Spectral asymptotics for infinite order pseudo-differential operators

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.


Introduction
In this article we study the spectral properties of global infinite order pseudo-differential operators. Our operator classes are intrinsically related to the ultradistributional framework so that the bounds on the derivatives of the symbols are controlled by Gevrey type weight sequences. Our aim is to establish Weyl asymptotic formulae for a large class of (hypoelliptic) ΨDOs of infinite order. It is worth mentioning that the Weyl asymptotics for the operators that we investigate here are not of power-log-type as in the finite order (distributional) setting, but of log-type, which in turn yields that the eigenvalues of infinite order ΨDOs, with appropriate assumptions, are "very sparse". As a by-product of our analysis, we also obtain Weyl asymptotic formulae for a class of finite order Shubin ΨDOs with some conditions on the symbols that are not the ones usually discussed in the literature.
The spaces of symbols and corresponding pseudo-differential operators involved in this work were introduced by Prangoski (see [18] for the symbolic calculus) and then extensively studied in several articles by himself and his coauthors; 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 p and M p , p ∈ N. The first one controls the smoothness, while the second one controls the growth at infinity of the symbols. These symbol classes are denoted by Γ pseudo-differential operators. Further results related to the symbolic calculus that will be employed in the article are stated in the Appendix (Section 8).
Section 4 is devoted to establishing the semi-boundedness of the Weyl quantisation a w of a positive hypoelliptic infinite order symbol a. This will be achieved with the aid of results on Anti-Wick quantisation from [16]. This result is interesting by itself because hypoellipticity in this setting allows the symbols to approach 0 sub-exponentially and thus generalises the familiar result for finite order operators. As a consequence, for hypoelliptic real-valued a such that |a(w)| → ∞ as |w| → ∞, one obtains that the closure A of the unbounded operator A on L 2 (R d ) generated by a w is self-adjoint and has a spectrum given by a sequence of eigenvalues λ n , n ∈ N, tending to ∞ or −∞, with eigenfunctions belonging to S * (R d ) and forming an orthonormal basis for L 2 (R d ).
We state in Section 5 our main results concerning Weyl asymptotic formulae and we postpone their proofs to Section 7, after developing the necessary machinery. We assume there that the symbol a satisfies elliptic type bounds with respect to a rather general comparison function f that is positive, increasing, and has suitable growth order. Theorem 5.1 gives the asymptotic behaviour of the spectral counting function N(λ) for infinite order symbols, which corresponds to f being of actual ultrapolynomial growth (and thus f increases faster than any power function at ∞). Even more, our method yields new interesting results for Shubin type ΨDOs of finite order. Theorem 5.2 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. [13,20]). Theorem 5.4 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.
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-valued hypoelliptic symbol a in Γ * ,∞ Ap,ρ such that a(w)/ ln |w| → +∞ as |w| → ∞. The main goal is the analysis of the semigroup T (t)f = ∞ j=0 e −tλ j (f, ϕ j )ϕ j , f ∈ L 2 (R d ), t ≥ 0, with infinitesimal generator −A (the closure of −a w in L 2 (R d )) where λ j and ϕ j are the eigenvalues and eigenfunctions of A. The crucial result to be shown here is that T (t), t ≥ 0, form a smooth family of operators continuously acting on S * (R d ). The proofs of these facts are rather lengthy and we devote a whole subsection to them. 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 of Shubin-Sobolev spaces that fill in the gaps between the Gelfand-Shilov type spaces S * (R d ) and L 2 (R d )), so we had to develop new techniques to overcome 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 a w as constructed in [17] modulo a smooth family of ultra-smoothing operators and use this to obtain the asymptotic formula This key asymptotic formula is the starting point for the proofs of our main theorems from Section 5 concerning Weyl asymptotic formulae; such proofs are the content of Section 7. The passage from asymptotics of the heat semigroup to Weyl formulae is accomplished using ideas from the theory of regular variation [1,11] and Tauberian tools.

Preliminaries
For x ∈ R d and α ∈ N d , we will use the notation x = (1+|x| 2 ) 1/2 , D α = D α 1 1 . . . D α d d , where D α j j = i −α j ∂ α j /∂x j α j . Following Komatsu [8], we work with some of the standard conditions (M. 1 (M.4) M 2 p /p! 2 ≤ (M p−1 /(p − 1)!) · (M p+1 /(p + 1)!), p ∈ Z + . Note that the Gevrey sequence M p = p! s , s > 1, satisfies all the conditions listed above. Given two weight sequences M p andM p , the notation M p ⊂M p (resp. M p ≺M p ) means that there are C, L > 0 (resp. for every L > 0 there is C > 0) such that M p ≤ CL pM p , ∀p ∈ N. For a multi-index α ∈ N d , M α stands for M |α| , |α| = α 1 + ... + α d . As usual ([8, Section 3]), we set m p = M p /M p−1 , p ∈ Z + , and if M p satisfies (M.1) and M p /C p → ∞, for any C > 0 (which obviously holds when M p satisfies (M.3) ′ ), its associated function is defined by M(ρ) = sup p∈N ln + ρ p /M p , ρ > 0. It is a non-negative, continuous, monotonically increasing function, vanishes for sufficiently small ρ > 0, and increases more rapidly than ln ρ n as ρ → ∞, for any n ∈ N. When M p = p! s , with s > 0, we have M(ρ) ≍ ρ 1/s . For a regular compact set K ⊆ R d (i.e. K = int K) and h > 0, E Mp,h (K) is the Banach space (abbreviated as (B)-space) of all ϕ ∈ C ∞ (int K) whose derivatives extend to continuous functions on K and satisfy sup α∈N d sup x∈K |D α ϕ(x)|/(h α M α ) < ∞ and D Mp,h K denotes its subspace of all smooth functions supported by K. For U ⊆ R d , we define as locally convex spaces (abbreviated as l.c.s.) and their strong duals, the corresponding spaces of ultradistributions of Beurling and Roumieu type, cf. [8,9,10].
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 p ) ≤ (k p ) if r p ≤ k p , ∀p ∈ Z + , and with it (R, ≤) becomes a directed set. For (r p ) ∈ R, consider the sequence N 0 = 1, N p = M p p j=1 r j , p ∈ Z + . It is easy to check that this sequence satisfies (M.1) and (M.3) ′ when M p does so and its associated function will be denoted by N rp (ρ), i.e. N rp (ρ) = sup p∈N ln + ρ p /(M p p j=1 r j ), ρ > 0. Note that for (r p ) ∈ R and k > 0 there is ρ 0 > 0 such that N rp (ρ) ≤ M(kρ), for ρ > ρ 0 .
A measurable function f on R d is said to have ultrapolynomial growth of class (M p ) (resp. of class {M p }) if e −M (h|·|) f L ∞ (R d ) < ∞ for some h > 0 (resp. for every h > 0). We have the following equivalent description of continuous functions of ultrapolynomial growth of class {M p }.
The following conditions are equivalent: The spaces of sub-exponentially decreasing ultradifferentiable function of Beurling and Roumieu type are defined as respectively. Their strong duals S ′(Mp) (R d ) and S ′{Mp} (R d ) are the spaces of tempered ultradistributions of Beurling and Roumieu type, respectively. When M p = p! s , s > [13]. If M p satisfies (M.2), the ultradifferential operators of class * act continuously on S * (R d ) and S ′ * (R d ) (for the definition of ultradifferential operators see [8]). These spaces are nuclear and the Fourier transform is a topological isomorphism on them. We refer to [6,15] for the topological properties of S * (R d ) and S ′ * (R d ). Here we recall that, when M p satisfies (M.2), the space where the projective limit is taken with respect to the natural order on R defined above and S Next, let E and F be l.c.s.; L(E, F ) stands for the space of continuous linear mappings from E to F ; when E = F , we write L(E). We employ the notation L b (E, F ) for the space L(E, F ) equipped with the topology of bounded convergence and, similarly, L p (E, F ) and L σ (E, F ) stand for L(E, F ) equipped with 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 and 0 ≤ k ≤ ∞, C k ((a, b); E) stands for the vector space of k times continuously differentiable E-valued functions on (a, b), while C k ([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].

ΨDOs of infinite order of Shubin type on
We discuss in this section properties of the classes of infinite order ΨDOs that we shall consider in the article; see also the Appendix for other important facts about their symbolic calculus. We refer to [18,4] and [17,Sections 3 and 4] for complete accounts. . Of course, we may assume that the constants c 0 and H appearing in (M.2) are the same for both sequences M p and A p . We assume that A p ⊂ M p . Let ρ 0 = inf{ρ ∈ R + | A p ⊂ M ρ p }; clearly 0 < ρ 0 ≤ 1. Throughout the rest of the article, ρ is a fixed number satisfying ρ 0 ≤ ρ ≤ 1, if the infimum is reached, or, otherwise ρ 0 < ρ ≤ 1. Clearly, we may also assume that A p ≤ c 0 L p M ρ p , where c 0 ≥ 1 is the constant from (M.2). For h, m > 0, define (following [18] is finite. As l.c.s., we define Then, Γ For τ ∈ R and a ∈ Γ * ,∞ Ap,ρ (R 2d ), the τ -quantisation of a is the operator Op τ (a), continuous on S * (R d ) given by the iterated integral: Let B ≥ 0 and h, m > 0. Following [18,17], In the above, we use the convention m 0 = 0 and hence, Q c Then, the spaces F S Ap,ρ (R 2d ; B), defined as a → j∈N a j , where a 0 = a and a j = 0, j ≥ 1, is continuous. We call this inclusion the canonical one. For B 1 ≤ B 2 , the mapping j p j → j p j| Q c is continuous. We also call this mapping canonical. Let F S * ,∞ Ap,ρ (R 2d ) = lim −→ B→∞ F S * ,∞ Ap,ρ (R 2d ; B), where the inductive limit is taken in an algebraic sense and the linking mappings are the canonical ones described above. Clearly, If j a j ∈ F S * ,∞ Ap,ρ (R 2d ; B) and n ∈ N, ( j a j ) n will just mean the function a n ∈ C ∞ (Q c Bmn ), while ( j a j ) <n denotes the function n−1 j=0 a j ∈ C ∞ (Q c Bm n−1 ). Furthermore, 1 denotes the element j a j ∈ F S * ,∞ Ap,ρ (R 2d ; B) given by a 0 (x, ξ) = 1 and a j (x, ξ) = 0, j ∈ Z + .
Recall, [18,Definition 3] that two sums, j∈N a j , j∈N b j ∈ F S * ,∞ Ap,ρ (R 2d ), are said to be equivalent, in notation j∈N a j ∼ j∈N b j , if there exist m > 0 and B > 0 (resp. there exist h > 0 and B > 0), such that for every h > 0 (resp. for every m > 0),

3.2.
Subordination. In the sequel, we will often use the notation w = (x, ξ) ∈ R 2d . Let Λ be an index set and {f λ | λ ∈ Λ} be a set of positive continuous functions on R 2d each with ultrapolynomial growth of class * . We say that a set Ap,ρ (R 2d ), in notation U (Λ) {f λ | λ ∈ Λ}, if the following estimate holds: there exists B ≥ B ′ such that for every h > 0 there exists C > 0 (resp. there exist h, C > 0) such that Whenever we want to emphasise that the estimate is valid for a particular B ≥ B ′ , we write When f λ = f , ∀λ ∈ Λ, we abbreviate the notation and simply write U f , and then say that U is subordinated to f . Clearly, for there exists a surjective mapping Σ : U → V such that the following estimate holds: there exists B ≥ B 1 such that for every h > 0 there exists C > 0 (resp. there exist h, C > 0) such that for all j a j ∈ U and the corresponding Σ( j a j ) = a ∈ V Again, when we want to emphasise the particular B for which this holds, we write V f U in F S * ,∞ Ap,ρ (R 2d ; B). If V f U and if we denote byṼ the image of V under the canonical inclusion Γ * ,∞ Ap,ρ (R 2d ) → F S * ,∞ Ap,ρ (R 2d ; 0), a → a + j∈Z + 0, then by specialising the above estimate for n = 1 together with the boundedness of V in Γ (Mp),∞ Ap,ρ (R 2d ; m) for some m > 0 (resp. in Γ {Mp},∞ Ap,ρ (R 2d ; h) for some h > 0) and the continuity and positivity of f , we derive thatṼ f in F S * ,∞ Ap,ρ (R 2d ; 0). In such a case, we slightly abuse notation and write V f . This estimate also implies Σ( j a j ) ∼ j a j . To see that given such an U ⊆ F S * ,∞ Ap,ρ (R 2d ; B) there always exists V f U, we can proceed as follows. Let ψ ∈ D (Ap) (R d ) in the (M p ) case and ψ ∈ D {Ap} (R d ) in the {M p } case respectively, such that 0 ≤ ψ ≤ 1, ψ(ξ) = 1 when ξ ≤ 2 and ψ(ξ) = 0 when ξ ≥ 3. Set χ(x, ξ) = ψ(x)ψ(ξ), χ n,R (w) = χ(w/(Rm n )) for n ∈ Z + and R > 0 and put χ 0,R (w) = 0. Given U ⊆ F S * ,∞ Ap,ρ (R 2d ; B) as above, for j a j ∈ U denote R( j a j )(w) = ∞ j=0 (1 − χ j,R (w))a j (w). If R > B, this is a well defined smooth function on R 2d , since the series is locally finite.
Ap,ρ (R 2d ) and the following estimate holds: there exists B = B(R) ≥ B ′ such that for every h > 0 there exists C > 0 (resp. there exist h, C > 0) such that We say that this U R is canonically obtained from U by {χ n,R } n∈N . Of course, here the mapping Σ : U → U R is just j a j → R( j a j ).
In what follows, we will frequently use the term " * -regularising set" for a subset of L(S ′ * (R d ), S * (R d )). Changing the quantisation and taking composition of ΨDOs with symbols in Γ * ,∞ Ap,ρ (R 2d ) always results in ΨDOs with symbols in the same class modulo * -regularising operators; we collect some of these facts in the Appendix and we refer to [17,18] for the complete theory.

Weyl quantisation. The sharp product in F S * ,∞
Ap,ρ (R 2d ; B). We recall in this and the next subsection results from [17] about the Weyl quantisation of symbols; we often write a w instead of Op 1/2 (a). Given It is easy to verify that j c j is a well defined element of F S * ,∞ Ap,ρ (R 2d ; B). If a ∈ Γ * ,∞ Ap,ρ (R 2d ), then a# j b j will denote the # product of the image of a under the canonical inclusion In particular, if a j and b j are real-valued for all j ∈ N and j a j # j b j = j b j # j a j , then c j are real-valued for all j ∈ N.
is a ring with the pointwise addition and multiplication given by #. Moreover, the multiplication is given by 1. The #-product of symbols corresponds to the composition of their Weyl quantisation (see the Appendix).

Hypoelliptic operators of infinite order
This section is devoted to hypoellipticity in the context of our symbol classes. Our main goal below is to establish a semi-boundedness result. In preparation, we start by discussing L 2 -realisations of the associated unbounded operators. .
Considering a w as a mapping on S ′ * (R d ), its restriction to the subspace {g ∈ L 2 (R d )| a w g ∈ L 2 (R d )} defines a closed extension of A which is called the maximal realisation of A. As standard, we denote by A the closure of A, also called the minimal realisation of A. Notice that the formal adjoint (a w ) * is in fact the pseudo-differential operatorā w and hence, it can be extended to a continuous operator on S ′ * (R d ). One can also consider the adjoint A * of A in L 2 (R d ). The following result gives the precise connection between A * and (a w ) * . Its proof is completely analogous to the one in the classical case for finite order ΨDOs and we omit it (see for example [13, Proposition 4.2.1, p. 160]).
Ap,ρ (R 2d ) with A and A * defined as above. Then A * coincides with the maximal realisation of (a w ) * , i.e. the domain of We now introduce the notion of hypoellipticity in Γ * ,∞ Ap,ρ .
. We say that a is Γ * ,∞ Ap,ρ -hypoelliptic (or, in short, simply hypoelliptic) if i) there exists B > 0 such that there are c, m > 0 (resp. for every m > 0 there is Operators with hypoelliptic symbols have parametrices and hence are globally regular; see the Appendix for the precise results.

4.1.
Semi-boundedness and the spectrum of operators with positive hypoelliptic Weyl symbols. Before we can say anything meaningful about the spectrum of operators with hypoelliptic positive Weyl symbols, we need to prove that such operators are always semi-bounded. This is a well know fact for finite order symbols. We prove here that it remains true even in the infinite order case. In order to appreciate more this result, the reader should keep in mind the operators can be of truly infinite order, i.e. the symbols are allowed to have ultrapolynomial growth; such operators then go beyond the classical Weyl-Hörmander calculus.
The proof heavily relies on the connection between the Weyl and the anti-Wick quantisation of symbols from Γ * ,∞ Ap,ρ (R 2d ) (see [16]). For a ∈ Γ * ,∞ Ap,ρ (R 2d ), we denote by A a its anti-Wick quantisation. By [16,Theorem 3.2], there exists a ∈ Γ * ,∞ Ap,ρ (R 2d ) and a * -regularising operator T such that b w = A a + T . By a careful inspection of the proof of the quoted result, one can find the explicit construction of a; it is given as follows. Start Since b is positive and hypoelliptic, the estimate (4.3) holds on the whole R 2d for b. Repeating the proof of [16, Theorem 3.2] verbatim and using (4.3) for b (which, as we mentioned, is valid on R 2d ), we obtain the following estimate: for every h > 0 there exists C > 0 (resp. there exist h, C > 0) such that where R ≥ 1 can be chosen to be the same for all j ∈ N and the following estimate holds: for every h > 0 there exists In the (M p ) case, fix 0 < h ′ < 1 and let C ′ > 1 be the constant for which (4.4) holds and in the {M p } case, let h ′ , C ′ > 1 be the constants for which this estimate holds. If we take large enough Using Proposition 4.4, Proposition 4.5 and Remark 8.7, we can prove the following spectral result in the same way as in the proof of [13, Theorem 4.2.9, p. 163].
Proposition 4.6. Let a ∈ Γ * ,∞ Ap,ρ (R 2d ) be a hypoelliptic real-valued symbol such that |a(w)| → ∞ as |w| → ∞ and let A be the unbounded operator on L 2 (R d ) defined by a w . Then the closure A of A is a self-adjoint operator having spectrum given by a sequence of real eigenvalues either diverging to +∞ or to −∞ according to the sign of a at infinity. The eigenvalues have finite multiplicities and the eigenfunctions belong to S * (R d ). Moreover, L 2 (R d ) has an orthonormal basis consisting of eigenfunctions of A.

The Weyl asymptotic formula for infinite order ΨDOs. Part I: statements of the main results
This section is dedicated to Weyl asymptotic formulae for a large class of infinite order hypoelliptic pseudo-differential operators. We state here our main results, their proofs are postponed to Section 7, after obtaining some auxiliary results on the spectrum of the heat parametrix of positive hypoelliptic symbols.
We consider throughout this section a real-valued hypoelliptic symbol a ∈ Γ * ,∞ Ap,ρ (R 2d ) such that a(w) → ∞ as |w| → ∞. If we denote as A the closure of the unbounded operator on L 2 (R d ) induced by its Weyl quantisation a w then we can apply Proposition 4.6 to obtain that the spectrum of the self-adjoint operator A is given by a sequence of real eigenvalues with finite multiplicities {λ j } j∈N which tends to ∞, where multiplicities are taken into account and the sequence is arranged in non-decreasing order λ 0 ≤ λ 1 ≤ λ 2 ≤ · · · ≤ λ j ≤ . . . . We denote the spectral counting function of the operator A = a w as Our goal is to show later the following three theorems on spectral asymptotics. For these results, we will suppose that the symbol a satisfies certain asymptotic bounds with respect to a comparison function f , which we assume throughout the rest of this section to be positive, strictly increasing, of ultrapolynomial growth of class * on some interval [Y, ∞), for some Y > 0, and absolutely continuous on each compact subinterval of [Y, ∞). Furthermore, we employ the notation and let Φ be a positive continuous function on the sphere S 2d−1 . Suppose that for each ε ∈ (0, 1) there are positive constants c ǫ , C ǫ , B ǫ > 0 such that Note that Theorem 5.1 deals with operators which are truly of infinite order because integration of (5.2) gives that w β = o(a(w)) for any β > 0.
The next theorem gives the Weyl asymptotic formula for a wider class of finite order pseudo-differential operators than the one that is usually discussed in the literature, see e.g. [13,Sect. 4.6]; in particular, our result is more general than [13, Theorem 4.6.1, p. 196] (see Example 5.8 below). The reader should also compare this with [20, Theorem 30.1, p. 224]; we work with different assumptions than in the quoted result and, on the other hand, we give a more explicit result concerning the asymptotic behaviour of N(λ).
We will derive the following "geometric" version of Theorems 5.1 and 5.2 where the asymptotic behaviour of N is given in terms of the symbol.
If one is only interested in upper O-estimates on N, the next theorem gives such bounds under much weaker assumptions on the symbol. and

and the bound (5.15) holds for each
Remark 5.5. If lim sup y→∞ yf ′ (y)/f (y) < ∞, Theorem 5.4 is also valid for a ∈ Γ m ρ (R 2d ) that is Γ m ρ -hypoelliptic and satisfies (5.12), as the proof given in Section 7 shows. Here we get that λ j is bounded from below by a constant multiple of f (j 1 2d ) for λ j > 0. In particular, this case applies to f (y) = y β ′ , where we obtain N(λ) = O(λ 2d/β ′ ) and λ j ≥ h β ′ j β ′ /(2d) , j ≥ j h , with the constants as in Theorem 5.4 (see also Example 5.8).
The rest of this section is devoted to some illustrative examples. The asymptotic formulae from Examples 5.6 and 5.7 prove a result that one might expect: the eigenvalues of a truly infinite order operator are "very sparse".
More generally, let f (y) =M (hy), whereM is the associated function of a sequence M p ⊂M p (resp. M p ≺M p ), andM p satisfies (M.1). Then [8] yf ′ (y)/f (y) =m(hy) → ∞. In this case, when Φ(ϑ) = 1 we obtain Example 5.7. We present in this example another nontrivial instance of a hypoelliptic pseudo-differential operator of infinite order. Let ν, l, s be three positive numbers such that 1 < ν < l < s and ν/l ≤ 1 − 1/s. Consider the entire function where h is a positive constant, and the symbol It is shown in [5,Sect. 3] that a ∈ Γ * ,∞ Ap,ρ (R 2d ) is hypoelliptic, where ν/l ≤ ρ ≤ 1 − 1/s, M p = p! l , and A p = p! ν . Denote as N the spectral counting function of the Weyl quantisation of a and let {λ j } ∞ j=0 be its sequence of eigenvalues. We will show that We start by noticing that, given any fixed 0 < ε < 1, we have bounds Example 5.8. If f (y) = y β ln α y, where β > 0, we have that yf ′ (y)/f (y) → β and σ(λ) ∼ (β α λ ln α λ) 1/β . Therefore, the conclusion of Theorem 5.2 reads in this case Likewise for the upper bound from Theorem 5.4.

The spectrum of the heat parametrix
Throughout this section we assume a is a hypoelliptic real-valued symbol in Γ * ,∞ Ap,ρ (R 2d ) such that a(w)/ ln |w| → ∞ as |w| → ∞. There exists B ≥ 1 such that the hypoellipticity condition (4.3) for a holds on Q c B and a(w) > 0, ∀w ∈ Q c B . Pickχ ∈ D (Ap) (R 2d ) (resp. χ ∈ D {Ap} (R 2d )) such that 0 ≤χ ≤ 1,χ = 1 on Q B 1 , for B 1 > B, andχ = 0 on the complement of a small neighbourhood of Q B 1 . Then b = (1 −χ)a +χ is positive on the whole R 2d and, in fact, it is a hypoelliptic symbol in Γ * ,∞ Ap,ρ (R 2d ) for which the hypoellipticity condition (4.3) holds globally on R 2d . 6.1. The heat parametrix of positive hypoelliptic symbols. For the symbol b constructed above, we can apply the theory given in [17,Subsection 7.2] for the construction of the heat parametrix. We have the following series of results.
There exist u j (t, w) ∈ C ∞ (R × R 2d ), j ∈ N, such that u 0 (t, w) = e −tb(w) and the following results hold.
Since the operator a w − b w = (a − b) w is * -regularising (by the definition of b), (6.2) implies We denote by A the unbounded operator on L 2 (R d ) defined by a w . We apply Proposition 4.6 and obtain that the spectrum of the self-adjoint operator A is given by a sequence of real eigenvalues {λ j } j∈N which tends to +∞, where the multiplicities are taken into account, and L 2 (R d ) has an orthonormal basis {ϕ j } j∈N consisting of eigenfunctions of A which all belong to S * (R d ) (ϕ j corresponds to λ j , j ∈ N). For each t ≥ 0, we define the following operator on L 2 (R d ) Obviously, the above series is unconditionally convergent and T (t) is continuous. Furthermore, T (t) is self-adjoint (one easily verifies that (T (t)g, g) ∈ [0, ∞), g ∈ L 2 (R d ), and hence it is positive) and T (0) = Id. Clearly, {T (t)} t≥0 is a C 0 -semigroup.
As it will become clear later, the analysis of this semigroup is one of the key ingredients in the proofs of the main results from Section 5. We will show: -T (t) belongs to L(S * (R d ), S * (R d )); -the mapping t → T (t), [0, ∞) → L b (S * (R d ), S * (R d )), is smooth; -T (t) and (u(t)) w are the same, modulo a smooth * -regularising family. As the proofs of these facts are rather lengthy, we devote a whole subsection to them. Remark 6.4. If a ∈ Γ m ρ (R 2d ) is a hypoelliptic real-valued symbol such that a(w) ≥ c w δ for some δ > 0, ∀|w| ≥ c, one can construct its heat parametrix as well. For this purpose, one can use the same construction as in [13,Theorem 4.5.1,p. 193] (although it is there given only for elliptic symbols). In fact, defining b ∈ Γ m ρ (R 2d ) to be positive on R 2d and equal to a outside of a compact neighbourhood of the origin, one can repeat the proof of the quoted result verbatim to find a symbol u(t, ·) ∈ Γ m ρ (R 2d ), t ≥ 0, which solves (6.2) (t 0 > 0 can be arbitrarily chosen), where u 0 (t, w) = e −tb(w) and u j is given as u j (t, w) = e −tb(w) 2j l=1 t l u l,j (w), j ∈ Z + , with symbols u l,j that satisfy the estimates |D α w u l,j (w)| ≤ C l,j,α (b(w)) l w −ρ(|α|+2j) , w ∈ R 2d . Notice then that (u(t)) w = (u(t, ·)) w satisfies the equation (6.3) for some vector-valued functionK ∈ C ∞ ([0, ∞); L b (S ′ (R d ), S(R d ))).

6.2.
The analysis of the semigroup T (t), t ≥ 0.
Observe now that for t > t 0 ≥ 0, g ∈ S ′ * (R d ) and ϕ ∈ S * (R d ), (6.21) implies Let B be a bounded subset of S * (R d ) and V a neighbourhood of zero in S * (R d ). Consider the neighbourhood of zero . We may of course assume V is the absolute polar B ′• of a bounded set , is differentiable and (d/dt)T (t) = −ã wT (t). As t → −ã wT (t) is continuous, t →T (t) is of class C 1 and now, the equality (d/dt)T (t) = −ã wT (t) readily implies that t →T (t) is in C ∞ ([0, ∞); L b (S * (R d ), S * (R d ))) and (d k /dt k )T (t) = (−1) k (ã w ) kT (t), k ∈ Z + .
Since T (t) solves (6.3) withK(t) = 0, we obtain Theorem 6.10 then implies that for each t > 0, the mapping s → T (t − s)K(s) belongs Similarly as in the proof of Proposition 6.9, one verifies Q(t)f ∈ S * (R d ) and, for each g ∈ S ′ * (R d ), Again, employing analogous techniques as in the proof of Proposition 6.9, one can prove f → Q(t)f ∈ L(S ′ * (R d ), S * (R d )), for each t ≥ 0. Using the properties of T (t) and K(t), one readily checks that the mapping (t, . Employing this fact together with (6.23) and the semigroup property of T (t), one can prove that t → Q(t), , is continuous. Now, reproducing the proof of [17,Lemma 7.15] verbatim, one gets the following result.
Notice that (6.1) together with a(w)/ ln |w| → +∞, as |w| → ∞, ensures that (u(t)) w is trace-class for each t > 0 (cf. [13,Theorem 4.4.21,p. 190]). Now, Lemma 6.11 ensures that T (t) is also trace-class for t > 0. As T (t) are self-adjoint, we conclude Tr The second integral is O(1) as t → 0 + (because of Corollary 6.12). Fix n > d/ρ, n ∈ Z + . Since u 0 (t, w) = e −tb(w) and b(w) = a(w) for w outside of a compact neighbourhood of the origin, we have In view of the second estimate in Lemma 6.2 (specialised for n = 0 and α = 0), the very last integral is O(1) as t → 0 + . Lemma 6.1 implies that there exists C ′ > 0 such that |u j (t, w)| ≤ Ce − t 4 b(w) w −2ρ , for all w ∈ R 2d , t ≥ 0, j = 1, . . . , n − 1. Using again b = a except in a compact neighbourhood of 0, we have To verify it, first notice that a ∈ Γ * ,∞ Ap,ρ (R 2d ) implies that there are m, C > 0 (resp. for every m > 0 there exists C > 0) such that a(w) ≤ Ce M (m|w|) , ∀w ∈ R 2d . Using this estimate (in the Roumieu case we can take m = 1 with the corresponding C > 0) and polar coordinates, we have Monotone convergence implies that the very last integral tends to ∞ as t → 0 + . We have shown: The next remark shows that (6.25) remains valid for hypoelliptic symbols of finite order.
Remark 6.14. Let a ∈ Γ m ρ (R 2d ) be a hypoelliptic real-valued symbol such that a(w) ≥ c w δ for some δ > 0, ∀|w| ≥ c, and consider its heat parametrix (u(t)) w = (u(t, ·)) w as constructed in Remark 6.4 and the C 0 -semigroup {T (t)} t≥0 as given by (6.4). The fact t → T (t) ∈ C ∞ ([0, ∞); L b (S(R d ), S(R d ))) can be proved far more easily in the distributional setting. To verify this, first notice that (a w ) j is hypoelliptic for each j ∈ Z + and denote its symbol by a j ∈ Γ jm ρ (R 2d ). Clearly |a j (w)| ≥ w δj away the origin. For each ϕ ∈ S(R d ), t ≥ 0 and j ∈ Z + , we have (a w ) j T (t)ϕ = T (t)(a w ) j ϕ ∈ L 2 (R d ). Because of [13, Theorem 2.1.16, p. 76], T (t)ϕ belongs to all Sobolev spaces H k Γ (R d ), k ∈ Z + , and thus T (t)ϕ ∈ S(R d ). Now, the closed graph theorem yields , in order to prove that t → T (t) is right continuous at t 0 it is enough to prove that for each k ∈ Z + , ε > 0 and bounded subset B of S(R d ), there exists η > 0 such that T (t)ϕ − T (t 0 )ϕ H k Γ ≤ ε, ∀t ∈ (t 0 , t 0 + η), ∀ϕ ∈ B. The a priori estimate in [13, Theorem 2.1.16, p. 76] yields that there exist C > 0 and j ∈ Z + such that Since T (t) → Id in L p (L 2 (R d ), L 2 (R d )) (by the Banach-Steinhaus theorem; {T (t)} t≥0 is a C 0 -semigroup) and B and (a w ) j (B) are precompact in S(R d ) and hence also in L 2 (R d ), we obtain that t → T (t) is right continuous at t 0 . Similarly, one proves that it is left continuous. The same a priori estimate proves that the set and, as H is equicontinuous, the Banach-Steinhaus theorem [19, Theorem 4.5, p. 85] gives the limit in the topology of precompact convergence. As S(R d ) is Montel, the limit holds in the strong topology. This immediately yields t → T (t) ∈ C ∞ ([0, ∞); L b (S(R d ), S(R d ))). Now one can obtain in the same way as above the validity of Lemma 6.11 and Corollary 6.12 in this case as well (of course, with S(R d ) and S ′ (R d ) in place of S * (R d ) and S ′ * (R d )).
Using the estimates for u(t, w) and u j (t, w) given in Remark 6.4, one readily obtains (6.24) and the asymptotic estimate (6.25) from Theorem 6.13 in the finite order case too.
7. The Weyl asymptotic formula for infinite order ΨDOs. Part II: proofs of the main results We now present the proofs of Theorems 5.1, 5.2, 5.4, and Corollary 5.3. In the sequel, we also use Vinogradov's notation for O-estimates, namely, g 1 (t) ≪ g 2 (t) as an alternative way of writing g 1 (t) = O(g 2 (t)).
For Theorems 5.1 and 5.2, and Corollary 5.3, we combine (5.2) and (5.7) into Let us verify that (7.1) implies that σ is a Karamata regular varying function [1] with index of regular variation 2d/β (= 0 if β = ∞), that is, that uniformly for α in compact subsets of (0, ∞). In fact, we have that for all λ (note that η(t) vanishes for t near 0). This easily yields (7.2). Similarly, the hypothesis (5.13) and the fact that σ is increasing imply that there are ν, C 1 > 0 such that In fact, we may take any ν > 0 such that 2d/ν < β ′ = lim inf y→∞ yf ′ (y)/f (y). For ν in this range, the inequality can be refined for large λ. Indeed, there is λ 0 = λ 0 (ν) such that The next starting point is the formula (6.25) from Theorem 6.13, which holds under all our three sets of hypotheses (see Remark 6.14 for the finite order case). As there are only finitely many possibly negative eigenvalues, we obtain (cf. (6.24)) Proof of Theorem 5.1. Let ε > 0 be arbitrary but fixed and set Using polar coordinates and the lower bound from (5.3), we have that where we have used the change of variables λ = f ((1 − ε)rΦ(ϑ)) which gives r 2d−1 dr = 1 2d Since σ is slowly varying (i.e. σ(αλ)/σ(λ) → 1 as λ → ∞), as follows from the Lebesgue dominated convergence theorem (the bound (7.3) holds here for every ν > 0 and C 1 depending only on ν). Thus, because σ(1/t) → ∞. But we can now take ε → 0 + to conclude lim sup Similarly, On the other hand, a small computation along the same lines as the above one shows that Inserting all this into (7.5), we conclude Notice that (5.5) is equivalent to (7.8). Finally, (5.6) follows from (5.5) and valid for every α ′ < α because of (5.2). This completes the proof of Theorem 5.1.
Proof of Theorem 5.2. Pick ε > 0 and find B so large that for all ϑ ∈ S 2d−1 and r > B. Note that Φ is continuous and thus Φ(ϑ) stays on a compact subset of (0, ∞). Using that (7.2) is valid uniformly for α on compact subsets of (0, ∞), we then obtain, Taking first t → 0 + and then ε → 0 + , we conclude that lim sup The estimate (7.7) remains valid in this case too. A similar analysis for the limit inferior, together with (7.5) and (7.7), leads to ∞ 0 e −tλ dN(λ) ∼ σ(1/t) We can apply once again the Karamata Tauberian theorem [1,11] to conclude that (5.9) holds.
Proof of Corollary 5.3. We only give the proof under the assumptions of Theorem 5.1, the proof of this corollary with the hypotheses from Theorem 5.2 is similar and the details are therefore left to the reader. By Theorem 5.1, we only need to show that where C ′ is given by (7.6). We show that lim sup λ→∞ 1 σ(λ) a(w)<λ dw ≤ C ′ ; one treats analogously the limit inferior to obtain the desired result and we thus omit the calculation. Fixing ε > 0, using the lower bound from ( The result now follows by taking ε → 0 + . Proof of Theorem 5.4. The lower bound (5.12) still applies to show (7.7). Combining this with the asymptotic estimate (7.5), we obtain (7.9) ∞ 0 e −tλ dN(λ) = 1 (2π) d R 2d e −ta(w) dw + o(σ(1/t)), t → 0 + .