Weyl asymptotic formulae and Sobolev spaces for infinite order pseudo-differential operators

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by  Stevan Pilipović

WEYL ASYMPTOTIC FORMULAE AND SOBOLEV SPACES FOR
INFINITE ORDER PSEUDO-DIFFERENTIAL OPERATORS
STEVAN PILIPOVIC´, BOJAN PRANGOSKI, AND JASSON VINDAS
Abstract. Westudyspectralpropertiesofaclassofglobalinfiniteorderpseudo-
differentialoperators.Weobtainformulaeconcerningtheasymptoticbehaviourof
the spectral counting functions of such operators. Unlike their finite order coun-
terparts, their spectral asymptotics are not of power-log-type but of log-type.
7
The ultradistributional setting of such operators of infinite order makes the the-
1
0 ory more complex so that the standard finite order global Weyl calculus cannot
2 be used in this context. As an application of the developed spectral analysis, we
introduceanewclassofinfiniteorderShubin-Sobolevtypespacesandderivereg-
n
a ularity results for solutions to elliptic infinite order pseudo-differentialequations.
J

7
2

] 1. Introduction
P
S In this article we study the spectral properties of global infinite order pseudo-
.
h differential operators. Our operator classes are intrinsically related to the ultradis-
t
a tributional framework so that the bounds on the derivatives of the symbols are
m
controlled by Gevrey type weight sequences. One of our aims is to establish Weyl
[ asymptotic formulae for a large class of (hypoelliptic) ΨDOs of infinite order. It is

1 worth mentioning that the Weyl asymptotics for the operators that we investigate
v here are not of power-log-type as in the finite order (distributional) setting, but of
7
log-type, which in turn yields that the eigenvalues of infinite order ΨDOs, with ap-
0
9 propriate assumptions, are “very sparse”. As a by-product of our analysis, we also
7
obtain Weyl asymptotic formulae for a class of finite order Shubin ΨDOs with some
0
. conditions on the symbols that are not the ones usually discussed in the literature.
1
0 Moreover, via an additional analysis of a class of elliptic infinite order symbols, we
7 introduce and study infinite order Shubin-Sobolev type spaces describing regularity
1
of solutions to elliptic pseudo-differential equations in the infinite order context.
:
v
The spaces of symbols and corresponding pseudo-differential operators involved
i
X in this work were introduced by Prangoski in his thesis (see [22] for the symbolic
r calculus) and then extensively studied in several articles by himself and his coau-
a
thors; we refer to works of Cappiello [2, 3] for similar symbol classes related to
SG-hyperbolic problems of finite order. The definition of these symbols classes is
linked to two Gevrey type weight sequences A and M , p N. The first one con-
p p
∈
trols the smoothness, while the second one controls the growth at infinity of the
symbols. These symbol classes are denoted by Γ(Mp),∞ and Γ{Mp},∞. The first one
Ap,ρ Ap,ρ
gives rise to operators acting continuously on Gelfand-Shilov type spaces of Beurl-
ing class (i.e. of (M )-class) and the second one on Gelfand-Shilov type spaces of
p
2010 Mathematics Subject Classification. 35P20,35S05, 46F05, 46E35,47D03.
Key words and phrases. Ultradistributions, infinite order pseudo-differential operators, Weyl
asymptotic formula, infinite order Sobolev spaces, hypoellipticity, heat parametrix.
The work of J. Vindas was supported by Ghent University, through the BOF-grant01N01014.
1
2 S. PILIPOVIC´,B. PRANGOSKI,AND J. VINDAS
Roumieu type (of {Mp}-class); we will employ ΓA∗,p∞,ρ as a common notation for both
cases. Since the symbols are allowed to grow sub-exponentially, i.e. ultrapolynomi-
ally, the corresponding ΨDOs are of infinite order and they go beyond the classical
Weyl-Ho¨rmander calculus.
The starting key point of our investigations is the semi-boundedness of ΨDOs
with positive hypoelliptic Weyl symbols in Γ∗,∞, which will be achieved with the
Ap,ρ
aid of the Anti-Wick calculus developed in [20]. This result is interesting by it-
self because hypoellipticity in this setting allows the symbols to approach 0 sub-
exponentially and thus generalises the familiar result for finite order operators. The
precise estimates on the heat parametrix recently given in [21] will enable us to
further study the heat semigroup associated to a hypoelliptic operator and extract
asymptotic information from it. Since the large growth of infinite-order symbols for-
bids the use of the well-known methods for global finite order ΨDOs (see [16, 24]),
this analysis requires various new techniques that are intrinsic to the infinite order
context. The passage from asymptotics of the heat semigroup to Weyl formulae
will be accomplished using ideas from the theory of regular variation [1, 14] and
Tauberian tools. Even more, the presented method yields new interesting results
for Shubin type ΨDOs of finite order.
In order to illustrate and motivate some of our results, we briefly discuss a
particular but interesting prototypical class of infinite order pseudo-differential op-
erators to which our results can be applied. Such operators arise as power series of
a classical elliptic Shubin differential operator aw = a(x,D) = a xαDβ
α+β m α,β
on Rd. It should be mentioned that in [10] the composition of|a| f|un|≤ction and a
P
pseudo-differential operator is treated in a completely different way in comparison
with our approach. Formally, given a suitable entire function P (an ultrapolyno-
mial with positive Taylor coefficients), we can try to associate two natural infinite
order operators to aw. The first instance is the operator P(aw), which might be
defined as an unbounded operator on L2(Rd) via functional calculus and results on
eigenexpansion characterisations of Gelfand-Shilov spaces with respect to globally
elliptic PDOs (cf. [8, 9, 25]). The second associated operator is given by the Weyl
quantisation of the composition of P with the symbol a, let us denote it as (P a)w.
◦
A priori, it is not even clear whether they are pseudo-differential operators, but we
will give sufficient conditions on the Taylor coefficients of P such that both P(aw)
and(P a)w give rise toinfinite order pseudo-differential operatorswith hypoelliptic
symbols◦ in our classes Γ∗,∞. Furthermore, under the same assumptions on P, we
Ap,ρ
will find explicit Weyl asymptotic formulae for operators of the form A = P(aw)+
1
“smaller order perturbations” and A = (P a)w+ “smaller order perturbations”.
2
◦
For example, in the special case of the harmonic oscillator H = a(x,D) = x 2 ∆
| | −
and the entire function P(λ) = 1 + ∞ (hλ)n/n2sn where the parameter s > 1,
n=1
writing N(λ) for the spectral counting function of an operator of above form A or
1
P
A , it will directly follow from our results that
2
e2ds
N(λ) (lnλ)2ds, λ .
∼ hds2ds2d(2s+1)d! → ∞
We now discuss the content and organisation of the article in more detail. Sec-
tion 2 gives some basic background material about Gelfand-Shilov type spaces of
ultradifferentiable functions and ultradistributions (Rd) and (Rd). In Subsec-
∗ ′∗
S S
tions 3.1, 3.2 and 3.3 we collect and explain some useful properties of the symbol
WEYL FORMULAE AND SOBOLEV SPACES FOR INFINITE ORDER ΨDOS 3
classes Γ∗,∞ and the corresponding global pseudo-differential operators of infinite
Ap,ρ
order Op (a), the τ-quantisation of the symbol a, acting on (Rd) and (Rd). In
τ S∗ S′∗
the case τ = 1/2 (the Weyl quantisation, denoted as aw = Op (a)), the symbolic
1/2
calculus is realised through the ring structure of the spaces of asymptotic expan-
sions FS∗,∞ where the product corresponds to the composition of operators (i.e.
Ap,ρ
the #-product; cf. [21, Section 3]). In Subsection 3.4 we employ our results on Anti-
Wick quantisation [20] to derive the semi-boundedness of the Weyl quantisation
of positive hypoelliptic infinite order symbols. As a consequence, for hypoelliptic
real-valued a such that a(w) as w , one obtains that the closure A
of the unbounded operat|or A|o→n L∞2(Rd)|ge|n→era∞ted by aw is self-adjoint and has a
spectrum given by a sequence of eigenvalues λj j N, each one with finite multiplic-
{ } ∈
ity. Moreover, the sequence λ tends to or , with eigenfunctions belonging
n
to (Rd) and forming an orthonormal b∞asis fo−r ∞L2(Rd).
∗
S
WestateinSection4ourmainresultsconcerningWeyl asymptoticformulaeand
we postpone their proofs to Section 7, after developing of the necessary machinery.
We assume there that the symbol a satisfies elliptic type bounds with respect to
rather general comparison function f that is positive, increasing and has suitable
growth order. Theorem 4.1 gives the asymptotic behaviour of the spectral counting
function N(λ) for infinite order symbols, which corresponds to f being of actual
ultrapolynomial growth (andthus f increases faster thananypower functionat ).
∞
Theorem 4.2 deals with the case of finite order Shubin type hypoelliptic symbols
that satisfy elliptic bounds but with certain growth conditions on f that appear
to be different from the ones treated in the literature (cf. [16, 24]). Theorem 4.4
provides an O-bound for N(λ) by requiring only knowledge on a lower bound for
the symbol. We present there also some illustrative examples.
Section 5 is devoted to the analysis of the concrete operators we have mentioned
above as motivating examples, that is, operators related to the composition of an
ultrapolynomial withclassical elliptic Shubin differential operatorsortheir symbols.
Theorem 5.4 gives a quite new inside about our symbol classes Γ∗,∞ by providing a
Ap,ρ
general method to construct infinite order ΨDOs out of Shubin elliptic polynomials.
In the Theorem 5.6 we obtain the spectral asymptotics for perturbations of the
operators considered in Theorem 5.4.
The heat kernel analysis needed for the proofs of the Weyl asymptotic formulae
for the class of operators under consideration is given in Section 6. We consider a
real-valuedhypoellipticsymbol ainΓ∗,∞ suchthata(w)/ln w + as w .
Ap,ρ | | → ∞ | | → ∞
ThemaingoalofthissectionistostudythesemigroupT(t)f = ∞j=0e−tλj(f,ϕj)ϕj,
f L2(Rd), t 0, with infinitesimal generator A (the closure of aw in L2(Rd))
∈ ≥ − P −
where λ and ϕ are the eigenvalues and eigenfunctions of A. The crucial result to
j j
be shown here is that T(t), t 0, act continuously on (Rd) and they form a
∗
smooth family of operators on ≥(Rd). The proofs of theseSfacts are rather lengthy
∗
S
and we devote a whole subsection to them (Subsection 6.2). It is important to
stress that the classical approach does not work here (cf. Remark 6.14; one of the
main reasons is the lack, at this point, of Shubin-Sobolev spaces that fill in the
gaps between the Gelfand-Shilov type spaces (Rd) and L2(Rd)), so we had to
∗
S
develop new techniques to solve the problems. Once we have these properties of the
semigroup T(t), t 0, we prove that it is equal to the heat parametrix of aw as
≥
constructed in [21] modulo a smooth family of ultra-smoothing operators and use
4 S. PILIPOVIC´,B. PRANGOSKI,AND J. VINDAS
this to obtain the asymptotic formula
∞ e−tλj = 1 e−ta(x,ξ)dxdξ +O e−4ta(x,ξ) dxdξ , t 0+.
(2π)d (x,ξ) 2ρ →
R2d R2d !
j=0 Z Z h i
X
Thiskeyasymptoticformulaisthestartingpointfortheproofsofourmaintheorems
from Section 4 concerning Weyl asymptotic formulae; such proofs are the content
of Section 7.
Section 8 is devoted to infinite order Shubin-Sobolev type spaces H (f)
A∗p,ρ
(called in the sequel Sobolev spaces), where the order is measured by a function f
that belongs to a certain class of “admissible” functions and that might grow ultra-
polynomially. When f(x,ξ) = (x,ξ) s, s R, such spaces reduce to the classical
Sobolev spaces Hs(Rd) of ordehr s, wihere ∈Γ is the class of Shubin symbols. These
Γ
spaces satisfy most of the familiar results that the classical finite order Sobolev
spaces enjoy (independence on the choice of the generating operator, a priori esti-
mates, duality, etc.); additionally, we investigate their connection to various spaces
of Gelfand-Shilov type. Furthermore, we discuss the Fredholmness of a Γ∗,∞-elliptic
Ap,ρ
ΨDOasamappingfromoneoftheseSobolevspacesintoanotherone.Asanapplica-
tion of these considerations, we conclude the article by discussing “ultra-regularity”
properties of our Sobolev spaces.
2. Preliminaries
For x Rd and α Nd, we will use the notation x = (1 + x 2)1/2,Dα =
D1α1...Ddαd∈, where Djαj ∈= i−αj∂αj/∂xjαj. h i | |
Following Komatsu [11], we work with some of the standard conditions (M.1),
(M.2), (M.3), (M.3) and (M.4) on sequences of positive numbers M , p N, for
′ p
∈
which we always assume M = 1. We only recall (M.4) :
0
(M.4) M2/p!2 (M /(p 1)!) (M /(p+1)!), p Z .
p ≤ p−1 − · p+1 ∈ +
Observe that (M.4) amounts to ask that M /p! satisfies (M.1). We mention that if
p
M satisfies (M.3), then (M.4) holds for an equivalent sequence to M [18, Prop.
p p
1.1]. Note that the Gevrey sequence M = p!s, s > 1, satisfies all the conditions
p
˜ ˜
listed above. Given two weight sequences M and M , the notation M M means
p p p p
⊂
that there are C,L > 0 such that M CLpM˜ , p N, while M M˜ means
p p p p
≤ ∀ ∈ ≺
that this inequality holds true for each L > 0 and a corresponding C = C > 0.
L
For a multi-index α Nd, M stands for M , α = α + ... + α . As usual
α α 1 d
([11, Section 3]), we s∈et m = M /M , p |Z| ,|an| d if M satisfies (M.1) and
p p p 1 + p
M /Cp , for any C > 0 (which o−bviou∈sly holds when M satisfies (M.3)),
p p ′
→ ∞
its associated function is defined by M(ρ) = sup ln ρp/M , ρ > 0. It is a non-
p N + p
negative, continuous, monotonically increasing fu∈nction, vanishes for sufficiently
small ρ > 0, and increases more rapidly than lnρn as ρ , for any n N. When
→ ∞ ∈
M = p!s, with s > 0, we have M(ρ) ρ1/s.
p
Let U be an open subset of Rd≍and K a regular compact subset of U. For
h > 0, Mp ,h(K) is the Banach space (abbreviated as (B)-space) of all (germs
{ }
E
odfe)noϕte∈s itCs∞su(Ubs)pathceatofsaatlilsfsymosuopthα∈fNudnscutpioxn∈sKs|uDpαpϕo(rxte)|d/(bhyαMKα.)W<e d∞efinaendasDloK{Mcapl}l,yh
convex spaces (abbreviated as l.c.s.) (Mp)(U), Mp (U), (Mp)(U), Mp (U) and
{ } { }
E E D D
their strong duals, the corresponding spaces of ultradistributions of Beurling and
Roumieu type, cf. [11, 12, 13].
WEYL FORMULAE AND SOBOLEV SPACES FOR INFINITE ORDER ΨDOS 5
We denote by R the set of all positive sequences which monotonically increase
to infinity. There is a natural order on R defined by (r ) (k ) if r k , p Z ,
p p p p +
and with it (R, ) becomes a directed set. ≤ ≤ ∀ ∈
For (r ) R≤, consider the sequence N = 1,N = M p r ,p Z . It is easy
p ∈ 0 p p j=1 j ∈ +
to check that this sequence satisfies (M.1) and (M.3) when M does so and its asso-
′ p
ciated function will be denoted by N (ρ), i.e. N (ρ) = suQp ln ρp/(M p r ),
rp rp p N + p j=1 j
ρ > 0. Note that for (r ) R and k > 0 there is ρ > 0 such∈that N (ρ) M(kρ),
p ∈ 0 rp ≤Q
for ρ > ρ .
0
A measurable function f on Rd is said to have ultrapolynomial growth of class
(M ) (resp. of class M ) if e M(h )f < for some h > 0 (resp. for every
p p − |·| L∞(Rd)
{ } k k ∞
h > 0). We have the following equivalent description of continuous functions of
ultrapolynomial growth of class M .
p
{ }
Lemma 2.1. ([21, Lemma 1.1]) Let B C(Rd). The following conditions are
⊆
equivalent: (i) For every h > 0 there exists C > 0 such that f(x) CeM(hx), for
| |
| | ≤
all x Rd, f B; (ii) Thereexist (rp) R and C > 0 such that f(x) CeNrp(|x|),
∈ ∈ ∈ | | ≤
for all x Rd, f B.
∈ ∈
We call an entire function P(z) = c zα, z Cd, an ultrapolynomial of
α Nd α ∈
class (M ) (resp. of class M ), whenever∈the coefficients c satisfy the estimate
p p α
c CLα/M , α N{d, fo}r some LP> 0 and C > 0 (resp. for every L > 0
α | | α
| | ≤ ∈
and some C = C(L) > 0). The corresponding operator P(D) = c Dα is an
α α
ultradifferential operator of the class (M ) (resp. of class M ) and, when M
p p p
{ } P
satisfies (M.2), it acts continuously on (Mp)(U) and (Mp)(U) (resp. on Mp (U)
{ }
E D E
and Mp (U)) and the corresponding ultradistribution spaces.
{ }
D
If M satisfies (M.1) and (M.3), for m > 0 we denote by Mp,m(Rd) the (B)-
p ′
space of all ϕ C (Rd) for which the norm sup mα eM(mS∞)Dαϕ /M
is finite. The s∈pace∞s of sub-exponentially decreasiαn∈gNdultr|a|dkifferen|·|tiablekfLu∞n(cRtdi)on oαf
Beurling and Roumieu type are defined as
(Mp)(Rd) = lim Mp,m Rd and Mp (Rd) = lim Mp,m Rd ,
{ }
S m←− S∞ S m−→0S∞
→∞ (cid:0) (cid:1) → (cid:0) (cid:1)
respectively. Their strong duals (Mp) and Mp are the spaces of tempered ultra-
′ ′{ }
S S
distributions of Beurling and Roumieu type, respectively. When M = p!s, s > 1,
p
Mp (Rd) is just the Gelfand-Shilov space s(Rd) [16]. If M satisfies (M.2), the
S{ } Ss p
ultradifferential operators of class act continuously on (Rd) and (Rd); these
∗ ′∗
∗ S S
spaces are nuclear and the Fourier transform is a topological isomorphism on them.
We refer to [6, 19] for the topological properties of (Rd) and (Rd). Here we re-
∗ ′∗
S S
call that, when M satisfies (M.2), the space Mp (Rd) is topologically isomorphic
p { }
S
to lim Mp,(rp)(Rd), where the projective limit is taken with respect to the natural
S
(r←p)−R ∞
∈
order on R defined above and Mp,(rp)(Rd) is the (B)-space of all ϕ C (Rd) for
∞
S∞ ∈
which the norm supα∈Nd keNrp(|·|)DαϕkL∞(Rd)/(Mα j|α=|1rj) is finite.
Next, let E and F be l.c.s.; (E,F) stands for the space of continuous lin-
L Q
ear mappings from E to F; when E = F, we write (E). We employ the nota-
L
tion (E,F) for the space (E,F) equipped with the topology of bounded con-
b
L L
vergence and, similarly, (E,F) and (E,F) stand for (E,F) equipped with
p σ
L L L
the topologies of precompact and simple convergence, respectively. Furthermore,
E ֒ F means that E is continuously and densely included in F. For (a,b) R
→ ⊆
6 S. PILIPOVIC´,B. PRANGOSKI,AND J. VINDAS
and 0 k , Ck((a,b);E) stands for the vector space of k times continuously
≤ ≤ ∞
differentiable E-valued functions on (a,b), while Ck([a,b);E) for the space of those
on [a,b), where the derivatives at a are to be understood as right derivatives; we
use analogous notations when considering functions over (a,b] or [a,b].
3. ΨDOs of infinite order of Shubin type on (Rd) and (Rd)
∗ ′∗
S S
WediscussinthissectionpropertiesoftheclassesofinfiniteorderΨDOsthatwe
shall consider in the article. In Subsections 3.1 and 3.2 we collect some useful facts
about their symbolic calculus and thesharp product; we refer to [22, 4] and[21, Sec-
tions 2 and 3] for a complete account. Subsection 3.3 deals with the L2-realisations
and the hypoellipticity. In Subsection 3.4 we prove the semi-boundedness of opera-
tors with positive hypoelliptic symbols.
3.1. Symbol classes and symbolic calculus. Let A and M be two weight
p p
sequences of positive numbers such that A = A = M = M = 1. We assume that
0 1 0 1
M satisfies (M.1), (M.2) and (M.3), and that A satisfies (M.1), (M.2), (M.3)
p p ′
and (M.4). Without loss of generality, we may assume that the constants c and
0
H appearing in (M.2) are the same for both sequences M and A . Moreover, we
p p
assume that A M . Let ρ = inf ρ R A Mρ ; clearly 0 < ρ 1.
p ⊂ p 0 { ∈ +| p ⊂ p} 0 ≤
Throughout the rest of the article, ρ is a fixed number satisfying ρ ρ 1, if the
0
≤ ≤
infimum is reached, or, otherwise ρ < ρ 1. Clearly, we may also assume that
0
≤
A c LpMρ, where c 1 is the constant from (M.2).
p ≤ 0 p 0 ≥
For h,m > 0, define ΓAMpp,,ρ∞(R2d;h,m) to be the (B)-space of all a ∈ C∞(R2d)
for which the norm
DαDβa(x,ξ) (x,ξ) ρα+ρβ e M(mξ)e M(mx)
sup sup ξ x h i | | | | − | | − | | .
hα+β A A
α,β Nd(x,ξ) R2d (cid:12) (cid:12) | | | | α β
∈ ∈ (cid:12) (cid:12)
is finite. As l.c.s., we define
Γ(Mp),∞(R2d;m) = limΓMp,∞(R2d;h,m); Γ(Mp),∞(R2d) lim Γ(Mp),∞(R2d;m);
Ap,ρ Ap,ρ Ap,ρ Ap,ρ
h←−0 m−→
→ →∞
Γ{Mp},∞(R2d;h) = lim ΓMp,∞(R2d;h,m); Γ{Mp},∞(R2d) = lim Γ{Mp},∞(R2d;h).
Ap,ρ Ap,ρ Ap,ρ Ap,ρ
m←−0 h−→
→ →∞
Then, Γ(Mp),∞(R2d;m) and Γ{Mp},∞(R2d;h) are (F)-spaces. The spaces Γ∗,∞(R2d)
Ap,ρ Ap,ρ Ap,ρ
are barrelled and bornological.
For τ ∈ R and a ∈ Γ∗A,p∞,ρ(R2d), the τ-quantisation of a is the operator Opτ(a),
continuous on (Rd) given by the iterated integral (first Fourier, then inverse
∗
S
Fourier transform):
1
(Op (a)u)(x) = ei(x y)ξa((1 τ)x+τy,ξ)u(y)dydξ.
τ (2π)d − −
Rd Rd
Z Z
Proposition 3.1. ([21, Proposition 2.1]) For each τ R, the bilinear mapping
(a,ϕ) 7→ Opτ(a)ϕ, Γ∗A,p∞,ρ(R2d) × S∗(Rd) → S∗(Rd), is∈hypocontinuous and it ex-
tends to the hypocontinuous bilinear mapping (a,T) 7→ Opτ(a)T, ΓA∗,p∞,ρ(R2d) ×
S′∗(Rd) → S′∗(Rd). The mappings a 7→ Opτ(a), ΓA∗,p∞,ρ(R2d) → Lb(S∗(Rd),S∗(Rd)),
Γ∗A,p∞,ρ(R2d) → Lb(S′∗(Rd),S′∗(Rd)) are continuous.
WEYL FORMULAE AND SOBOLEV SPACES FOR INFINITE ORDER ΨDOS 7
Let t 0. We denote Q = (x,ξ) R2d x < t, ξ < t and Qc = R2d Q .
≥ t ∈ |h i h i t \ t
If 0 t 1, then Q = and Qc = R2d. Let B 0 and h,m > 0. Denote by
≤ ≤ t ∅ (cid:8) t ≥ (cid:9)
FSAMpp,ρ,∞(R2d;B,h,m) the vector space of all formal series ∞j=0aj(x,ξ) such that
a C (intQc ), DαDβa (x,ξ) can be extended to a continuous function on
j ∈ ∞ Bmj ξ x j P
Qc for all α,β Nd and
Bmj ∈
DαDβa (x,ξ) (x,ξ) ρα+ρβ+2jρe M(mξ)e M(mx)
supsup sup ξ x j h i | | | | − | | − | | < .
j N α,β (x,ξ) Qc (cid:12) (cid:12)h|α|+|β|+2jAαAβAjAj ∞
∈ ∈ Bmj (cid:12) (cid:12)
In the above, we use the convention m = 0 and hence, Qc = R2d. With this
0 Bm0
norm, FSMp,∞ R2d;B,h,m becomes a (B)-space. As l.c.s., we define
Ap,ρ
F(cid:0)S(Mp),∞(R2d(cid:1);B,m) = limFSMp,∞(R2d;B,h,m),
Ap,ρ Ap,ρ
h←−0
→
FS(Mp),∞(R2d;B) = lim FS(Mp),∞(R2d;B,m),
Ap,ρ Ap,ρ
m−→
→∞
FS{Mp},∞(R2d;B,h) = lim FSMp,∞(R2d;B,h,m),
Ap,ρ Ap,ρ
m←−0
→
FS{Mp},∞(R2d;B) = lim FS{Mp},∞(R2d;B,h).
Ap,ρ Ap,ρ
h−→
→∞
Then, the spaces FS(Mp),∞(R2d;B,m) and FS{Mp},∞(R2d;B,h) are (F)-spaces and
Ap,ρ Ap,ρ
FS∗,∞(R2d;B) is barrelled and bornological. The inclusion mapping Γ∗,∞(R2d)
Ap,ρ Ap,ρ →
FSA∗,p∞,ρ(R2d;B), defined as a 7→ j Naj, where a0 = a and aj = 0, j ≥ 1, is
continuous. We call this inclusion th∈e canonical one. For B B , the mapping
1 2
jpj 7→ jpj|QcB2mj, FSA∗,p∞,ρ(R2dP;B1) → FSA∗,p∞,ρ(R2d;B2) is c≤ontinuous. We also
call this mapping canonical.
P P
Let FS∗,∞(R2d) = lim FS∗,∞(R2d;B), where the inductive limit is taken in an
Ap,ρ Ap,ρ
B−→
algebraic sense and the →lin∞king mappings are the canonical ones described above.
Clearly, FS∗,∞(R2d) is non-trivial.
Ap,ρ
If jaj ∈ FSA∗,p∞,ρ(R2d;B) and n ∈ N, ( jaj)n will just mean the function
an ∈ CP∞(QcBmn), while ( jaj)<n denotes thPe function jn=−01aj ∈ C∞(QcBmn−1).
Furthermore, 1 denotes thPe element jaj ∈ FSA∗,p∞,ρ(R2d;PB) given by a0(x,ξ) = 1
and a (x,ξ) = 0, j Z .
j +
∈ P
Definition 3.2. ([22, Definition 3]) Sums, j Naj, j Nbj ∈ FSA∗,p∞,ρ(R2d), are
∈ ∈
said to be equivalent, in notation a b , if there exist m > 0 and
j N j ∼P j N jP
B > 0 (resp. there exist h > 0 and B∈> 0), such∈that for every h > 0 (resp. for
P P
every m > 0),
DαDβ (a (x,ξ) b (x,ξ)) (x,ξ) ρα+ρβ+2nρ
ξ x j<n j − j h i | | | |
sup sup sup
n∈Z+ α,β (x,ξ)∈QcBmn (cid:12)(cid:12)(cid:12) P h|α|+|β|+2nAαAβA(cid:12)(cid:12)(cid:12)nAn ·
e M(mξ)e M(mx) < .
− | | − | |
· ∞
In the sequel, we will often use the notation w = (x,ξ) R2d.
∈
Let Λ be an index set and f λ Λ be a set of positive continuous functions
λ
{ | ∈ }
8 S. PILIPOVIC´,B. PRANGOSKI,AND J. VINDAS
on R2d each with ultrapolynomial growth of class . We say that a set U(Λ) =
∗
jaj(λ) λ ∈ Λ ⊆ FSA∗,p∞,ρ(R2d;B′) is subordinated to {fλ|λ ∈ Λ} in FSA∗,p∞,ρ(R2d),
inn notation U(Λ)o- f λ Λ , if the following estimate holds: there exists B B
P (cid:12) λ ′
(cid:12) { | ∈ } ≥
such that for every h > 0 there exists C > 0 (resp. there exist h,C > 0) such that
Dαa(λ)(w) w ρ(α+2j)
w j h i | |
supsup sup sup C.
λ Λ j N α N2dw Qc (cid:12)(cid:12) h|α|+2jAα(cid:12)(cid:12)+2jfλ(w) ≤
∈ ∈ ∈ ∈ Bmj (cid:12) |(cid:12)|
Whenever we want to emphasise that the estimate is valid for a particular B B ,
′
we write U(Λ) - {fλ|λ ∈ Λ} in FSA∗,p∞,ρ(R2d;B). When fλ = f, ∀λ ∈ Λ, we abb≥revi-
ate the notation and simply write U - f, and then say that U is subordinated to
f. Clearly, for U ⊆ FSA∗,p∞,ρ(R2d;B1) such that U - f, there exists B ≥ B1 such that
the image of U under the canonical mapping FSA∗,p∞,ρ(R2d;B1) → FSA∗,p∞,ρ(R2d;B) is
a bounded subset of FS(Mp),∞(R2d;B,m) for some m > 0 (resp. a bounded subset
Ap,ρ
of FS{Mp},∞(R2d;B,h) for some h > 0).
Ap,ρ
Given U ⊆ FSA∗,p∞,ρ(R2d;B1) with U - f, we say that a bounded set V in
Γ(Mp),∞(R2d;m) for some m > 0 (resp. in Γ{Mp},∞(R2d;h) for some h > 0) is sub-
Ap,ρ Ap,ρ
ordinated to U under f, in notations V - U, if there exists a surjective mapping
f
Σ : U V such that the following estimate holds: there exists B B such that
1
→ ≥
for every h > 0 there exists C > 0 (resp. there exist h,C > 0) such that for all
a U and the corresponding Σ( a ) = a V
j j ∈ j j ∈
P Dα a(w)P a (w) w ρ(α+2n)
w − j<n j h i | |
sup sup sup C.
n∈Z+α∈N2dw∈QcBmn (cid:12)(cid:12)(cid:12) (cid:16) h|α|+P2nA|α|+2nf((cid:17)w(cid:12)(cid:12)(cid:12)) ≤
Again, when we want to emphasise the particular B for which this holds, we write
V -f U in FSA∗,p∞,ρ(R2d;B). If V -f U and if we denote by V˜ the image of V
under the canonical inclusion Γ∗A,p∞,ρ(R2d) → FSA∗,p∞,ρ(R2d;0), a 7→ a+ j Z+0, then
by specialising the above estimate for n = 1 together with the bounde∈dness of V
P
in Γ(Mp),∞(R2d;m) for some m > 0 (resp. in Γ{Mp},∞(R2d;h) for some h > 0) and
Ap,ρ Ap,ρ
the continuity and positivity of f, we derive that V˜ - f in FS∗,∞(R2d;0). In such
Ap,ρ
a case, we slightly abuse notation and write V - f. This estimate also implies
Σ( jaj) ∼ jaj. To see that given such an U ⊆ FSA∗,p∞,ρ(R2d;B) there always
exists V - U, we can proceed as follows. Let ψ (Ap)(Rd) in the (M ) case and
P f P ∈ D p
ψ Ap (Rd) in the M case respectively, such that 0 ψ 1, ψ(ξ) = 1 when
{ } p
∈ D { } ≤ ≤
ξ 2andψ(ξ) = 0when ξ 3.Setχ(x,ξ) = ψ(x)ψ(ξ),χ (w) = χ(w/(Rm ))
n,R n
fhori n≤∈ Z+ and R > 0 andhpiut≥χ0,R(w) = 0. Given U ⊆ FSA∗,p∞,ρ(R2d;B) as above,
for jaj ∈ U denote R( jaj)(w) = ∞j=0(1−χj,R(w))aj(w). If R > B, this is a
well defined smooth function on R2d, since the series is locally finite.
P P P
Proposition 3.3. ([21, Proposition 2.2]) Let U = a(λ) λ Λ be a subset
j j ∈
of FSA∗,p∞,ρ(R2d;B′) that is subordinated to {fλ|λ ∈ Λ}niPn FSA∗,p∞(cid:12)(cid:12),ρ(R2d)o. There exists
R0 > B′ such that for each R ≥ R0, UR = R( jaj(λ)) λ ∈ Λ ⊆ ΓA∗,p∞,ρ(R2d) and
the following estimate holds: there exists B =n B(PR) ≥ B(cid:12)′ such tohat for every h > 0
(cid:12)
WEYL FORMULAE AND SOBOLEV SPACES FOR INFINITE ORDER ΨDOS 9
there exists C > 0 (resp. there exist h,C > 0) such that
Dα R( a(λ))(w) a(λ)(w) w ρ(α+2n)
w j j − j<n j h i | |
sup sup sup sup C.
λ∈Λn∈Z+α∈N2dw∈QcBmn (cid:12)(cid:12)(cid:12) (cid:16) P h|α|+2nAP|α|+2nfλ(w) (cid:17)(cid:12)(cid:12)(cid:12) ≤
If in addition fλ = f, ∀λ ∈ Λ, then UR is bounded in ΓA(Mp,pρ),∞(R2d;m) for some
m > 0 (resp. bounded in ΓA{Mp,pρ},∞(R2d;h) for some h > 0) and hence UR -f U.
We say that this UR is canonically obtained from U by χn,R n N. Of course,
{ } ∈
here the mapping Σ : U U is just a R( a ).
→ R j j 7→ j j
Proposition3.4. ([21,Proposition2.3P]) LetV be aPboundedsubsetofΓ(Mp),∞(R2d;m˜)
Ap,ρ
for some m˜ > 0 (resp. of Γ{Mp},∞(R2d;h˜) for some h˜ > 0). Assume that there exist
Ap,ρ
B,m > 0 such that for every h > 0 there exists C > 0 (resp. there exist B,h > 0
such that for every m > 0 there exists C > 0) such that
Dαa(w) w ρ(α+2n)
sup sup sup sup | w |h i | | C.
a∈V n∈Z+α∈N2dw∈QcBmn h|α|+2nA|α|+2neM(m|w|) ≤
Then, Op (a) a U is an equicontinuous subset of ( (Rd), (Rd)) for each
τ R.{ τ | ∈ } L S′∗ S∗
∈
In what follows, we will frequently use the term “ -regularising set” for a subset
of ( (Rd), (Rd)). ∗
′∗ ∗
L S S
Proposition 3.5. ([21, Proposition 2.4]) Let U1 ⊆ FSA∗,p∞,ρ(R2d;B) be such that
U - f, for some continuous positive function f with ultrapolynomial growth of
1
class and let τ,τ R. For each a U and j N, define
∗ 1 ∈ j j ∈ 1 ∈
p (x,ξ) = (τ1 −Pτ)|β|∂βDβa (x,ξ), (x,ξ) Qc .
j,a β! ξ x k ∈ Bmj
k+β=j
X| |
Then, U = jpj,a j aj ∈ U1 is a subset of FSA∗,p∞,ρ(R2d;B) and U - f. There
exists R > 0n, which can be chosenoarbitrarily large, such that
P (cid:12) P
(cid:12)
Op (R( a )) Op (R( p )) a U
τ1 j j − τ j j,a j j ∈ 1
is an equiconti(cid:8)nuous subPset of L(S′∗(Rd),PS∗(Rd)).(cid:12)(cid:12)MPoreover, {(cid:9)R( jaj) jaj ∈
U - U and R( p ) a U - U.
1} f 1 { j j,a | j j ∈ 1} f P (cid:12) P
(cid:12)
3.2. WeylquantisaPtion. ShParp product and ring structureofFS∗,∞(R2d;B).
Ap,ρ
We recall in this and the next subsection results from [21] about the Weyl quanti-
sation of symbols; we often write aw instead of Op (a).
1/2
For jaj, jbj ∈ FSA∗,p∞,ρ(R2d;B) we define their sharp product, denoted as
a # b , via the formal series c = a # b where
j j Pj j P j j j j j j
Pc (x,Pξ) = (−1)|β|∂αPDβa (x,Pξ)∂βDαbP(x,ξ), (x,ξ) Qc .
j α!β!2l ξ x s ξ x k ∈ Bmj
s+k+l=j α+β=l
X | X|
It is easy to verify that j cj is a well defined element of FSA∗,p∞,ρ(R2d;B). If a ∈
Γ∗A,p∞,ρ(R2d), then a# jbPj will denote the # product of the image of a under the
P
10 S. PILIPOVIC´,B. PRANGOSKI,AND J. VINDAS
canonical inclusion Γ∗A,p∞,ρ(R2d) → FSA∗,p∞,ρ(R2d;B) and jbj. The same convention
applies if b Γ∗,∞(R2d) or if both a,b Γ∗,∞(R2d).
∈ Ap,ρ ∈ Ap,ρ P
Remark 3.6. If jaj, jbj ∈ FSA∗,p∞,ρ(R2d;B) and jcj = jaj# j bj, then
c = b # a . In particular, if a and b are real-valued for all j N and
j j j j Pj j P j j P P P ∈
a # b = b # a , then c are real-valued for all j N.
Pj j Pj j Pj j j j j ∈
P As inPthe distPributionPal setting, in the infinite order case the composition of two
pseudo-differential operators is again a pseudo-differential operator with symbol in
the same class, but there is always an additional -regularising operator.
∗
Theorem 3.7. ([21, Theorem 3.1]) Let U1,U2 ⊆ FSA∗,p∞,ρ(R2d;B) be such that U1 -
f1 and U2 - f2 in FSA∗,p∞,ρ(R2d;B) for some continuous positive functions f1 and f2
with ultrapolynomial growth of class . Then:
∗
i) U1#U2 - f1f2 in FSA∗,p∞,ρ(R2d;B).
ii) Let V - U , with Σ : U V the surjective mapping, k = 1,2. There
k fk k k k → k
exists R > 0, which can be chosen arbitrarily large, such that
Op Σ ( a ) Op Σ ( b ) Op R( a # b )
1/2 1 j j 1/2 2 j j − 1/2 j j j j
n (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
P P a PU , Pb U (cid:12)
j j ∈ 1 j j ∈ 2(cid:12)
o
is an equicontinuous subset of ( (Rd), P(Rd)) and P
′∗ ∗
L S S
(3.1) R( a # b ) a U , b U - U #U .
j j j j j j ∈ 1 j j ∈ 2 f1f2 1 2
n o
Corollary 3.8P. ([21, CPorolla(cid:12)(cid:12)rPy 3.1]) Let UP1,U2 ⊆ FSA∗,p∞,ρ(R2d;B) with U1 - f1 and
U - f for some continuous positive functions of ultrapolynomial growth of class .
2 2
∗
For a U and b U denote c = a # b U #U . Then,
j j ∈ 1 j j ∈ 2 j j,a,b j j j j ∈ 1 2
there exists R > 0, which can be chosen arbitrarily large, such that
P P P P P
awbw cw a = R( a ), b = R( b ), c = R( c )
− j j j j j j,a,b
n o
is an equicontinuous s(cid:12)ubset ofP( (Rd), (PRd)) and (3.1P) holds.
(cid:12) ′∗ ∗
L S S
Remark 3.9. Corollary 3.8 is applicable when U and U are bounded subsets of
1 2
Γ(Mp),∞(R2d;m) for some m > 0 (resp. of Γ{Mp},∞(R2d;h) for some h > 0). In this
Ap,ρ Ap,ρ
case, the corollary reads: there exists R > 0, which can be chosen arbitrary large,
such that awbw Op (R(a#b)) a U , b U is equicontinuous -regularising
{ − 1/2 | ∈ 1 ∈ 2} ∗
set and {R(a#b)|a ∈ U1, b ∈ U2} is bounded in ΓA(Mp,pρ),∞(R2d;m) for some m > 0
(resp. of Γ{Mp},∞(R2d;h) for some h > 0, cf. Lemma 2.1).
Ap,ρ
Proposition 3.10. ([21, Proposition3.1]) For each B 0, FS∗,∞(R2d;B) is a ring
≥ Ap,ρ
with the pointwise addition and multiplication given by #. Moreover, the multipli-
cation # : FS∗,∞(R2d;B) FS∗,∞(R2d;B) FS∗,∞(R2d;B) is hypocontinuous.
Ap,ρ × Ap,ρ → Ap,ρ
3.3. Realisations in L2(Rd). Hypoelliptic operators of infinite order. We
now discuss hypoellipticity in the context of our symbol classes and L2-realisations
of the associated unbounded operators.