h £ L 1 (f2,//) such that \-^u(x,u>)\ < h(w) for all x £ I and u> £ fi. Then the function x i-» Jn u(x,u>) /i(du>) is differentiate, the function to i-> -^n{x,ijj) is an element in L 1 (fi!,/i) for all x £ 1, and — I u{x,u)

fi{du) = I -^(x,u>)

Ai(dw).

(2.87)

We need various results on the integration of functions with values in vector o

spaces. Let I C R be a closed interval such t h a t 1 ^ 0 . Further let (X, \\.\\x) be a Banach space. We denote by C(I; X) the space of all continuous functions u : I —¥ X, and C 1 (I; X) is the space of all these functions which are continuously o

differentiable in I and have onesided derivatives at the endpoints of I. For any compact interval [a, b], a ^ b, we may define the Riemann integral f u(t) dt for u £ C([a, b]; X), and then, as usual, we may define the improper Riemann integrals /»oo

/ Ja

pb

u(t)dt,

/ J— 00

/-oo

u{t)dt

and

/ J — 00

u(t) dt

36

Chapter 2 Essentials from Analysis

for suitable functions u : I —• X, I = [a, oo), I = (—oo, 6] or (—oo, oo), respectively. The following result is taken from the book [88], p.9, of St.Ethier and Th. Kurtz. Lemma 2.3.24 A. Let u G C(l;X) such that Jj ||u(t)|| x dt < oo. Then the integral J. u(£) di exists and we have the estimate

fu(t)dt

< [\\u(t)\\xdt.

(2.88)

In particular, for every compact interval I, every u G C(I;X) is integrable. B. For u G C1([a, b]; X) we have

jT (j|U(i))di = u(6)-u(a).

(2.89)

C. Let (A, D(A)) be a closed operator on X and u G C(I;X) such that u(£) G D(A) for all t G I, Au G C(I; X) and u as well as Au are integrable (over I). It follows that J, u(t) dt G D(A) and we have k(f

u(t) dt j = f Au(t) dt.

(2.90)

Another way for denning an integral for functions u : I —> X is due to S. Bochner [37]. We follow the book [315] of K. Yosida. Let {£l,A,n) be a measure space and u : S l - ^ I a finitely valued function such that ' k

u(u,) = ^ x n »

(2.91)

for Xj G X and Clj C 0, //($!,•) < oo. We call functions of type (2.91) fi-Xelementary functions. The //-integral of u is now denned by

J

A;

u(w) /i(dw) := J2 xolJL X where (Q, A, (J,) is a measure space and X is a topological vector space. For details we refer to W. Rudin's monograph [255]. We suppose that X* separates points in X, i.e. for x, y G X, x =£ y, there exists u G X* such that u(a;) ^ u(y). Now let f: 0 —• X be a function and for u G X* define the function u(f) : Q, —> C (or R) by u(f)(w) — u(f(w)). Suppose that u(f) is /Li-integrable. If there is some x G X such that u(x) = f u(f)(w) /z(du;)

(2.94)

Jn for all u G X*, we define the /x-integral of f by x=

I fd/i.

(2.95)

Jn A criterion for the existence of the /x-integral (2.95) is Theorem 2.3.26 Let X and X* be as above and for a compact Hausdorff space G let (G,B(G),fi) be a probability space, i.e. {G,B{G),JJL) is a measure space and fJ-(G) = 1. If i : G —>• X is continuous and if the convex hull of f(G) C X has a compact closure in X, then the ^-integral x = JQidfi exists.

2.4

Convexity

We need only a few notions and results from convexity theory, but also some concrete applications. Our standard references are the lecture notes [17] of H. Bauer, G. Choquet's lectures on analysis [57]-[59], once again W. Rudin's monograph [255], and for Choquet theory we refer to the book of R. Phelps [238]. Let X be a vector space and K C X. The set K is said to be convex if for all x,y G K and A G [0,1] we have Aa; + (1 - X)y £ K. The intersection of arbitrarily many convex sets is convex again, hence for any set H C X we may define its convex hull conv(H) as the smallest convex set containing H, i.e. conv(tf) = p | {K D H I K convex}.

(2.96)

38

Chapter 2 Essentials from Analysis

Let K C X be a non-empty convex set. A point XQ € K is called an extreme point oi K ii K \ {x0} is convex too. Equivalently, xo is an extreme point whenever xo = Xx + (1 — A)y for some x,y & K and A € (0,1) it follows t h a t XQ = x = y. T h e set of all extreme points of K is denoted by ext(K). A subset C c X i s called a cone with vertex at 0 G X , if for all A > 0 it follows t h a t AC C C. We call C a peafced cone if C n ( - C ) = {0}. Further, C is said to be a cone with base if there exists a hyperplane H in X such t h a t 0 g H and for every x G C \ {0} the intersection of {Az | A > 0 } and i7 is non-empty. T h e set B := C C\ H is called a 6ase of C. A cone is said to be convex if it is convex as a subset of X. It follows t h a t a cone C is convex if and only i f C + C c C . A central question is whether we may describe a convex set by using only its extreme points. An important first result is the theorem of M. Krein and P. Milman. T h e o r e m 2.4.1 (Krein—Milman) Every compact convex set K ^ 0 in a locally convex topological vector space is the closure of the convex hull of its extreme points, i.e. K = conv(ext(K)).

(2.97)

Suppose t h a t X is a locally convex topological vector space with dual space X*. Further, let K C X be a non-empty compact set and [i a probability measure on K, i.e. a Radon measure on the Borel sets of K such t h a t fx(K) = 1. We call x € X a barycentre of /U if u(x) = /

u(y) /i(dy)

(2.98)

JK

for all u e l ' . T h e o r e m 2.4.2 ( C h o q u e t ) Let K C X be a non-empty compact convex set and suppose that the induced topology on K is metrizable. Then for every x G K there exists a probability measure ^ on K with barycentre x such that s u p p / i C ext(K). Choquet's theorem may be used to obtain concrete representation results. Let i f be a compact Hausdorff space. Consider a linear subspace H C C(K; 1R) such t h a t 1 G H and denote by H* the dual of H (with the weak-*-topology).

39

2.4 Convexity The set A(H):={leH*

\l(l) = l=\\l\\}

(2.99)

is a compact convex set in H* and every functional lx(i) :— i(x), x G K and i £ H, belongs to A(H). We call H point separating if for all £1,2:2 G K, xi ^ X2, there exists f G H such that i(x\) ^ f(x2). The set dHK := {x G K I lx G ext{k{H)) }

(2.100)

is called the Choquet boundary of iJ. Theorem 2.4.3 Suppose that H is point separating. A point x belongs to djjK if and only if /x = ex is the only probability measure with i(x) = f f(y) M(dy)

(2.101)

JK

for all { € H. Theorem 2.4.4 Let H be a point separating subspace of C(K;W) such that 1 G H. For any I G H* exists a measure fi on K such that l{() = f t(x) n(dx)

(2.102)

JK

for all f G H. Moreover, fJ.(G) = 0 for any Baire set G such that GtldtfK

= 0.

Recall that the Baire sets form the smallest cr-field in K such that all elements f G C(K;W) are measurable. In presenting Theorem 2.4.2-Theorem 2.4.4 we used the monograph [276], but note that there are no proofs for these results in [276]. Proofs were given in [238]. Finally in this section we want to discuss shortly Hausdorff's moment problem. Let (ci/)„gN0 be a sequence of real numbers and put Jc„ := c„ + i — cv. We call the sequence (C 1/ )^ 6 N 0 completely monotone if and only if {-l)k6kcv

>0

for all k,v G N 0 .

(2.103)

40

Chapter 2 Essentials from Analysis

Theorem 2.4.5 (Hausdorff) A sequence {cv)v&io *s completely monotone if and only if there exists a measure on [0,1] such that cu = f tv n(dt), v e No, Jo

(2.104)

holds. A proof of Theorem 2.4.5 is given in [313], pp. 148-154.

2.5

Analytic Functions

The reader is of course assumed to have some knowledge in the theory of complex-valued functions of one complex variable. As a standard reference we mention the monograph [2] of L. Ahlfors which is to our opinion still one of the best books in the field. Furthermore, we would like to mention R. Burckel's book [51] which is ideal in both precision and historical scholarship. From the theory of complex-valued functions of one variable we quote only the theorem of G. Herglotz, see [122], pp.508-511, and Hadarmard's threeline-theorem, see [51], p.147. Theorem 2.5.1 A. (Herglotz) Denote the unit disk in C by D and let f : D —> C be a function. This function is analytic with Re f > 0 if and only if it has the representation

f(z) =ic+ T ^±^

M(dC),

z € D,

(2.105)

where /i is a finite measure on (—n, w] andc = lmf(0) € R. The representation is unique. B. (Hadamard) Let Cl :— {x + iy | 0 < x < 1, y G R} and Cl its closure. Further let f be a bounded continuous function on fi which is analytic in Cl. Then the function M 7 := sup {|f(7 + ij/)| | y e R } satisfies M1 < M^1'Ml for 0 < 7 < 1. Here we want to discuss shortly some basic results from the theory of complex-valued functions of several complex variables and the theory of functions of one variable with values in some topological vector spaces. As standard references for several complex variable theory we mention L. Hormander's book

2.5

Analytic Functions

41

[151] or the book of St. Krantz [189]. A good reference for functions of one complex variable with values in a topological vector space is W. Rudin's book [255], we refer also to the monograph [179] of T. Kato. Let G C C™ be an open set and f : G - > C a continuous function. We call f analytic (or holomorphic) in G if f is analytic in each of its variables, i.e. for any point a = (a\,..., an) £ G the functions Zj H> gj(zj)

:= f(ai,...,

a,j + zj:...,

an), l X be a weakly analytic function. Then the following assertions hold: 1. f is a strongly analytic

function;

2. if j is a closed path in G, such that ind 7 (w) — 0 for every w 0 G,

then

we have f[(z)dz

= 0,

(2.111)

and if z £ G and ind 7 (z) = 1

f(z) = J_ I l^L dw. v ;

2niJ7w-

z

(2.112)

2.6

43

Functions and Distributions

Obviously, formula (2.111) is Cauchy's theorem and formula (2.112) is Cauchy's integral formula in this new situation. From (2.111) it follows t h a t if 71 and 72 are two closed paths in G such t h a t for all w £ G we have i n d 7 l ( w ) = ind 7 2 (w), it follows t h a t f f(z) dz=

f

f(z) dz.

(2.113)

Moreover, the function f is strongly continuous.

2.6

Functions and Distributions

T h e purpose of this section is to collect various material for functions and distributions as it is needed in later chapters. In principle it should be possible to find these results in L.Hormander's monograph [152]. As further standard reference we mention F. Friedlander [97], W. Rudin [255], and C. Zuily [317]. In our presentation we often used the book [165], First let us introduce some spaces of functions and their topologies. T h e Schwartz space S ( R " ) consists of all functions u € C°°(R n ) such t h a t for all m i , m2 € No R » l i m j ( u ) := sup ((1 + \x\2)m^2 I 6 R

"

V

\dau(x)\)

< 00

(2.114)

\ • u belongs to (CQ) ^ 0. Then u o T is a non-zero constant which does not belong to '(G) having a continuous extension to C°°(G), i.e. £'(G) C V(G). In case of G = R n we also have £ ' ( R n ) C '(G) is by definition the complement of the largest open set G C G on which u coincides with 0 £ V(G). In the sense of this definition S'(G) is the space of all elements u € V'(G) having a compact support. Let us collect some examples. For u G Ljoc(G) a distribution is defined by u() := fG(x)u(x) dx, 4> G C Q ° ( G ) . Moreover, for a locally finite measure \i on B(G) we may define fi() := JG (j>(x) fi(dx) which gives also a distribution. In each of these cases we identify u or jj, with the corresponding distribution. Now we have V(G)

C L/ o c (G) C P ' ( G ) ,

(2.120)

and MC(G)

C V'{G).

(2.121)

Furthermore, it follows t h a t L P ( R " ) , 1 < p < oo, all polynomials and all measurable, polynomially bounded functions belong to '(G)

(dau)() := ( - l ) H u ( d a 0 )

(2.122)

(tf>u)(0:=u(^)

(2.123)

and

2.6

47

Functions and Distributions

for all S Cg 0 (G r ). T h e convolution of functions with distributions can be denned in several situations. T h e o r e m 2 . 6 . 9 For a distribution fined by

u and a function

the convolution

{u*)(x):=u((x-.)) in each of the following

andGG^(G);

2. u e £ ' ( G )

and4>€C°°(G);

3. u G S'(R ) Moreover,

(2.124)

situations:

1. u e P ' ( G )

n

is de-

and G «S(R n ).

in each case u * is a C°° -function

and we have

da(u*) = (dau)*(f> = u*(da), and for u G • G^ is a diffeomorphism, G\, G^ C R™ being open sets. Clearly, for G Cg°(G,2) it follows that o T G Cg°(Gi), thus in the notation introduced before, the pullback T*0 belongs to Co°(Gi). For u G V'{G2) we define the pullback T*u G V(G{) by

^^:=U(^W(T_1)V)'

(2J31)

for all G Co°(Gi). Obviously, det(dT) denotes the Jacobi determinant of the differential of T. Whenever T is such that T*4> G 0 define HA : R™ -> R n by x M- Xx. Definition 2.6.11 A distribution u G X>'(R™ \ {0}) is called homogeneous of degree p £ R if H£(u) = Apu

(2.132)

holds. Note that for u G L}oc(Rn \ {0}) we find with 4> G Cg°(Rn \ {0}) (H^u)(^») = A - " ( u , ( H * / ^ ) ) = A_n /

u(x)

'(Rn \ {0}) be a homogeneous tension to V'{Rn). Then u belongs to {x) n(dx)

(2.134)

JG

for allege

Cg°(G).

(Recall t h a t measures are always non-negative in our definition). In order to apply Theorem 2.6.13.B, let's say in case G = R n , it is sufficient to show t h a t u(|(£| 2 ) > 0 for all G Cg°(R n ). T h e problem is of course t h a t in general the square root of a function G Co°(R™), (j) > 0, need no longer belong to Co°(]R n ). But one can smooth it out by using the Friedrichs mollifier in an appropriate way. Let Gj C Rnj be two open sets and u^- G C(Gj), j — 1,2. On Gi x G 2 C R™1 x R" 2 we define the function ui u 2 by (ui ®u 2 )(a;i,a;2) : = u i ( a j i ) • u 2 ( x 2 ) , Xj G Gj. We call ui®U2 the tensor product of ui and U2- T h e space C 0 X > ( G I ) ( 2 ) C Q D ( G 2 ) is the set of all finite linear combinations Y^T ,- 2= i Uji ®Uj 2 where Ujk G Co°(Gfc). T h e tensor product generalises to distributions. T h e o r e m 2 . 6 . 1 4 Let Uj G V(Gj), j = 1,2. Then there exists a unique distribution u = u i ® u 2 G £>'(Gi x G 2 ) called the tensor product of u i with u 2 such that u(4>i ® fa) = ui(