Higher Order Terms and Discontinuous Symbols
Dimitri Gioev
Stockholm 2001 Doctoral Dissertation Royal Institute of Technology
Department of Mathematics
fentlig granskning f¨or avl¨aggande av filosofie doktorsexamen fredagen den 20 april 2001 kl 14.00 i sal E1, Kungl Tekniska H¨ogskolan, Lindstedtsv¨agen 3, Stockholm.
ISBN 91–7283–072–7 TRITA-MAT–01–MA–03 ISSN 1401–2278
ISRN KTH/MAT/R–03/01–SE c
Dimitri Gioev, April 2001
Universitetsservice US AB, Stockholm 2001
The Ph.D. thesis is mainly concerned with two generalizations of Szeg¨o limit theorem, in a “smooth” and in a “discontinuous” setting. The first generalization has common combinatorial background with certain prob- lems for random walks.
In the first chapter, we explicitly compute the third term in Szeg¨o asymp- totic formula for Zoll manifolds. This includes the case of the operator of multiplication by a smooth function on the sphere in any dimension. A pos- sible application is an expression for a regularized determinant of a mul- tiplication operator, or a zeroth order pseudodifferential operator, on the two dimensional sphere. Further asymptotic terms can also be calculated by our method, however the corresponding computation becomes very in- volved. We also calculate moments and certain joint distributions for maxi- mal non-negative excursions of random walks with independent identically distributed steps.
The main tool in both proofs is a new combinatorial identity, which is called generalized Hunt–Dyson formula, abbreviated as gHD, and is equiv- alent to Bohnenblust–Spitzer theorem (BSt).
In the second chapter, we prove gHD, which is a formula for the quantity X
σ∈Sm
h max(0, xσ1, xσ1 + xσ2, · · · , xσ1 + · · · + xσm)n
− max(0, xσ1, xσ1 + xσ2, · · · , xσ1 + · · · + xσm−1)ni ,
where the summation is taken over all permutations of m real variables x1, · · · , xm, and n is an arbitrary natural power. In the case of power n = 1 the latter identity reduces to the classical Hunt–Dyson formula (HD). After that we establish a connection between gHD and BSt. More precisely, we derive BSt from gHD and vice versa providing the former with a new proof.
In the third chapter, a one term Szeg¨o type asymptotic formula with a sharp remainder estimate for a class of integral operators with symbols hav- ing discontinuities in both position variable and momentum is established.
In this case a logarithmic factor appears in the asymptotics.
2000 Mathematics Subject Classification. Primary: 58J40, 58J37, 58J52, 47B35, 35Pxx, 35S05. Secondary: 60C05, 60G50, 05A19, 05E05.
ISBN 91–7283–072–7 • TRITA-MAT–01–MA–03 • ISSN 1401–2278 • ISRN KTH/MAT/R–03/01–SE
iii
Introduction 3 0.1 Higher Szeg¨o terms, random walks, combinatorics . . . . 4 0.2 Generalized Hunt–Dyson formula (gHD), and its equiva-
lence with Bohnenblust–Spitzer theorem (BSt) . . . 11 0.3 Szeg¨o limit theorem for operators with discontinuous sym-
bols . . . 12 0.4 Acknowledgments . . . 15 1 Higher order terms in Szeg¨o asymptotic formula and proper-
ties of random walks as corollaries of a certain combinatorial
identity 19
1.1 Introduction and main results . . . 20 1.1.1 Brief summary . . . 20 1.1.2 First problem: Higher order terms in Szeg¨o asymp-
totic formula on Zoll manifolds . . . 21 1.1.3 Second problem: Maximal non-negative excursion
of a random walk, an explicit formula for its mo- ments, joint distribution of of the former with the position at a smaller time, and convergence to the
supremum functional of a Brownian motion . . . . 35 1.1.4 Organization of Chapter 1 . . . 40 1.2 Formulation of gHD, and of an equivalent version of BSt . 42 1.3 The abstract scheme for Szeg¨o problem on Zoll manifolds 45 1.4 The full third Szeg¨o term and the remainder estimate . . . 51 1.4.1 Computation of the sum (1.4.3) . . . 52 1.4.2 Computation of the sum (1.4.5) . . . 53 1.4.3 Computation of the sum (1.4.4) . . . 54
v
1.4.4 End of the proof of Theorem 1.1.1 . . . 56
1.4.5 Proof of Lemma 1.4.1 . . . 57
1.5 Calculation of the expression (1.4.6) . . . 65
1.6 How the basic functors ˜Φj, j ∈ N, do arise . . . . 69
1.6.1 An abstract application of gHD and generalized con- volutions . . . 69
1.6.2 Computation of higher order Szeg¨o functionals cor- responding to the contribution from the principal symbol of Bκ1 · · · Bκm . . . 71
1.6.3 Calculation of the moment of an arbitrary order of the maximal non-negative excursion of a random walk: proof of Theorem 1.1.3 . . . 73
1.7 Contribution of the symmetric part of the subprincipal sym- bol of Bκ1· · · Bκm . . . 75
1.8 Contribution of the non-symmetric part of the subprincipal symbol of Bκ1 · · · Bκm . . . 79
1.9 Construction of the invariant formulas for the functors ˜Φj, Φj and Fj, j ∈ N . . . . 85
1.10 Proofs of the results on random walks . . . 87
1.10.1 Joint distribution of the maximal non-negative ex- cursion with the position at a smaller time: proof of Theorem 1.1.2 . . . 87
1.10.2 Convergence to the supremum functional of a Brow- nian motion: proof of Corollary 1.1.4 . . . 88
2 Generalized Hunt–Dyson formula and Bohnenblust–Spitzer the- orem 91 2.1 Introduction and main results . . . 92
2.2 Further notations and auxiliary statements . . . 98
2.3 Independent proof of generalized Hunt–Dyson formula, The- orem 2.1.1 . . . 101
2.4 Proof of the linking step between gHD and BSt, Corollary 2.1.2 . . . 110
2.5 Derivation of BSt from gHD, the equivalence of the two . . 116
2.6 Proofs of the auxiliary statements . . . 119
3 Szeg¨o limit theorem: sharp remainder estimate for integral op- erators with discontinuous symbols in higher dimensional case 123 3.1 Introduction and statement of the main results . . . 124 3.1.1 Self-adjoint Aλ,Γ . . . 125 3.1.2 Not self-adjoint Aλ,Γ . . . 126 3.1.3 The crucial estimate of a certain Hilbert–Schmidt
norm . . . 128 3.2 Proofs in the self-adjoint case, proof of the crucial estimate 129 3.3 Proofs in the non self-adjoint case . . . 136 3.4 Proofs of the auxiliary statements . . . 139 3.4.1 Proof of the first geometrical proposition . . . 139 3.4.4 Proof of the estimate (3.2.12), and also of the aux-
iliary result on the average decay of certain Fourier
transforms . . . 140 3.4.3 Proof of the second geometrical proposition . . . . 143
Bibliography 146
3
0.1 Higher Szeg ¨o terms, random walks, combinatorics
Let M = S1 and denote by Pλ, λ ∈ N, the projector onto the set of the eigenfunctions of the operator p
−d2/dx2 corresponding to the eigenvalues less than or equal to λ, {eikx : |k| ≤ λ}. Assume that B is the operator of multiplication by a sufficiently smooth positive function b(x). The Szeg¨o limit theorem [Sz1] states that
log det(PλBPλ) : = Tr Pλlog(PλBPλ)Pλ
= Tr(Pλlog(B)Pλ) + o(λ), λ → ∞. (1.1) The strong Szeg¨o limit theorem [Sz2] gives a two term asymptotics
Tr Pλlog(PλBPλ)Pλ
= Tr(Pλlog(B)Pλ)
+ Υ(Sz)2 (log; B) + o(1), λ → ∞, (1.2) where Υ(Sz)2 (log; B) is a functional depending on the operator B and the func- tion f (t) = log t.
The formula (1.1) gives the first order asymptotics of determinants of large Toeplitz matrices. It has been conjectured for a positive continuous b by G. P´olya [P] and established for b ∈ L1(S1) by G. Szeg¨o [Sz1].
Almost forty years later Szeg¨o proved the two term formula (1.2) for a positive function b ∈ C1+α(S1), α > 0. This generalization has played an important role in the study of the two-dimensional Ising model by L. On- sager and others. The idea to consider this generalization has been com- municated to Szeg¨o by Onsager via S. Kakutani, according to [MPW]. In the latter work, Onsager’s famous result on spontaneous magnetization of a two-dimensional Ising ferromagnet is rigorously proved via Szeg¨o strong limit theorem. Onsager himself did not publish this derivation, which orig- inated the work [Sz2] (apart from a letter reprinted in [B¨o1]).
An alternative proof of (1.2) has been given by M. Kac in [K]. This proof is based on the following purely combinatorial identity which is called Hunt–Dyson formula, or HD. For arbitrary κ1, · · · , κm ∈ R and any j = 1, · · · , m denote
m∗j(κ1, · · · , κm) := max(0, κ1, κ1 + κ2, · · · , κ1 + · · · + κj), (1.3)
and m∗0(κ1, · · · , κm) := 0. The formula HD states that for any κ1, · · · , κm ∈ R and j = 1, · · · , m one has
X
τ∈Sm
m∗j(κτ1, · · · , κτm) − m∗j−1(κτ1, · · · , κτm)
= X
τ∈Sm
(κτ1 + · · · + κτj)+
j
(1.4)
where Sm is the group of all permutations τ of the numbers 1, · · · , m, and
(κ)+ := max(0, κ), κ ∈ R. (1.5)
Note that the expression on the right in (1.4) involves the maximum of zero and only one sum of the type as on the right in (1.3). The formula HD (1.4) has first been established by G. A. Hunt. An alternative, short and elegant, proof of HD by F. J. Dyson is reproduced in [K].
Suppose that κ1, · · · , κm ∈ R are the steps of a random walk starting at the origin. Then the random variable (1.3) represents the maximal non- negative excursion of the random walk during the period from 0 up to j time units. Assume that the steps of the random walk are independent and identi- cally distributed. It has also been shown in [K] there that the expectation of the maximal non-negative excursion (1.3) can easily be computed via HD (1.4), this argument is due to K. L. Chung.
Further applications of HD in generalizations of the strong Szeg¨o limit theorem are the following. In [O1], K. Okikiolu has established the latter for M = S2 and S3, where Pλis the projection on the eigenfunctions of √
−∆
corresponding to the eigenvalues ≤ λ, and in [O2] for the case M = Td, d ∈ N, and Pλbeing the projection on the trigonometric polynomials whose frequencies lie in λΩ, Ω ⊂ Rd satisfies certain regularity conditions. In the works [GO1, GO3] by V. Guillemin and K. Okikiolu, the two term Szeg¨o asymptotics is established for any Zoll manifold M (that is, a compact closed manifold all of whose geodesics are simple and have the same period, for instance M = Sd, d ∈ N). In [GO2], the strong Szeg¨o limit theorem is generalized to the case of M being a compact manifold with “almost no” periodic geodesics. In the article [LRS] by A. Laptev, D. Robert and Yu. Safarov the functional Υ(Sz)2 ( f ; B) from (1.2) is explicitly computed for an arbitrary analytic f with f (0) = 0. In the works [O1, O2, GO1, GO2, GO3] the combinatorial identity HD plays the crucial role.
It turns out that an abstract scheme suggested in [GO1] for M being a Zoll manifold, can be further exploited, to compute the third and higher order terms in the Szeg¨o asymptotics. In Chapter 1, we explicitly calculate the third Szeg¨o term and explain a method to compute further asymptotic terms. Our first main result is the following (Theorem 1.1.1). Let M be a Zoll manifold of dimension d ∈ N. Let −∆ be the Laplace–Beltrami operator on M. It is known [DG, CdV] that its spectrum has band structure.
Denote by Pλ, λ ∈ N, the projector onto the span of the eigenfunctions of √
−∆ corresponding to the eigenvalues in the first λ bands. Assume that f is an analytic on C function with f (0) = 0. Then for an arbitrary pseudodifferential operator B of order zero on M the following three term Szeg¨o type asymptotic formula holds
Tr( f (PλBPλ) ) − Tr( Pλf (B)Pλ) = λd−1· Υ(Sz)2 ( f ; B) + λd−2 · Υ(Sz)3 ( f ; B) + O(λd−3), λ → ∞.
(1.6)
The asymptotics (1.6) holds for instance in the case M = Sd, d ∈ N, and B being the operator of multiplication by a smooth function on Sd. A possible application of (1.6) for f (t) = log t and d = 2 is the following. In this case the third Szeg¨o term is constant, as λ → ∞. The constant term in Tr( Pλlog (B)Pλ) can also be computed, see Remark 1.1.7. The sum of these two can be viewed as a regularized determinant of the operator of multiplication by a smooth function B, or any pseudodifferential operator B of zeroth order, on S2. Here the convex hull of the spectrum of B does not contain 0.
In (1.6), the functional Υ(Sz)2 depends only on the principal symbol of B, whereas the functional Υ(Sz)3 is a sum of three functionals
Υ(Sz)3 ( f ; B) = Υ(0)3 ( f ; B) + Υ(0,Poi)3 ( f ; B) + Υ(0,sub)3 ( f ; B). (1.7) Here Υ(0)3 depends only on the principal symbol of B, Υ(0,Poi)3 also depends only on the principal symbol of B and contains certain Poisson brackets, and Υ(0,sub)3 depends on both principal and subprincipal symbol of B. The functional Υ(0,Poi)3 has the most complicated and “non-terminating” struc- ture. However the latter vanishes in several special cases.
The heart of the proof of (1.6) is a new combinatorial formula, which is a generalization of HD (1.4) to the case of the maximums raised to an
arbitrary power n ∈ N. That formula is our second main result and we call it generalized Hunt–Dyson formula, or gHD. We introduce some notations and then formulate gHD. Let m ∈ N and let κ1, · · · , κm be arbitrary real numbers. For any permutation τ ∈ Sm, any j ∈ N and for any integers
k1 ≥ 1, · · · , kj ≥ 1, k1 + · · · + kj = m, we introduce the notation ¯κτ := (κτ1, · · · , κτm) and
k1(¯κτ) := κτ1 + · · · + κτk1 k2(¯κτ) := κτk1+1 + · · · + κτk1+k2
· · ·
kj(¯κτ) := κτk1+···+k j−1+1 + · · · + κτk1+···+k j−1+k j.
(1.8)
Each of kl(¯κτ), l = 1, · · · , j, is a sum of kl permuted variables out of κτ1, · · · , κτm so that each of the permuted variables enters exactly one sum.
Then gHD (Theorem 1.2.2) states that for an arbitrary power n ∈ N, X
τ∈Sm
m∗m(¯κτ)n
− (m∗m−1(¯κτ))n
= X
τ∈Sm
min(m,n)X
j=1
1 j!
X
k1,··· ,k j≥1 k1+···+k j=m
X
l1,··· ,l j≥1 l1+···+l j=n
n l1, · · · , lj
!
× (k1(¯κτ))l+1
k1 · . . . · (kj(¯κτ))l+j kj ,
(1.9)
where (a)l+ := ((a)+)l, a ∈ R, l ∈ N.
We see that in the case n = 1 one has
j = 1, k1 = m, l1 = 1, and (1.9) gives the formula HD (1.4)
X
τ∈Sm
m∗m(¯κτ) − m∗m−1(¯κτ)
= X
τ∈Sm
(κτ1 + · · · + κτm)+ m
= (m − 1)! (κ1 + · · · + κm)+.
After having discovered and proved the formula gHD (1.9) we realized that it is related to the celebrated combinatorial result due to H. F. Bohnen- blust which appeared in an article by F. Spitzer on random walks [S1, The- orem 2.2]. We call this combinatorial fact Bohnenblust–Spitzer theorem, or
BSt. It states that the set of maximums of zero and sums of accumulating lengths of m real variables permuted in all possible m! ways coincides with the set of sums of positive parts of the sums of those m real variables, ar- ranged according to the cyclic representations of all m! permutations. This statement becomes clear if we consider a simple example. Let us choose m = 2 and any κ1, κ2 ∈ R. The symmetric group S2 consists of two permu- tations. Let us rewrite these via the cyclic representations
S2 =
( 1 2 1 2
!
, 1 2 2 1
! )
=
(1)(2), (1 2) . In this case BSt gives
max(0, κ1, κ1 + κ2), max(0, κ2, κ2 + κ1)
=
(κ1)++ (κ2)+, (κ1 + κ2)+ .
Note that a certain maximum of zero and accumulating sums of the per- muted variables does not need to equal the element of the set on the right- hand side corresponding to the cyclic representation of that permutation.
The statement of BSt is merely that the whole sets are identical, a fact which is indeed miraculous.
Consider a random walk with independent identically distributed real- valued steps X1, · · · , Xm starting at the origin (we will not mention other types of random walks). The position of the random walk after j = 1, · · · , m steps is Sj := X1+ · · · + Xj. Denote the length of the maximal non-negative excursion from during the period up to j = 1, · · · , m time units by
Tj := m∗j(X1, · · · , Xm) = max(0, S1, · · · , S j). (1.10) The result BSt enabled Spitzer to compute the characteristic function of the random variable Tm, m ∈ N, and even the joint distribution of the latter with the final position Sm. The work [S1] was revolutionary, because the com- binatorial principle BSt behind the properties of the maximal non-negative excursions of random walks was unclear, although quite elaborate formulas involving partitions had been obtained before by E. Sparre Andersen [SA].
It turns out that the formula gHD can be derived from BSt, and vice versa. The statement of gHD and a convenient form of BSt are given in Section 1.2. Our independent proof of gHD, together with a derivation of BSt from it, is given in Chapter 2 (see Section 0.2 for details).
In the construction of the functional Υ(Sz)2 corresponding to the second Szeg¨o asymptotic term, and also in the expressions for the functionals Υ(0)3 and Υ(0,Poi)3 which are the parts of the third Szeg¨o term, certain functors Φj, j ∈ N, appear. These functors act on set of all analytic functions f (t) on C with f (0) = 0, which is denoted by A1. For each j ∈ N, the action of the functor Φj on an arbitrary f ∈ A1, denoted by Φj[ f ], is a map from Cj to C. We demand that the functor Φj, j ∈ N, is linear
Φj[α f + βg](x1, · · · , xj) = αΦj[ f ](x1, · · · , xj)
+ βΦj[g](x1, · · · , xj), (1.11) for all α, β ∈ C, f, g ∈ A1 and x1· · · , xj ∈ C, and prescribe its action on the monomials f (t) = tm, m ∈ N, by
Φj[tm](x1, · · · , xj) = X
l1,··· ,l j≥1 l1+···+l j=m
xl11 l1
· · · xljj lj
. (1.12)
An expression for Φj, j ∈ N, is obtained in Section 1.9. For an arbitrary f ∈ A1
Φj[ f (t)](x1, · · · , xj) = Z x1
0
· · · Z xj
0
Φ˜ j[t− jf (t)](ξ1, · · · , ξj) dξ1· · · dξj, (1.13) where the auxiliary functor ˜Φj, j ∈ N, also acts linearly on analytic on C functions. The action of ˜Φj, j ∈ N, on a monomial tm, m = 0, 1, 2, · · · , is given by
Φ˜ j[tm](ξ1, · · · , ξj) := X
l1,··· ,l j≥0 l1+···+l j=m
ξ1l1 · · · ξljj. (1.14)
The latter is nothing but the mth complete symmetric function evaluated at the point (ξ1, · · · , ξj, 0, · · · ), see [M, I.2]. In Section 1.9, an “invariant”
formula for ˜Φj acting an any analytic on C function f is obtained, and hence the formula for (1.13).
We compute also the bivariate characteristic function of the maximal non-negative excursion Tm (1.10) and the position of the random walk S j
at a smaller time j = 1, · · · , m − 1 (Theorem 1.1.2). This problem has only a “partial” symmetry, in contrast with the case j = m considered by Spitzer. The resulting expression involves the introduced in (1.13) functors
Φj, j ∈ N, which appear also in the part of the third Szeg¨o term given by the “non-symmetric” functor Υ(0,Poi)3 .
The formula gHD is very suitable for calculation of the moment of the maximal non-negative excursion of a random walk Tm of any order n ∈ N (Theorem 1.1.3). In fact we designed the former exactly for that purpose, although in the Szeg¨o problem context. This information can in principle be extracted from Spitzer’s formula for the characteristic function E
eiαTm , but our direct calculation is perhaps easier. We find the resulting formula (1.46) in Chapter 1
E
(Tm)n
=
min(m,n)X
j=1
X
p1,··· ,p j≥0 1·p1+···+n·pn=n
p1+···+pn= j
B(p1, · · · , pn)
× X
k1,··· ,k j≥1 k1+···+k j≤m
E(Sk1)+ k1
p1
· · ·
E(Skj)n+ kj
pn ,
(1.15)
involving the Bell polynomial coefficients, curious in itself. The latter are defined by
B(p1, · · · , pn) := n!
(1!)p1 · · · (n!)pn p1! · · · pn!
for non-negative integers p1, · · · , pn which are the numbers of parts 1, · · · , n in the partition, so that 1 · p1+ · · · + n · pn = n. In (1.15), p1+ · · · + pn is the total number of parts in the partition.
The formula (1.15) can be rewritten as (1.43), Chapter 1, E
(Tm)n
=
min(m,n)X
j=1
1 j!
X
l1,··· ,l j≥1 l1+···+l j=n
n l1, · · · , lj
!
× X
k1,··· ,k j≥1 k1+···+k j≤m
E(Sk1)l+1 k1
· · · E(Skj)l+j kj
.
(1.16)
In the case n = 1 the latter gives the classic formula of Kac, Hunt, Sparre Andersen and Spitzer
E{Tm} = Xm
k=1
E(Sk)+ k
.
An application of the formula (1.16) is given in Corollary 1.1.4. It is shown that for a random walk with independent identically distributed steps, which for simplicity are assumed to have normal distribution, and an arbitrary n ∈ N one has
m→∞lim E
(m−1/2Tm)n
= 2n/2π−1/2Γn + 1 2
. (1.17)
In this case the Fr´echet–Shohat theorem is applicable (see Remark 1.1.13).
Therefore the result (1.17) on the convergence of the moments implies the uniform on R convergence of the distributions, as m → ∞,
P
m−1/2Tm < x
→
√2/πR x
0 e−y2/2dy, x ≥ 0
0, x < 0
= P
0≤s≤1maxB(s) < x ,
where B(s) is the normalized Brownian motion with B(0) = 0, because the limiting distribution is continuous on R. The result of Corollary 1.1.4 is not new, and follows of course from the general theory of convergence of scaled random walks to the Brownian motion (Donsker invariance principle and Skorokhod representation), see for instance [BD].
0.2 Generalized Hunt–Dyson formula (gHD), its equivalence with Bohnenblust–Spitzer theorem (BSt)
In Chapter 2, we present an independent proof of gHD which is purely com- binatorial. After that we derive BSt from gHD, and vice versa, providing the former with a new proof.
We discovered and proved gHD being unaware of BSt. The proof of gHD in Chapter 2 starts with a generalization of F. J. Dyson’s observation on which the proof of HD (n = 1) in [K] is based. Surprisingly enough, this idea works for any power n ∈ N at all, and enables us to reduce the power from n to n − 1, and use the induction argument with respect to the power n ∈ N. The base of induction is the identity HD, n = 1. The proof of the inductive step is technically involved. In particular, the Cauchy and Cayley identities from the theory of partitions, and a version of the principle of inclusion and exclusion, are referred to.
The statement and an idea of the original proof of BSt due to Bohnen- blust appeared in [S1]. The detailed proof was published by R. Farrell only nine years later [Far].
At least two other proofs of BSt are known: an analytic approach due to G. Baxter (via a similar operator identity and Wiener–Hopf factorization, see [B1, B2] and [B3]), and an algebraic by J. G. Wendel [We1, We2].
0.3 Szeg ¨o limit theorem for operators with discontinuous sym- bols
Another way of generalizing Szeg¨o type asymptotic formulas is to study the asymptotics of Tr f (Bλ), λ → ∞, where the integral operator Bλ has the kernel
KBλ(x, y) = λ 2π
dZ
eiλξ·(x−y)σ(B)(x, ξ) dξ (3.1) of pseudodifferential type, with the difference that the symbol σ(B) is non- smooth (for instance having jump discontinuities in both x and ξ) and f is sufficiently regular (for example having bounded second derivative). Here and belowR
stands forR
Rd.
The problems of this type have been studied in dimension one and dis- crete case (Toeplitz determinants) by R. E. Hartwig and M. E. Fisher [HF], E. Basor [Ba], A. B¨ottcher [B¨o2], and others (we refer to [W7] for fur- ther references and the history of the question). The continuous case (3.1) for dimension one (Wiener–Hopf determinants) has been considered by H. Widom [W6] (a two term asymptotics), E. Basor and H. Widom [BaW]
(a three term asymptotics). A more difficult problem is the study of the derivative with respect to λ of the logarithm of the Fredholm determinant for the integral operator with the kernel
sin λ(x − y) x − y
acting on a union of intervals on the real line. This problem plays an im- portant role in the theory of random matrices. Recently, a full asymptotic expansion has been obtained by P. Deift, A. Its and X. Zhou [DIZ]. This result generalizes a two term formula rigorously proved first by H. Widom [W8], and dating back to the work of F. J. Dyson [D].
In the higher dimensional case not much seems to be known. Probably the only result for the case when the symbol σ(B)(x, ξ) has discontinuities in both x and ξ is a two term asymptotic formula [W7], for a special choice of the discontinuity of one of the variables (the characteristic function of a half-space).
Logarithmic asymptotic terms are typical for the problems of this kind.
In Chapter 3, we obtain an analog of one term Szeg¨o limit theorem with correct order remainder in higher dimensional case. In that problem the leading term is of Weyl type, and the remainder has order λd−1log λ (dis- continuous case) or λd−β, 0 < β < 1 (fractal case), as λ → ∞.
Assume that
σ(x, ξ) ∈ C∞0 (R2d).
Let Ω be a bounded domain in Rd, d ≥ 2, with C1 boundary. Let Γ ⊂ Rd be a set of finite Lebesgue measure |Γ| < ∞ with a possibly fractal boundary such that that for certain 0 < β ≤ 1 and c > 0
|{x ∈ Rd : dist(x, ∂Γ) ≤ }| ≤ c β, 0 < < 1. (3.2) This condition will be satisfied if the boundary ∂Γ has Minkowski dimen- sion d − β, 0 < β ≤ 1, and the respective Minkowski content of ∂Γ is finite, see [Col, Fal] for details. Denote by H the Hilbert space L2(Rd) with the usual scalar product. Let Aλ,Γ be an operator in H with Schwartz kernel
KAλ,Γ(x, y) = λ 2π
dZ
eiλξ·(x−y)χΓ(ξ) σ(x, ξ) dξ, λ ≥ 2. (3.3) Let PΩ be the operator of multiplication by the characteristic function of Ω
PΩf (x) = χΩ(x) f (x), f ∈ H.
Introduce the operator
Bλ = PΩAλ,ΓPΩ. (3.4)
Note that the symbol of Bλ has discontinuities in both position variables x, y due to the multiplication by χΩ, and momentum ξ, due to the factor χΓ(ξ) corresponding to the multiplication by the characteristic function of Γ on the Fourier transform side.
The main result of Chapter 3 is the sharp remainder estimate, for λ ≥ 2, Tr
PΩ f (PΩAλ,ΓPΩ) PΩ− PΩ f (Aλ,Γ) PΩ
≤ C(Ω, Γ, σ) · ˜C( f ) ·
λd−β, 0 < β < 1 λd−1log λ, β = 1
(3.5)
in the following two settings. In the first case (Theorem 3.1.1) the op- erator Aλ,Γ is required to be self-adjoint, but the function f is quite gen- eral. More precisely, we assume f00 ∈ L∞(I) where I is the convex hull of
{spec Aλ,Γ}λ≥2 ⊂ R. In that case we refer to an abstract result by A. Laptev and Yu. Safarov [LS] to obtain
Tr
PΩ f (PΩAλ,ΓPΩ) PΩ − PΩ f (Aλ,Γ) PΩ
≤ 1
2 · k f00kL∞(I) · kPΩAλ,Γ(I − PΩ)k2S2, (3.6) where kPΩAλ,Γ(I − PΩ)kS2 denotes the Hilbert–Schmidt norm.
In the second case (Theorem 3.1.4) the operator Aλ,Γ need not to be self- adjoint, but the function f is assumed to be analytic on a neighborhood of
{z : |z| < C(σ)}, C(σ) := sup
λ≥2
kAλ,Γk, (3.7)
and satisfy f (0) = 0. Here k · k stands for the operator norm in H, and 0 < C(σ) < ∞ (see Section 3.1). In that case we use the commutator argument of H. Widom [W5] to get
Tr
f (PΩAλ,ΓPΩ) − PΩ f (Aλ,Γ) PΩ
≤ (C(σ))−1 · f∗ C(σ)
· kPΩAλ,Γ(I − PΩ)k2S2, (3.8) where f∗(x) := P∞
m=1m|cm|xm−1 for the analytic function f (z) = P∞
m=1cmzm. In both cases the following estimate (Theorem 3.2.2) plays the crucial role
kPΩAλ,Γ(I − PΩ)k2S2 ≤ C(Ω, Γ, σ) ·
λd−β, 0 < β < 1
λd−1log λ, β = 1, (3.9) for λ ≥ 2. The estimate (3.9) does not require self-adjointness of the opera- tor Aλ,Γ.
In Corollaries 3.1.2, 3.1.3 and 3.1.5 we compute in certain special cases the leading term of the asymptotics of Tr
PΩf (PΩAλ,ΓPΩ)PΩ)
in (3.5). It is of Weyl type and equals Tr
PΩ f (Aλ,Γ)PΩ) .
One should expect a stronger result to hold. A two term asymptotic formula which generalizes (3.5) was conjectured for any dimension and proven in the case of d = 1 and matrix-valued symbol σ by H. Widom in [W6]. In [W7] the higher dimensional case for scalar σ has been taken up in a special situation of one of the sets Γ and Ω being a half-space and the other compact with smooth boundary. In that case for an analytic on the neighborhood of {z : |z| ≤ maxx,ξ∈Rd |σ(x, ξ)|} function f with f (0) = 0 and a
scalar-valued σ a two-term asymptotic formula is valid with remainder term O(λd−1), as λ → ∞. The proof in [W7] is technically involved and reduces the problem to the one-dimensional case [W6]. The case of both Γ and Ω being compact with smooth boundary is still open. The results [W6, W7]
indicate that one should not expect a better remainder estimate in (3.5).
Note that when β > 0 is small, the volume (3.2) of the boundary layer along ∂Γ decays slowly, as its thickness tends to zero. This fact is reflected in a faster growth of the remainder term which has in that case order λd−β, as λ → ∞. If β = 1 (for example ∂Γ is of class C1), the remainder term is of a smaller order λd−1log λ. However one can not get rid of the logarithmic factor even if ∂Γ and ∂Ω are of class C∞, because the symbol of Bλ has discontinuities in both the position variables x, y and the momentum ξ.
In Section 3.4, we prove an auxiliary result a result which concerns the average decay of the Fourier transform of a L2 function whose L2 modulus of continuity is H¨older continuous with power β/2, 0 < β ≤ 1. This auxil- iary result is needed for the proof of the estimate (3.9), and is a version of [BCT, Lemma 2.10] (there β = 1).
0.4 Acknowledgments
Thesis
It is a great pleasure for me to express a deep gratitude to my Teacher and scientific advisor Ari Laptev for posing the problems, constant attention to the work and his invaluable help. The credit for forming me as a mathe- matician goes entirely to him.
I would like to thank Percy Deift for bringing the article [RS], where I found a reference to [S1], to my attention. From [S1], I learned about Bohnenblust–Spitzer theorem, two months after having discovered and prov- ed the generalized Hunt–Dyson formula.
I am grateful to Kurt Johansson who suggested a simpler way to con- struct the invariant formula for the functors Φj, j ∈ N, in Section 1.9.
I wish to thank Michael Shapiro and Stanislav Smirnov for useful com- ments on the introductory section to Chapter 2, and also for their general support. Thanks are also due to Oleg Safronov for a useful discussion con- cerning Section 3.4.
Royal Institute of Technology in Stockholm
It is a pleasure for me to thank the Department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, and the Faculty, for providing me with excellent working conditions, wonderful atmosphere, and generous financial support during the whole period of my Ph.D. studies.
I wish to thank to Tommy Ekola who shared the office with me during the course of my Ph.D. studies, for being a perfect neighbor, and especially for his great help and competent advice in producing the layout of the thesis using the latest version of the LATEX typesetting system.
I wish also to thank the administrative staff at the Department, and espe- cially Siv Sandvik, Leena Druck, and Anja Orest.
Teachers
I am grateful to Dmitri G. Vassiliev, David E. Edmunds, and Yuri G. Sa- farov, for their scientific and moral support during my year as an exchange student at the University of Sussex.
I would like to thank Gregory A. Seregin, Anatoly A. Aksenov, and Vladimir E. Nomofilov (St. Petersburg Technical University).
Friends
I use this opportunity to thank Dmitri Apassov, my friend since the years at the Physical–Technical School No. 566, St. Petersburg. I would like to thank my friends in Stockholm, Vadim Belenky for the support, and also for his biting humor, and Sergej Nikolajev and his wife Camilla Lindgren.
Family
This thesis is dedicated to the memory of my Grandparents. My Grandfa- ther, Professor of Biochemistry Dimitri A. Golubentsev, was the first ex- ample of a scientist in my life. My Grandmother, Valentina N. Pavlova, graduated as an engineer from the same Technical University at St. Peters- burg as myself, and later on worked as a teacher of mathematics. They meant so much to me, especially my Grandmother who spent the last third of her life educating and cultivating me. It was she who taught me to make technical drawings and to solder electronic devices, explained what the log- arithm is and how to use the four-digit tables of elementary functions by V. M. Bradis.
Finally, I thank my family, for the support which made this work pos- sible, especially my uncle Nicolay D. Golubentsev, whose attention and help I have always felt, Tora Hedin for her love and understanding, and my Mother Natalia D. Barkova. She taught me discipline and concentration, and always understood my decisions. Her work as a lecturer in Medicine makes me a third generation teacher.
Higher order terms in Szeg ¨o asymptotic formula and
properties of random walks as corollaries of a certain
combinatorial identity
19
1.1 Introduction and main results
1.1.1 Brief summary
We start by explaining the background of our work and its relation to the known results. After that we give all the exact definitions and formulations.
In the work [K] a certain combinatorial formula for the maximum of zero and accumulating sums of real variables (which is now known as Hunt–
Dyson combinatorial identity (HD)) is used in elegant and simple proofs of two results in pure analysis. The first result is the strong (two-term) Szeg¨o limit theorem in the unit circle. The second is a formula for the expectation of the maximal non-negative excursion of a random walk with independent identically distributed steps (we will not consider other types of random walks).
In [S1] an essentially deeper, and indeed miraculous, combinatorial fact concerning the maximum of zero and accumulating sums (which is now known as Bohnenblust–Spitzer theorem (BSt)) is stated. It is applied to cal- culate the characteristic function of the above maximal non-negative excur- sion. Also the bivariate characteristic function of the position of a random walk at a given time and the maximal excursion during the period from zero up to that time is computed. To our knowledge a connection between BSt and Szeg¨o problem has not been noticed before. In this work we obtain HD as a simple corollary of BSt, although it has probably been understood by F. Spitzer (see [S1]).
In the present work we find two new applications of BSt in pure analysis.
The first one is a computation of the third and higher order terms in Szeg¨o asymptotic formula for pseudodifferential operators (ΨDO) on Zoll mani- folds. We explicitly compute the third asymptotic term. This result includes in particular the case of the operator of multiplication by a smooth function on the sphere in any dimension. A possible application is an expression for a regularized determinant of a multiplication operator, or a zeroth order pseudodifferential operator, on the two dimensional sphere.
The main technical tool here is a new combinatorial formula, which is called generalized Hunt–Dyson formula (gHD). We discovered and proved gHD being unaware of BSt. It turns out that gHD and BSt can be derived from each other, and in particular the derivation of gHD from BSt is some- what shorter than the proof “from scratch” in Chapter 2. At least three other proofs of BSt are known, the original geometrical one by H. F. Bohnenblust [S1, Far], an analytical by G. Baxter [B3], and an algebraic by J. Wendel
[We2]. The new proof of BSt via gHD uses a generalization of F. J. Dyson’s argument on which the proof of the usual HD is based, and is purely com- binatorial.
The second application of BSt we give is a computation of the bivariate characteristic function of the position of a random walk at a given time p ∈ N and the maximal excursion during the period from zero up to some larger time p + q, q ∈ N. We give also an explicit formula for the moments of all orders of the maximal excursion which involves the Bell polynomials coefficients. For the the first moment our formula gives the result of M. Kac [K]. Also that formula allows us to reprove that the distribution function of the properly scaled maximal excursion uniformly converges to that of the supremum functional of the normalized Brownian motion.
The main difficulty is that both problems have only a limited grade of symmetry. It turns out, however, that it is possible to make use of that
“partial” symmetry. This explains the similarity between the expression for the non-symmetric contribution to the third Szeg¨o term and the bivariate characteristic function mentioned above. We remark that the problem of finding the joint distribution of the position of a random walk at a given time, and maximal excursion during a smaller time interval, possesses a
“full” symmetry, and is a trivial consequence of a result from [S1].
We conclude that, as in [K], the two “purely analytic” problems have the same combinatorial background, although essentially more complicated.
1.1.2 First problem: Higher order terms in Szeg ¨o asymptotic formula on Zoll manifolds
We start by describing the Szeg¨o problem in the ΨDO setting. After that we explain briefly the difficulties that appear, and how we overcome these.
In the end of this subsection we make a few historical remarks.
Let M be a Zoll manifold, that is a compact, closed Riemannian manifold all of whose geodesics are closed and simple with length 2π, of dimension d ∈ N. The simplest example of such a manifold is the standard sphere Sd, d ∈ N. Denote by (x, ξ) the points of the cotangent bundle T∗M. For any l ∈ Z denote by Ψl(M) the space of classical ΨDO of order l on M. The full symbol of any G ∈ Ψl(M) can be written in local coordinates as a sum P−∞
k=lσk(G)(x, ξ), where each σk(G)(x, ξ), k ∈ Z, k ≤ l, is homogeneous in ξ of degree k. In particular, σl(G)(x, ξ) is called the principal symbol. The
subprincipal symbol of G is defined [DH, (1.2)] by sub(G)(x, ξ) = σl−1(G)(x, ξ) − (2i)−1
Xd k=1
∂xk∂ξkσl(G)(x, ξ). (1.1) Both principal and subprincipal symbol are well-defined on T∗M.
Let ∆ be the Laplace–Beltrami operator on M. Then [DG, CdV] there exist a constant α and A−1 ∈ Ψ−1(M), which commutes with ∆, so that the operator
A := √
−∆ − (α/4) + A−1 (1.2)
has the spectrum N. Also A ∈ Ψ1(M) and is positive and elliptic. Let a1(x, ξ) := σ1(A)(x, ξ) and denote by
Θ :=
Xd j=1
∂a1
∂ξj
∂
∂xj
− ∂a1
∂xj
∂
∂ξj
(1.3)
the corresponding Hamiltonian vector field. Define the cotangent sphere by S∗M := {(x, ξ) ∈ T∗M : |ξ|x = 1},
where | · |x is the Riemannian length. Denote by πµ, µ ∈ N, the projection onto the eigenspace of A corresponding to the eigenvalue µ, and set πµ := 0, µ = 0, −1, −2, · · · . Write
Pλ = Xλ
µ=1
πµ, λ ∈ N. (1.4)
Let A1 denote the set of all analytic functions on C with no constant term A1 := n
f (t) : f (t) = X∞ m=1
cmtm, t ∈ Co .
Let B ∈ Ψ0(M). In particular B can be the operator of multiplication by a smooth function b(x) ∈ C∞(M), in which case the symbol of B does not depend on ξ. We study the asymptotic expansion of
Tr( f (PλBPλ) ), λ → ∞, (1.5) for any f ∈ A1. In [GO1], which is our standard reference, V. Guillemin and K. Okikiolu obtain a two-term asymptotic formula for (1.5), where f (t)
is either an arbitrary monomial tm, m ∈ N, or f (t) = log(1 + t). This is a generalization of the strong Szeg¨o limit theorem.
In particular it is shown in [GO1] that the leading term of the asymp- totics (1.5) is
Tr( Pλf (B)Pλ). (1.6)
We see that the function f enters the expression for the leading term in a very simple manner.
In the article [LRS] by A. Laptev, D. Robert and Yu. Safarov, an “invari- ant” expression for the second Szeg¨o term as a functional of an arbitrary f ∈ A1 has been suggested. The function f enters the expression for the second Szeg¨o term in a more complicated way than in (1.6). More pre- cisely, a functor W2 whose action on an arbitrary f ∈ A1 is a map C2 → C, is involved. This action is given by
W2[ f ](x1, x2) := 1 2
Z x1
0
Z x2
0
f0(ξ1) − f0(ξ2)
ξ1− ξ2 dξ1dξ2. (1.7) We are concerned with the question of calculating the third and further terms in Szeg¨o type expansions (1.5), and constructing the correspond- ing “higher order” functionals and functors. One should also mention in this connection an equivalent to W2 functor U constructed by H. Widom [W6, W7]. It appears in a two term Szeg¨o type asymptotic formula for operators with discontinuous symbols and is given by
U[ f ](a, b) :=
Z 1
0
f (1 − t)a + tb
− (1 − t) f (a) + t f (b)
t(1 − t) dt, (1.8)
where f ∈ A1 and a, b ∈ C.
In Chapter 1, we compute the third Szeg¨o term for an arbitrary Zoll manifold M and all B ∈ Ψ0(M) and f ∈ A1, see Theorem 1.1.1. We discuss also a method of calculation of further asymptotic terms. The expression for the third Szeg¨o term we obtain is much more complicated than the one for the second Szeg¨o term. Not going into details here, we merely mention that it contains two parts corresponding to a “symmetric” and a “partially symmetric” contribution. The expression for the part of the third Szeg¨o term coming from the “partially symmetric” contribution is especially lengthy.
This part however vanishes in several special cases. Also the structure of just that part is very similar to the expression for the bivariate characteristic function for the position of a random walk at a given time and the maximal
non-negative excursion over a larger time interval that we obtain in Theorem 1.1.2.
In the case of M = S1 and B ∈ C∞(M), the third Szeg¨o term given by Theorem 1.1.1 equals zero. This agrees with the known fact that in the two-term Szeg¨o asymptotic formula for the operator of multiplication by a smooth function B on the unit circle, the remainder decays rapidly, as λ → ∞ (see [W1]). However for a pseudodifferential operator B ∈ Ψ0(S1) the third Szeg¨o term does not vanish, and decays as λ−1, as λ → ∞.
It is important that on a Zoll manifold M there exists a full asymptotic expansion of
Tr(πλG), λ → ∞, (1.9)
for any G ∈ Ψ0(M). This result is due to Y. Colin de Verdi`ere [CdV]. A convenient form of (1.9) can be found in [GO1, Lemma 0.2]. The abstract scheme introduced in [GO1] uses (1.9) to reduce the calculation of the sec- ond Szeg¨o term in (1.5) for an arbitrary monomial f (t) = tm, m ∈ N, to a use of a combinatorial identity due to K. L. Chung, G. Hunt, F. J. Dyson and M. Kac. Following [GO1] we call this identity Hunt–Dyson combinatorial formula, HD. This idea goes back to [K] (for M = S1), and to [O1] for the sphere in dimension 2 and 3. The formula HD states that for arbitrary κ1, · · · , κm ∈ R and j = 1, · · · , m one has
X
τ∈Sm
m∗, j(κτ1, · · · , κτm) − m∗, j−1(κτ1, · · · , κτm)
= X
τ∈Sm
−(κτ1 + · · · + κτj)− j
(1.10)
where for j = 1, · · · , m
m∗, j(κ1, · · · , κm) := min(0, κ1, κ1 + κ2, · · · , κ1 + · · · + κj), (1.11) and m∗,0(κ1, · · · , κm) := 0. Here Sm is the group of all permutations τ of the numbers 1, · · · , m, and
−(κ)− := min(0, κ), κ ∈ R. (1.12) Note that the expression on the right in (1.10) involves the minimum of zero and only one sum of the type as on the right in (1.11). Also the formula (1.10) with m∗, j replaced by
m∗j(κ1, · · · , κm) := max(0, κ1, κ1 + κ2, · · · , κ1 + · · · + κj), (1.13)
and with −(κ)− replaced by
(κ)+ := max(0, κ), κ ∈ R, (1.14) holds true. From (1.10) one can easily obtain a formula for
X
τ∈Sm
m∗,m(κτ1, · · · , κτm)
by producing a telescopic summation over j = 1, · · · , m.
The idea behind the “purely combinatorial” proofs of “purely analytic results” in [K, GO1, GO2, GO3, O1, O2] (and in [S1]) is the following.
Let us consider the discrete case. Assume that F(κ1, · · · , κm) is a symmetric with respect to all permutations of the arguments function defined on Zm. Let Km be a symmetric subset of Zm. Then
X
(κ1,··· ,κm)∈Km
m∗,m(κ1, · · · , κm) F(κ1, · · · , κm)
= X
(κ1,··· ,κm)∈Km
1 m!
X
τ∈Sm
m∗,m(κτ1, · · · , κτm)
F(κ1, · · · , κm),
(1.15)
and the symmetrized factor has a relatively simple structure, which makes it possible to compute the sum over Km.
If one tries to apply the method of [GO1] to calculate further Szeg¨o asymptotic terms, then two difficulties come in. The first one is that there appear expressions of the type as on the left in (1.15) with a symmetric F, but m∗,m raised to a power n ∈ N, n ≥ 2. What one needs then is a generalization of HD for the expression of the type
X
τ∈Sm
m∗,m(κτ1, · · · , κτm)n ,
or X
τ∈Sm
h m∗,m(κτ1, · · · , κτm)n
− m∗,m−1(κτ1, · · · , κτm)ni
, (1.16)
for all n ∈ N, n ≥ 2. We have found a formula for (1.16), which will be called generalized Hunt–Dyson formula, abbreviated as gHD. After having discovered and proved this purely combinatorial formula, we realized that it is equivalent to the celebrated Bohnenblust–Spitzer theorem, or BSt. This result of H. F. Bohnenblust appeared in an article by F. Spitzer [S1, Theo- rem 2.2]. In Section 1.2, we state gHD, and a convenient for our purposes
version of BSt. In Chapter 2, we give an independent proof of gHD, and show the equivalence of BSt from gHD, providing the former with a new proof. Formally, to calculate the characteristic function of a random vari- able is the same thing as to calculate its moments of all orders (and not only the expectation as in [K]). Therefore the equivalent results BSt and gHD are much stronger than HD.
We discuss now the second difficulty that comes in when one tries to calculate further Szeg¨o asymptotic terms. It is that there appear expressions of the type as on the left in (1.15) not only with m∗,m raised to some power n ∈ N, but with a function F, which is generally speaking non-symmetric, and exhibits only a certain type of partial symmetry. This is a much more serious obstacle, and we are forced to modify the combinatorial argument to exploit the existing partial symmetry. The resulting expressions become lengthy.
Let us make necessary definitions and formulate the result. For a given B ∈ Ψ0(M) define its “Fourier coefficients,” in accordance with [GO1], by
Bκ = X∞
µ=1
πµ+κ B πµ
= 1 2π
Z 2π
0
eiκte−itABeitAdt, κ ∈ Z.
(1.17)
Each Bκ ∈ Ψ0(M), and one has a “Fourier decomposition”
B = X∞ κ=−∞
Bκ. (1.18)
Also by Egorov’s theorem
σ0(Bκ)(x, ξ) = 1 2π
Z 2π
0
σ0(B)(x, ξ; t) eiκtdt, (1.19) where σ0(B)(x, ξ; t) is the composition of σ0(B) with the shift along the trajectory determined by the flow Θ starting at (x, ξ). For any G ∈ Ψl(M), l ∈ Z, introduce the Guillemin–Wodzicki residue notation
Res(G) = (2π)−d Z
S∗M
σ−d(G)(x, ξ) dx dξ, (1.20) where the integrand is always well-defined.
