Spectral asymptotics for infinite order pseudodifferential operators
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Abstract
We study spectral properties of a class of global infinite order pseudodifferential 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 powerlogtype but of logtype. 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.
Keywords
Weyl asymptotic formula Spectral asymptotics Infinite order pseudodifferential operators Hypoellipticity Heat parametrix UltradistributionsMathematics Subject Classification
35P20 35S05 46F05 47D031 Introduction
In this article we study the spectral properties of global infinite order pseudodifferential 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) \(\Psi \)DOs of infinite order. It is worth mentioning that the Weyl asymptotics for the operators that we investigate here are not of powerlogtype as in the finite order (distributional) setting, but of logtype, which in turn yields that the eigenvalues of infinite order \(\Psi \)DOs, with appropriate assumptions, are “very sparse”. As a byproduct of our analysis, we also obtain Weyl asymptotic formulae for a class of finite order Shubin \(\Psi \)DOs with some conditions on the symbols that are not the ones usually discussed in the literature.
The spaces of symbols and corresponding pseudodifferential 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 SGhyperbolic problems of finite order. The definition of these symbols classes is linked to two Gevrey type weight sequences \(A_p\) and \(M_p\), \(p\in \mathbb N\). The first one controls the smoothness, while the second one controls the growth at infinity of the symbols. These symbol classes are denoted by \(\Gamma ^{(M_p),\infty }_{A_p,\rho }\) and \(\Gamma ^{\{M_p\},\infty }_{A_p,\rho }\). The first one gives rise to operators acting continuously on Gelfand–Shilov spaces of Beurling type (i.e. of \((M_p)\)class) and the second one on Gelfand–Shilov spaces of Roumieu type (of \(\{M_p\}\)class); we will employ \(\Gamma ^{*,\infty }_{A_p,\rho }\) as a common notation for both cases. Since the symbols are allowed to grow subexponentially, i.e. ultrapolynomially, the corresponding \(\Psi \)DOs are of infinite order and they go beyond the classical Weyl–Hörmander calculus.
The article is organised as follows. Section 2 gives some basic background material about the Gelfand–Shilov type spaces \(\mathcal S^*(\mathbb R^d)\) and \(\mathcal S'^*(\mathbb R^d)\). We collect and explain in Sect. 3 some useful properties of the symbol classes \(\Gamma ^{*,\infty }_{A_p,\rho }\) and the corresponding global pseudodifferential operators. Further results related to the symbolic calculus that will be employed in the article are stated in the “Appendix” (Sect. 8).
Section 4 is devoted to establishing the semiboundedness of the Weyl quantisation \(a^{w}\) of a positive hypoelliptic infinite order symbol a. This will be achieved with the aid of results on antiWick quantisation from [16]. This result is interesting by itself because hypoellipticity in this setting allows the symbols to approach 0 subexponentially and thus generalises the familiar result for finite order operators. As a consequence, for hypoelliptic realvalued a such that \(a(w)\rightarrow \infty \) as \(w\rightarrow \infty \), one obtains that the closure \(\overline{A}\) of the unbounded operator A on \(L^2(\mathbb R^d)\) generated by \(a^w\) is selfadjoint and has a spectrum given by a sequence of eigenvalues \(\lambda _n,n\in \mathbb N,\) tending to \(\infty \) or \(\,\infty \), with eigenfunctions belonging to \(\mathcal S^*(\mathbb R^d)\) and forming an orthonormal basis for \(L^2(\mathbb R^d)\).
We state in Sect. 5 our main results concerning Weyl asymptotic formulae and we postpone their proofs to Sect. 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 \(N(\lambda )\) for infinite order symbols, which corresponds to f being of actual ultrapolynomial growth (and thus f increases faster than any power function at \(\infty \)). Even more, our method yields new interesting results for Shubin type \(\Psi \)DOs 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 Obound for \(N(\lambda )\) by requiring only knowledge on a lower bound for the symbol. We present there also some illustrative examples.
2 Preliminaries
For a regular compact set \(K\subseteq \mathbb R^d\) (i.e. \(K=\overline{\mathrm {int}\, K}\)) and \(h>0\), \(\mathcal E^{M_p,h}(K)\) is the Banach space (abbreviated as (B)space) of all \(\varphi \in C^{\infty }(\mathrm {int}\,K)\) whose derivatives extend to continuous functions on K and satisfy \(\sup _{\alpha \in \mathbb N^d}\sup _{x\in K}D^{\alpha }\varphi (x)/(h^{\alpha }M_{\alpha })<\infty \) and \(\mathcal D^{M_p,h}_K\) denotes its subspace of all smooth functions supported by K. For \(U\subseteq \mathbb R^d\), we define as locally convex spaces (abbreviated as l.c.s.) \(\mathcal E^{(M_p)}(U), \mathcal E^{\{M_p\}}(U),\mathcal D^{(M_p)}(U),\mathcal D^{\{M_p\}}(U)\) and their strong duals, the corresponding spaces of ultradistributions of Beurling and Roumieu type, cf. [8, 9, 10].
We denote by \(\mathfrak {R}\) the set of all positive sequences which monotonically increase to infinity. There is a natural order on \(\mathfrak {R}\) defined by \((r_p)\le (k_p)\) if \(r_p\le k_p\), \(\forall p\in \mathbb Z_+\), and with it \((\mathfrak {R},\le )\) becomes a directed set. For \((r_p)\in \mathfrak {R}\), consider the sequence \(N_0=1\), \(N_p=M_p\prod _{j=1}^{p}r_j\), \(p\in \mathbb Z_+\). 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 \(N_{r_p}(\rho )\), i.e. \(N_{r_{p}}(\rho )=\sup _{p\in \mathbb N} \ln _+ \rho ^{p}/(M_p\prod _{j=1}^{p}r_j)\), \(\rho > 0\). Note that for \((r_{p})\in \mathfrak {R}\) and \(k > 0 \) there is \(\rho _{0} > 0\) such that \(N_{r_{p}} (\rho ) \le M(k \rho )\), for \(\rho > \rho _{0}\).
A measurable function f on \(\mathbb R^d\) is said to have ultrapolynomial growth of class \((M_p)\) (resp. of class \(\{M_p\}\)) if \(\Vert e^{M(h\cdot )}f\Vert _{L^{\infty }(\mathbb R^d)}<\infty \) for some \(h>0\) (resp. for every \(h>0\)). We have the following equivalent description of continuous functions of ultrapolynomial growth of class \(\{M_p\}\).
Lemma 2.1
[17, Lemma 2.1] Let \(B\subseteq C(\mathbb R^d).\) The following conditions are equivalent : (i) For every \(h>0\) there exists \(C>0\) such that \(f(x)\le Ce^{M(hx)}\), for all \(x\in \mathbb R^d,\,f\in B;\,(ii)\) There exist \((r_p)\in \mathfrak {R}\) and \(C>0\) such that \(f(x)\le Ce^{N_{r_p}(x)},\) for all \(x\in \mathbb R^d,\,f\in B\).
Next, let E and F be l.c.s.; \(\mathcal {L}(E,F)\) stands for the space of continuous linear mappings from E to F; when \(E=F\), we write \(\mathcal {L}(E)\). We employ the notation \(\mathcal {L}_b(E,F)\) for the space \(\mathcal {L}(E,F)\) equipped with the topology of bounded convergence and, similarly, \(\mathcal {L}_p(E,F)\) and \(\mathcal {L}_{\sigma }(E,F)\) stand for \(\mathcal {L}(E,F)\) equipped with the topologies of precompact and simple convergence, respectively. Furthermore, \(E\hookrightarrow F\) means that E is continuously and densely included in F. For \((a,b)\subseteq \mathbb R\) and \(0\le k\le \infty \), \( C^{k}((a,b);E)\) stands for the vector space of k times continuously differentiable Evalued 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].
3 \(\Psi \)DOs of infinite order of Shubin type on \(\mathcal S^*(\mathbb R^d)\) and \(\mathcal S'^*(\mathbb R^d)\)
We discuss in this section properties of the classes of infinite order \(\Psi \)DOs that we shall consider in the article; see also the “Appendix” for other important facts about their symbolic calculus. We refer to [4, 18] and [17, Sections 3 and 4] for complete accounts.
3.1 Symbol classes and symbolic calculus
Let \(A_p\) and \(M_p\) be two weight sequences of positive numbers such that \(A_0=A_1=M_0=M_1=1\). We assume that \(M_p\) satisfies (M.1), (M.2) and (M.3), and that \(A_p\) satisfies (M.1), (M.2), \((M.3)'\) and (M.4). 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 \(A_p\subset M_p\). Let \(\rho _0=\inf \{\rho \in \mathbb R_+\,A_p\subset M_p^{\rho }\}\); clearly \(0<\rho _0\le 1\). Throughout the rest of the article, \(\rho \) is a fixed number satisfying \(\rho _0\le \rho \le 1\), if the infimum is reached, or, otherwise \(\rho _0< \rho \le 1\). Clearly, we may also assume that \(A_p\le c_0 L^pM_p^{\rho }\), where \(c_0\ge 1\) is the constant from (M.2).
Let \(\displaystyle FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})={\mathop {\mathop {\lim }\nolimits _{{\longrightarrow }}}\limits _{\qquad B\rightarrow \infty }} FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\), where the inductive limit is taken in an algebraic sense and the linking mappings are the canonical ones described above. Clearly, \( FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) is nontrivial.
If \(\sum _j a_j\in FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) and \(n\in \mathbb N\), \((\sum _j a_j)_n\) will just mean the function \(a_n\in C^{\infty }(Q_{Bm_n}^c)\), while \((\sum _j a_j)_{<n}\) denotes the function \(\sum _{j=0}^{n1} a_j\in C^{\infty }(Q_{Bm_{n1}}^c)\). Furthermore, \(\mathbf {1}\) denotes the element \(\sum _j a_j\in FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) given by \(a_0(x,\xi )=1\) and \(a_j(x,\xi )=0\), \(j\in \mathbb Z_+\).
3.2 Subordination
In the sequel, we will often use the notation \(w=(x,\xi )\in \mathbb R^{2d}\).
Proposition 3.1
We say that this \(U_{R}\) is canonically obtained from U by \(\{\chi _{n,R}\}_{n\in \mathbb N}\). Of course, here the mapping \(\Sigma {:}\,U\rightarrow U_R\) is just \(\sum _j a_j\mapsto R(\sum _j a_j)\).
Proposition 3.2
In what follows, we will frequently use the term “\(*\)regularising set” for a subset of \(\mathcal {L}(\mathcal S'^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\). Changing the quantisation and taking composition of \(\Psi \)DOs with symbols in \(\Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) always results in \(\Psi \)DOs 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.
3.3 Weyl quantisation. The sharp product in \( FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\)
We recall in this and the next subsection results from [17] about the Weyl quantisation of symbols; we often write \(a^w\) instead of \(\mathrm {Op}_{1/2}(a)\).
Remark 3.3
If \(\sum _j a_j,\sum _j b_j\in FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) and \(\sum _j c_j=\sum _j a_j\#\sum _jb_j\), then \(\sum _j\overline{c_j}=\sum _j\overline{b_j}\#\sum _j\overline{a_j}\). In particular, if \(a_j\) and \(b_j\) are realvalued for all \(j\in \mathbb N\) and \(\sum _ja_j\#\sum _jb_j=\sum _jb_j\#\sum _ja_j\), then \(c_j\) are realvalued for all \(j\in \mathbb N\).
Proposition 3.4
[17, Proposition 4.5] For each \(B\ge 0\), \( FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) is a ring with the pointwise addition and multiplication given by \(\#\). Moreover, the multiplication \(\#: FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\times FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\rightarrow FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) is hypocontinuous.
The multiplicative identity of \( FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};B)\) is given by \(\mathbf {1}\). The \(\#\)product of symbols corresponds to the composition of their Weyl quantisation (see the “Appendix”).
4 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 semiboundedness result. In preparation, we start by discussing \(L^2\)realisations of the associated unbounded operators.
Lemma 4.1
Given \(a\in \Gamma ^{*,\infty }_{A_p,\rho }(\mathbb R^{2d})\), let us denote by A the unbounded operator on \(L^2(\mathbb R^d)\) with domain \(\mathcal S^*(\mathbb R^d)\) defined as \(A \varphi =a^w\varphi \), \(\varphi \in \mathcal S^*(\mathbb R^d)\). Considering \(a^w\) as a mapping on \(\mathcal S'^*(\mathbb R^d),\) its restriction to the subspace \(\{g\in L^2(\mathbb R^d)\, a^wg\in L^2(\mathbb R^d)\}\) defines a closed extension of A which is called the maximal realisation of A. As standard, we denote by \(\overline{A}\) the closure of A, also called the minimal realisation of A. Notice that the formal adjoint \((a^w)^*\) is in fact the pseudodifferential operator \(\bar{a}^w\) and hence, it can be extended to a continuous operator on \(\mathcal S'^*(\mathbb R^d)\). One can also consider the adjoint \(A^*\) of A in \(L^2(\mathbb R^d)\). 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 \(\Psi \)DOs and we omit it (see for example [13, Proposition 4.2.1, p. 160]).
Proposition 4.2
Let \(a\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) with A and \(A^*\) defined as above. Then \(A^*\) coincides with the maximal realisation of \((a^w)^*\), i.e. the domain of \(A^*\) is \(D(A^*)=\{g\in L^2(\mathbb R^d)\, (a^w)^*g\in L^2(\mathbb R^d)\}\) and \(A^*g=(a^w)^*g\), \(\forall g\in D(A^*)\).
We now introduce the notion of hypoellipticity in \(\Gamma ^{*,\infty }_{A_p,\rho }\).
Definition 4.3
 (i)there exists \(B>0\) such that there are \(c,m>0\) (resp. for every \(m>0\) there is \(c>0\)) such that$$\begin{aligned} a(x,\xi )\ge c e^{M(mx)M(m\xi )},\quad (x,\xi )\in Q^c_B, \end{aligned}$$(4.2)
 (ii)there exists \(B>0\) such that for every \(h>0\) there is \(C>0\) (resp. there are \(h,C>0\)) such that$$\begin{aligned} \left D^{\alpha }_{\xi }D^{\beta }_x a(x,\xi )\right \le C\frac{h^{\alpha +\beta }a(x,\xi )A_{\alpha }A_{\beta }}{\langle (x,\xi )\rangle ^{\rho (\alpha +\beta )}},\,\, \alpha ,\beta \in \mathbb N^d,\, (x,\xi )\in Q^c_B. \end{aligned}$$(4.3)
Operators with hypoelliptic symbols have parametrices and hence are globally regular; see the “Appendix” for the precise results.
Proposition 4.4
[17, Proposition 5.4] Let a be hypoelliptic and A be the corresponding unbounded operator on \(L^2(\mathbb R^d)\) defined above. Then the minimal realisation \(\overline{A}\) coincides with the maximal realisation. Moreover, \(\overline{A}\) coincides with the restriction of \(a^w\) on the domain of \(\overline{A}\). If additionally a is realvalued, then \(\overline{A}\) is a selfadjoint operator on \(L^2(\mathbb R^d)\).
4.1 Semiboundedness 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 semibounded. 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.
Proposition 4.5
Let \(b\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) be positive hypoelliptic symbol. Then, there exists \(C>0\) such that \((b^w\varphi ,\varphi )\ge C\Vert \varphi \Vert ^2_{L^2(\mathbb R^d)}\), \(\forall \varphi \in \mathcal S^*(\mathbb R^d)\).
Proof
Using Propositions 4.4, 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 \(a\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) be a hypoelliptic realvalued symbol such that \(a(w)\rightarrow \infty \) as \(w\rightarrow \infty \) and let A be the unbounded operator on \(L^2(\mathbb R^d)\) defined by \(a^w\). Then the closure \(\overline{A}\) of A is a selfadjoint operator having spectrum given by a sequence of real eigenvalues either diverging to \(+\infty \) or to \(\infty \) according to the sign of a at infinity. The eigenvalues have finite multiplicities and the eigenfunctions belong to \(\mathcal S^*(\mathbb R^d)\). Moreover, \(L^2(\mathbb R^d)\) has an orthonormal basis consisting of eigenfunctions of \(\overline{A}\).
5 The Weyl asymptotic formula for infinite order \(\Psi \)DOs. Part I: statements of the main results
This section is dedicated to Weyl asymptotic formulae for a large class of infinite order hypoelliptic pseudodifferential operators. We state here our main results, their proofs are postponed to Sect. 7, after obtaining some auxiliary results on the spectrum of the heat parametrix of positive hypoelliptic symbols.
Theorem 5.1
Note that Theorem 5.1 deals with operators which are truly of infinite order because integration of (5.2) gives that \(\langle w \rangle ^{\beta }=o(a(w))\) for any \(\beta >0\).
The next theorem gives the Weyl asymptotic formula for a wider class of finite order pseudodifferential 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 \(N(\lambda )\).
Theorem 5.2
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.
Corollary 5.3
If one is only interested in upper Oestimates on N, the next theorem gives such bounds under much weaker assumptions on the symbol.
Theorem 5.4
Remark 5.5
If \(\limsup _{y\rightarrow \infty } yf'(y)/f(y)<\infty \), Theorem 5.4 is also valid for \(a\in \Gamma _{\rho }^m(\mathbb R^{2d})\) that is \(\Gamma _{\rho }^m\)hypoelliptic and satisfies (5.12), as the proof given in Section 7 shows. Here we get that \(\lambda _{j}\) is bounded from below by a constant multiple of \(f(j^{\frac{1}{2d}})\) for \(\lambda _j> 0\). In particular, this case applies to \(f(y)=y^{\beta '}\), where we obtain \(N(\lambda )=O(\lambda ^{2d/\beta '})\) and \(\lambda _{j}\ge h^{\beta '} j^{\beta '/(2d)}\), \(j\ge j_{h}\), 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”.
Example 5.6
Example 5.7
Example 5.8
6 The spectrum of the heat parametrix
Throughout this section we assume a is a hypoelliptic realvalued symbol in \(\Gamma ^{*,\infty }_{A_p,\rho }(\mathbb R^{2d})\) such that \(a(w)/\ln w\rightarrow \infty \) as \(w\rightarrow \infty \). There exists \(B\ge 1\) such that the hypoellipticity condition (4.3) for a holds on \(Q^c_B\) and \(a(w)>0\), \(\forall w\in Q^c_B\). Pick \(\tilde{\chi }\in \mathcal D^{(A_p)}(\mathbb R^{2d})\) [resp. \(\tilde{\chi }\in \mathcal D^{\{A_p\}}(\mathbb R^{2d})\)] such that \(0\le \tilde{\chi }\le 1\), \(\tilde{\chi }=1\) on \(Q_{B_1}\), for \(B_1>B\), and \(\tilde{\chi }=0\) on the complement of a small neighbourhood of \(\overline{Q_{B_1}}\). Then \(b=(1\tilde{\chi })a+\tilde{\chi }\) is positive on the whole \(\mathbb R^{2d}\) and, in fact, it is a hypoelliptic symbol in \(\Gamma ^{*,\infty }_{A_p,\rho }(\mathbb R^{2d})\) for which the hypoellipticity condition (4.3) holds globally on \(\mathbb R^{2d}\).
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 \(u_j(t,w)\in C^{\infty }(\mathbb R\times \mathbb R^{2d})\), \(j\in \mathbb N\), such that \(u_0(t,w)=e^{tb(w)}\) and the following results hold.
Lemma 6.1
Notice that for each \(R>0\), the function \(u(t,w)=\sum _{n=0}^{\infty } (1\chi _{n,R}(w))u_n(t,w)=R(\sum _j u_j)(t,w)\) is in \( C^{\infty }(\mathbb R\times \mathbb R^{2d})\).
Lemma 6.2
Theorem 6.3

T(t) belongs to \(\mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\);

the mapping \(t\mapsto T(t)\), \([0,\infty )\rightarrow \mathcal {L}_b(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\), is smooth;

T(t) and \((\mathbf {u}(t))^w\) are the same, modulo a smooth \(*\)regularising family.
Remark 6.4
6.2 The analysis of the semigroup T(t), \(t\ge 0\)
Lemma 6.5
The infinitesimal generator of \(\{T(t)\}_{t\ge 0}\) is \(\overline{A}\).
Proof
Let \(c>0\) be large enough such that \(\lambda _j>c+1\), \(j\in \mathbb N\), and \(\tilde{a}(w)=a(w)+c>0\), \(w\in \mathbb R^{2d}\). Then \(\tilde{a}\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) is hypoelliptic and we denote by \(\tilde{A}\) the corresponding unbounded operator on \(L^2(\mathbb R^d)\). Notice that \(\sigma (\overline{\tilde{A}})\subseteq \{\lambda \in \mathbb R\, \lambda > 1\}\) and \(\overline{\tilde{A}}\) is selfadjoint (see Proposition 4.4).
Proposition 6.6
Lemma 6.7
 (i)\(q_z:=R(\sum _j q^{(z)}_j)\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\), \(z\in \mathbf {P}\), and for every \(h>0\) there exists \(C>0\,(\)resp. there exist \(h,C>0)\) such that$$\begin{aligned} \left D^{\alpha }_w q_z(w)\right \le \frac{Ch^{\alpha }A_{\alpha }}{\tilde{a}_z(w)\langle w\rangle ^{\rho \alpha }},\,\, w\in \mathbb R^{2d},\,\alpha \in \mathbb N^{2d},\, z\in \mathbf {P}; \end{aligned}$$(6.15)
 (ii)
the set \(\{(1+z)q^w_z\, z\in \mathbf {P}\}\) is equicontinuous in both \(\mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\) and \(\mathcal {L}(\mathcal S'^*(\mathbb R^d),\mathcal S'^*(\mathbb R^d))\).
Proof
The estimate (6.10) implies \(\{\sum _j q^{(z)}_j\, z\in \mathbf {P}\}\precsim \{1/\tilde{a}_z\, z\in \mathbf {P}\}\) in \( FS _{A_p,\rho }^{*,\infty }(\mathbb R^{2d};0)\). Thus, we can apply Proposition 3.1 to obtain the existence of \(R'>0\) such that for each \(R\ge R'\), \(q_z:=R(\sum _j q^{(z)}_j)\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\) and (6.15) is valid when \(w\in Q^c_{Bm_1}=Q^c_B\), for some \(B=B(R)>0\). There exists \(j_0\in \mathbb Z_+\) such that \(q_z(w)=\sum _{n=0}^{j_0}(1\chi _{n,R}(w))q^{(z)}_n(w)\), for all \(w\in Q_B\), \(z\in \mathbf {P}\). Because of (6.10) we can conclude the validity of (6.15) when \(w\in Q_B\) as well, and the proof of (i) is complete.
Fix \(R\ge R'\) and consider \(q_z=R(\sum _j q^{(z)}_j)\), \(z\in \mathbf {P}\). As a direct consequence of (6.15) and (6.8) [resp. (6.9)], we have \(\{(1+z)q_z\, z\in \mathbf {P}\}\precsim e^{M(m\xi )}e^{M(mx)}\) (resp. \(\{(1+z)q_z\, z\in \mathbf {P}\}\precsim e^{N_{k_p}(\xi )}e^{N_{k_p}(x)}\)). Hence, Proposition 8.1 proves (ii). \(\square \)
Fix \(R>0\) for which the conclusions in Proposition 6.6 and Lemma 6.7 hold and denote \(q_z=R(\sum _jq^{(z)}_j)\in \Gamma _{A_p,\rho }^{*,\infty }(\mathbb R^{2d})\), \(z\in \mathbf {P}\). Since \(\sigma (\overline{\tilde{A}})\subseteq \{\lambda \in \mathbb R\, \lambda > 1\}\), it follows that \((z+\overline{\tilde{A}})\) is injective for each \(z\in \mathbf {P}\). Hence, the operator \(\tilde{a}_z^w{:}\,\mathcal S^*(\mathbb R^d)\rightarrow \mathcal S^*(\mathbb R^d)\) is injective, as well. Moreover, for given \(\varphi \in \mathcal S^*(\mathbb R^d)\), there exists \(g\in L^2(\mathbb R^d)\) such that \((z+\overline{\tilde{A}})g=\varphi \) (as \(z\in \rho (\overline{\tilde{A}})\)), i.e. \(\tilde{a}_z^wg=\varphi \). Since \(\tilde{a}_z\) is hypoelliptic, it is globally regular and hence \(g\in \mathcal S^*(\mathbb R^d)\). Thus \(\tilde{a}_{z}^w\) is a continuous bijection on \(\mathcal S^*(\mathbb R^d)\). As \(\mathcal S^{(M_p)}(\mathbb R^d)\) is an (F)space and \(\mathcal S^{\{M_p\}}(\mathbb R^d)\) is a \(({ DFS})\)space, it follows that \(\mathcal S^*(\mathbb R^d)\) is a Pták space (see [19, Sect. IV. 8, p. 162]). The Pták homomorphism theorem [19, Corollary 1, p. 164] implies that \(\tilde{a}_{z}^w\) is topological isomorphism on \(\mathcal S^*(\mathbb R^d)\), for each \(z\in \mathbf {P}\).
Lemma 6.8
The operators \((\tilde{a}_z^w)^{1}\), \(z\in \mathbf {P}\), are continuous on \(\mathcal S^*(\mathbb R^d)\) and they extend to continuous operators on \(\mathcal S'^*(\mathbb R^d)\). The set \(\{(1+z)(\tilde{a}_z^w)^{1}\,z\in \mathbf {P}\}\) is equicontinuous in \(\mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\) and in \(\mathcal {L}(\mathcal S'^*(\mathbb R^d),\mathcal S'^*(\mathbb R^d))\). Furthermore, for each \(z\in \mathbf {P}\), \((\tilde{a}_z^w)^{1}\) is exactly the restriction of \((z+\overline{\tilde{A}})^{1}\) to \(\mathcal S^*(\mathbb R^d)\).
Proposition 6.9
For each \(t\ge 0\), \(\tilde{T}(t)\in \mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\). Moreover, the mapping \(t\mapsto \tilde{T}(t)\) belongs to \( C^{\infty }([0,\infty );\mathcal {L}_b(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d)))\) and its derivatives are given by \((d^k/dt^k)\tilde{T}(t)=(1)^k(\tilde{a}^w)^k\tilde{T}(t)\), \(t\ge 0\), \(k\in \mathbb Z_+\).
Proof
In an analogous fashion one proves that for each \(t>0\), the mappings \(\varphi \mapsto I_2(t,\varphi )\) and \(\varphi \mapsto I_3(t,\varphi )\) belong to \(\mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\) and the mappings \(t\mapsto I_2(t,\cdot )\) and \(t\mapsto I_3(t,\cdot )\), \((0,\infty )\rightarrow \mathcal {L}_b(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\), are continuous.
As a direct consequence of the previous proposition we then have,
Theorem 6.10
We have \(T(t)\in \mathcal {L}(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d))\) for each \(t\ge 0\). Moreover, the mapping \(t\mapsto T(t)\) belongs to \( C^{\infty }([0,\infty );\mathcal {L}_b(\mathcal S^*(\mathbb R^d),\mathcal S^*(\mathbb R^d)))\) and one has \((d^k/dt^k) T(t)=(1)^k(a^w)^kT(t)\), \(t\ge 0\), \(k\in \mathbb Z_+\).
Lemma 6.11
The mapping \(t\mapsto \mathbf {Q}(t)\) belongs to \( C^{\infty }([0,\infty );\mathcal {L}_b(\mathcal S'^*(\mathbb R^d),\mathcal S^*(\mathbb R^d)))\).
Denoting the Weyl symbol of \(\mathbf {Q}(t)\) by Q(t, w), this lemma together with the property of symbols of operators in \(\mathcal {L}(\mathcal S'^*(\mathbb R^d),\mathcal S^*(\mathbb R^d)))\) (cf. [18, Propositions 2 and 3]) imply:
Corollary 6.12
The mapping \(t\mapsto Q(t,\cdot )\) belongs to \( C^{\infty }([0,\infty );\mathcal S^*(\mathbb R^{2d}))\).
Theorem 6.13
The next remark shows that (6.25) remains valid for hypoelliptic symbols of finite order.
Remark 6.14
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 \(\Psi \)DOs. 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 Oestimates, namely, \(g_{1}(t)\ll g_{2}(t)\) as an alternative way of writing \(g_{1}(t)=O(g_{2}(t))\).
We first make some comments that apply to all cases simultaneously. A preliminary observation is that \(f(y)/y^{\delta } \rightarrow \infty \) as \(y\rightarrow \infty \) for any \(0<\delta <\displaystyle {\mathop {\liminf }\nolimits _{y\rightarrow \infty }}yf'(y)/f(y)\) as follows by integrating (5.13) which holds in the three cases. It then follows from (5.3), (5.8), or (5.12) that \(a(w)/ w ^{\delta } \rightarrow \infty \) as \(w\rightarrow \infty \). Incidentally, this also implies that \(f'(y)>0\) a.e. on \([Y_1,\infty )\), for some large enough \(Y_1\ge Y\) and additionally \(f(y)>1\) on \([Y_1,\infty )\). Without loss of generality, we may assume \(Y_1=Y>1\). We conclude that \(\sigma \) is absolutely continuous on every compact interval contained in \([f(Y),\infty )\). We extend \(\sigma \) to [0, f(Y)] as a positive nondecreasing absolutely continuous function with \(\sigma (\lambda )=1\) near \(\lambda =0\). Note also that \(\sigma (\lambda )\rightarrow \infty \) as \(\lambda \rightarrow \infty \). We now derive some regular variation properties of \(\sigma \).
Proof of Theorem 5.1
Proof of Theorem 5.2
The classical argument quoted above in the proof of Theorem 5.1 easily gives \( \sigma (\lambda _{j})\sim j/C,\, j\rightarrow \infty ,\) with \(C= d^{1}(2\pi )^{d1}\pi \int _{\mathbb {S}^{2d1}}(\Phi (\vartheta ))^{2d/\beta } d\vartheta \). This immediately implies \((j/C)^{\frac{1}{2d}}\sim f^{1}(\lambda _j)\), as \(j\rightarrow \infty \). Note that (5.7) yields that f is regularly varying of index \(\beta \), i.e., \( f(\alpha \lambda )\sim \alpha ^{\beta }f(\lambda ),\) \(\lambda \rightarrow \infty ,\) uniformly for \(\alpha >0\) on compacts of \((0,\infty )\). Using this, \( \lambda _{j}=f((j/C)^{\frac{1}{2d}}(1+o(1)))\sim C^{\frac{\beta }{2d}}f(j^{\frac{1}{2d}}), \) which is (5.10). \(\square \)
Proof of Corollary 5.3
Proof of Theorem 5.4
Notes
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