EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS AND ULTRADISTRIBUTIONS

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1 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS AND ULTRADISTRIBUTIONS APARAJITA DASGUPTA AND MICHAEL RUZHANSKY Abstract. In this paper we give a goba characterisation of casses of utradifferentiabe functions and corresponding utradistributions on a compact manifod X. The characterisation is given in terms of the eigenfunction expansion of an eiptic operator on X. This extends the resut for anaytic functions on compact manifod by Seeey [See69], and the characterisation of Gevrey functions and Gevrey utradistributions on compact Lie groups and homogeneous spaces by the authors [DR14a]. 1. Introduction Let X be a compact anaytic manifod and et E be an anaytic, eiptic, positive differentia operator of order ν. Let {φ } and {λ } be respectivey the eigenfunctions and eigenvaues of E, i.e. Eφ = λ φ. Then acting by E on a smooth function f = f φ we see that it is anaytic if and ony if that there is a constant C > 0 such that for a k 0 λ 2k f 2 νk! 2 C 2k+2. Consequenty, Seeey has shown in [See69] that f = f φ is anaytic if and ony if the sequence {A λ1/ν f } is bounded for some A > 1. The aim of this paper is to extend Seeey s characterisation to casses more genera than that of anaytic functions. In particuar, the characterisation we obtain wi cover Gevrey spaces extending aso the characterisation that was obtained previousy by the authors in [DR14a] in the setting of compact Lie groups and compact homogeneous spaces. The characterisation wi be given in terms of the behaviour of coefficients of the series expansion of functions with respect to the eigenfunctions of an eiptic positive pseudo-differentia operator E on X, simiar to Seeey s resut in [See69], with a reated construction in [See65]. Interestingy, our approach aows one to define anaytic or Gevrey functions even if the manifod X is ony smooth. Goba characterisations as obtained in this paper have severa appications. For exampe, the Gevrey spaces appear naturay when deaing with weaky hyperboic probems, such as the wave equation for sums of squares of vector fieds satisfying Hörmander s condition, aso with the time-dependent propagation speed of ow reguarity. In the setting of compact Lie groups the goba characterisation of Gevrey Date: September 24, Mathematics Subect Cassification. Primary 46F05; Secondary 22E30. Key words and phrases. Gevrey spaces; utradistributions; Komatsu casses. The second author was supported by the EPSRC Grant EP/K039407/1 and by the Leverhume Research Grant RPG

2 2 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY spaces that has been obtained by the authors in [DR14a] has been further appied to the we-posedness of Cauchy probems associated to sums of squares of vector fieds in [GR14]. In this setting the Gevrey spaces appear aready in R n and come up naturay in energy inequaities. More generay, our argument wi give a characterisation of functions in Komatsu type casses resembing but more genera than those introduced by Komatsu in [Kom73, Kom77]. Consequenty, we give a characterisation of the corresponding dua spaces of utradistributions. We discuss Roumieu and Beuring type or inective and proective imit, respectivey spaces as we as their duas and -duas in the sense of Köthe [Köt69] which aso turn out to be perfect spaces. In the periodic setting or in the setting of functions on the torus different function spaces have been intensivey studied in terms of their Fourier coefficients. Thus, periodic Gevrey functions have been discussed in terms of their Fourier coefficients by Taguchi [Tag87] this was recenty extended to genera compact Lie groups and homogeneous spaces by the authors in [DR14a], see aso Decroix, Haser, Piipović and Vamorin [DHPV04] for the periodic setting. More genera periodic Komatsu-type casses have been considered by Gorbachuk [Gor82], with tensor product structure and nucearity properties anaysed by Petzsche [Pet79]. See aso Piipović and Prangoski [PP14] for the reation to their convoution properties. Reguarity properties of spaces of utradistributions have been studied by Piipović and Scarpaezos [PS01]. We can refer to a reativey recent book by Carmichae, Kamiński and Piipović [CKP07] for more detais on these and other properties of utradistributions on R n and reated references. The paper is organised as foows. In Definition 2.1 we introduce the cass Γ {Mk }X of utradifferentiabe functions on a manifod X, and in Theorem 2.3 we give severa aternative reformuations and further properties of these spaces. In Theorem 2.4 we characterise these casses in terms of eigenvaues of a positive eiptic pseudodifferentia operator E on X. Consequenty, we show that since the spaces of Gevrey functions γ s on X correspond to the choice M k = k! s, in Coroary 2.5 we obtain a goba characterisation for spaces γ s X. These resuts are proved in Section 3. Furthermore, in Theorem 2.7 we describe the eigenfunction expansions for the corresponding spaces of utradistributions. This is achieved by first characterising the -dua spaces in Section 4 and then reating the -duas to topoogica duas in Section 5. In Theorem 2.7 we describe the counterparts of the obtained resuts for the Beuring-type spaces of utradifferentiabe functions and utradistributions. We wi denote by C constants, taking different vaues sometimes even in the same formua. We aso denote N 0 := N {0}. The authors woud ike to thank Jens Wirth for a discussion. 2. Formuation of resuts on compact manifods Let X be a compact C manifod of dimension n with a fixed measure. We fix E to be a positive eiptic pseudo-differentia operator of an integer order ν N on X, and we write E Ψ ν +ex in this case. The eigenvaues of E form a sequence {λ }, and for each eigenvaue λ we denote the corresponding eigenspace by H. We may assume that λ s are ordered as 0 < λ 1 < λ 2 <. The space H L 2 X consists

3 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 3 of smooth functions due to the eipticity of E. We set d := dim H, H 0 := ker E, λ 0 := 0, d 0 := dim H 0. Since the operator E is eiptic, it is Fredhom, hence aso d 0 <. It can be shown see [DR14c, Proposition 2.3] that for a, and that d C1 + λ n ν 2.1 d 1 + λ q < if and ony if q > n ν. 2.2 =1 We denote by e k, 1 k d, an orthonorma basis in H. For f L 2 X, we denote its Fourier coefficients by f, k := f, e k L 2. We wi write f, 1. f := C d, 2.3. f, d for the whoe Fourier coefficient corresponding to H, and we can then write 1/2 d f HS = f f 1/2 = f, k 2. We note the Panchere formua k=1 f 2 L 2 X = f, k 2 = d =0 k=1 f 2 HS. 2.4 We refer to [DR14c], and for further deveopments in [DR14b] for a discussion of different properties of the associated Fourier anaysis. Here we note that the eipticity of E and the Panchere formua impy the foowing characterisation of smooth functions in terms of their Fourier coefficients: f C X N C N : f, k C N λ N for a 1, 1 k d. 2.5 If X and E are anaytic, the resut of Seeey [See69] can be reformuated we wi give a short argument for it in the proof of Coroary 2.5 as f is anaytic L > 0 C : f, k Ce Lλ1/ν for a 1, 1 k d. 2.6 The first aim of this note is to provide a characterisation simiar to 2.5 and 2.6 for casses of functions in between smooth and anaytic functions, namey, for Gevrey functions, and for their dua spaces of utradistributions. However, the proof works for more genera casses than those of Gevrey functions, namey, for casses of functions considered by Komatsu [Kom73], as we as their extensions described beow. =0

4 4 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY We now define an anaogue of these casses in our setting. Let {M k } be a sequence of positive numbers such that M.0 M 0 = 1 M.1 stabiity M k+1 AH k M k, k = 0, 1, 2,... For our characterisation of functiona spaces we wi be aso assuming condition M.2 M 2k AH 2k M 2 k, k = 0, 1, 2,... We note that conditions M.1+M.2 are the weaker version of the condition assumed by Komatsu, namey, the condition M k AH k min M qm k q, k = 0, 1, 2,..., 0qk which ensures the stabiity under the appication of utradifferentia operators. The assumptions M.0, M.1 and M.2 are weaker than those imposed by Komatsu in [Kom73, Kom77] who was aso assuming M.3 M k 1 k=1 M k <. Thus, in [Kom73, Kom77, Kom82] Komatsu investigated casses of utradifferentiabe functions on R n associated to the sequences {M k }, namey, the space of functions ψ C R n such that for every compact K R n there exist h > 0 and a constant C such that sup ψx Ch M. 2.7 x K Sometimes one aso assumes ogarithmic convexity, i.e. the condition ogarithmic convexity Mk 2 M k 1M k+1, k = 1, 2, 3,... This is usefu but not restrictive, namey, one can aways find an aternative coection of M k s defining the same cass but satisfying the ogarithmic convexity condition, see e.g. Rudin [Rud74, 19.6]. We aso refer to Rudin [Rud74] and to Komatsu [Kom73] for exampes of different casses satisfying 2.7. These incude anaytic and Gevrey functions, quasi-anaytic and non-quasi-anaytic functions characterised by the Denoy-Careman theorem, and many others. Given a space of utradifferentiabe functions satisfying 2.7 we can define a space of utradistributions as its dua. Then, among other things, in [Kom77] Komatsu showed that under the assumptions M.0, ogarithmic convexity, M.1 and M.3, f is an utradistribution supported in K R n if and ony if there exist L and C such that fξ C expmlξ, ξ R n, 2.8 and in addition for each ɛ > 0 there is a constant C ɛ such that fζ C ɛ exph K ζ + ɛ ζ, ζ C n, 2.9 where fζ = e iζx, fx is the Fourier-Lapace transform of f, Mr = sup k N og rk M k and H K ζ = sup x K Im x, ζ. There are other versions of these estimates given by, for exampe, Roumieu [Rou63] or Neymark [Ney69]. Moreover, by further strengthening assumptions M.1, M.2 and M.3 one can prove a version of these conditions without the term ɛ ζ in 2.9, see [Kom77, Theorem 1.1] for the precise formuation.

5 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 5 We now give an anaogue of Komatsu s definition on a compact C manifod X. Definition 2.1. The cass Γ {Mk }X is the space of C functions φ on X such that there exist h > 0 and C > 0 such that we have E k φ L 2 X Ch νk M νk, k = 0, 1, 2, We can make severa remarks concerning this definition. Remark Taking L 2 -norms in 2.10 is convenient for a number of reasons. But we can aready note that by embedding theorems and properties of the sequence {M k } this is equivaent to taking L -norms, or to evauating the corresponding action of a frame of vector fieds instead of the action of powers of a singe operator E on functions, see Theorem This is aso equivaent to casses of functions beonging to the corresponding function spaces in oca coordinate charts, see see Theorem 2.3, v. In order to ensure that we cover the cases of anaytic and Gevrey functions we wi be assuming that X and E are anaytic. 3 The advantage of Definition 2.1 is that we do not refer to oca coordinates to introduce the cass Γ {Mk }X. This aows for definition of anaogues of anaytic or Gevrey functions even if the manifod X is ony smooth. For exampe, by taking M k = k!, we obtain the cass Γ {k!} X of functions satisfying the condition E k φ L 2 X Ch νk M νk, k = 0, 1, 2, If X and E are anaytic, we wi show in Coroary 2.5 that this is precisey the cass of anaytic functions 1 on X. However, if X is ony smooth, the space of ocay anaytic functions does not make sense whie definition given by 2.11 sti does. We aso note that such space Γ {k!} X is sti meaningfu, for exampe it contains constants if E is a differentia operator, as we as the eigenfunctions of the operator E. We summarise properties of the space Γ {Mk }X as foows: Theorem 2.3. We have the foowing properties: i The space Γ {Mk }X is independent of the choice of an operator E Ψ ν ex, i.e. φ Γ {Mk }X if and ony if 2.10 hods for one and hence for a eiptic pseudo-differentia operators E Ψ ν ex. ii We have φ Γ {Mk }X if and ony if there exist constants h > 0 and C > 0 such that E k φ L X Ch νk M νk, k = 0, 1, 2, iii Let 1,, N be a frame of smooth vector fieds on X so that N =1 2 is eiptic. Then φ Γ {Mk }X if and ony if there exist h > 0 and C > 0 such that φ L X Ch M, 2.13 for a muti-indices, where = 1 1 = K. K K with 1 1,, K N and 1 An aternative argument coud be to use Theorem 2.4 for the characterisation of this cass in terms of the eigenvaues of E, and then use Seeey s resut [See69] showing that this is the anaytic cass.

6 6 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY iv We have φ Γ {Mk }X if and ony if there exist h > 0 and C > 0 such that φ L 2 X Ch M, 2.14 for a muti-indices as in iii. v Assume that X and E are anaytic and that there is a constant C > 0 such that M k Ck! for a k. Then the cass Γ {Mk }X is preserved by anaytic changes of variabes, and hence is we-defined on X. Moreover, in every oca coordinate chart, it consists of functions ocay beonging to the cass Γ {Mk }R n. The important exampe of the situation in Theorem 2.3, Part v, is that of Gevrey casses γ s X of utradifferentiabe functions, when we have γ s X = Γ {Mk }X with the constants M k = k! s for s 1. By Theorem 2.3, Part v, this is the space of Gevrey functions on X, i.e. functions which beong to the Gevrey casses γ s R n in a oca coordinate charts, i.e. such that there exist h > 0 and C > 0, such that ψ L R n Ch! s, for a ocaisations ψ of the function φ on X, and for a muti-indices. If s = 1, this is the space of anaytic functions. For the sequence {M k }, we define the associated function as Mr := sup og rνk, r > 0, k N M νk and we may set M0 := 0. We briefy estabish a simpe property of eigenvaues λ usefu in the seque, namey that for every q, L > 0 and δ > 0 there exists C > 0 such that we have λ q e δm Lλ 1/ν Indeed, from the definition of the function M it foows that λ q e δm Lλ 1/ν C uniformy in λ q M δ νp L νpδ λ pδ for every p N. In particuar, using this with p such that pδ = q + 1, we obtain λ q e δm Lλ 1/ν M δ νp L νq+1 λ C uniformy in 1, impying Now we characterise the cass Γ {Mk }X of utradifferentiabe functions in terms of eigenvaues of operator E. Uness stated expicity, we usuay assume that X and E are ony smooth i.e. not necessariy anaytic. Theorem 2.4. Assume conditions M.0, M.1, M.2. Then φ Γ {Mk }X if and ony if there exist constants C > 0, L > 0 such that where Mr = sup k og rνk M νk. φ HS C exp{ MLλ 1/ν } for a 1,,

7 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 7 For the Gevrey cass γ s X = Γ {k! s }X of Gevrey-Roumieu utradifferentiabe functions we have Mr r 1/s. Indeed, using the inequaity tn N! et we get [ ] r νk r 1/s Mr = sup og k νk! sup s [ og sup og e sr1/s] = sr 1/s. s! On the other hand, first we note the inequaity inf p N 2p2ps r 2p exp s 8e r1/s, see [DR14a, Formua 3.20] for the proof. Using the inequaity k + ν k+ν 4ν k k k for k ν, an anaogous proof yieds the inequaity inf p N νpνps r νp exp s 4νe r1/s. This and the inequaity p! p p impy expmr = sup k r νk νk! s = 1 inf k {r 1/s νk νk!} s 1 inf k {r 1/s νk νk νk } 1 s exp{ s s 4νe r1/s } = exp 4νe r1/s. Combining both inequaities we get s 4νe r1/s Mr sr 1/s Consequenty, we obtain Coroary 2.5. Let X and E be anaytic and et s 1. We have φ γ s X if and ony if there exist constants C > 0, L > 0 such that φ HS C exp Lλ 1 sν for a 0. In particuar, for s = 1, we recover the characterisation of anaytic functions in 2.6. We now turn to the eigenfunction expansions of the corresponding spaces of utradistributions. Definition 2.6. The space Γ {M k } X is the set of a inear forms u on Γ {M k }X such that for every ɛ > 0 there exists C ɛ > 0 such that hods for a φ Γ {Mk }X. uφ C ɛ sup ɛ M 1 ν sup x X E φx We can define the Fourier coefficients of such u by ] ûe k := ue k and û := ûe := [ue k d Theorem 2.7. Assume conditions M.0, M.1, M.2. We have u Γ {M k }X if and ony if for every L > 0 there exists K = K L > 0 such that û HS K exp MLλ 1/ν hods for a N. k=1.

8 8 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY The spaces of utradifferentiabe functions in Definition 2.1 can be viewed as the spaces of Roumieu type. With natura modifications the resuts remain true for spaces of Beuring type. We summarise them beow. We wi not give compete proofs but can refer to [DR14a] for detais of such modifications in the context of compact Lie groups. The cass Γ Mk X is the space of C functions φ on X such that for every h > 0 there exists C h > 0 such that we have E k φ L 2 X C h h νk M νk, k = 0, 1, 2, The counterpart of Theorem 2.3 hods for this cass as we, and we have Theorem 2.8. Assume conditions M.0, M.1, M.2. We have φ Γ Mk X if and ony if for every L > 0 there exists C L > 0 such that φ HS C L exp{ MLλ 1/ν } for a 1. For the dua space and for the -dua, the foowing statements are equivaent i v Γ M k X; ii v [Γ Mk X] ; iii there exists L > 0 such that we have exp MLλ 1/ν v HS < ; =1 iv there exist L > 0 and K > 0 such that v HS K exp MLλ 1/ν hods for a N. The proof of Theorem 2.8 is simiar to the proof of the corresponding resuts for the spaces Γ {Mk }X, and we omit the repetition. The ony difference is that we need to use the Köthe theory of sequence spaces at one point, but this can be done anaogous to [DR14a], so we may omit the detais. Finay, we note that given the characterisation of -duas, one can readiy proof that they are perfect, namey, that [Γ {Mk }X] = [Γ {Mk }X] and [ΓMk X] = [Γ Mk X], 2.18 see Definition 4.1 and condition 4.1 for their definition. Again, once we have, for exampe, Theorem 2.4 and Theorem 4.2, the proof of 2.18 is purey functiona anaytic and can be done amost identicay to that in [DR14a], therefore we wi omit it. 3. Proofs First we prove Theorem 2.3 carifying the definition of the cass Γ {Mk }X. In the proof as we as in further proofs the foowing estimate wi be usefu: e L X Cλ n 1 2ν for a This estimate foows, for exampe, from the oca Wey aw [Hör68, Theorem 5.1], see aso [DR14b, Lemma 8.5].

9 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 9 Proof of Theorem 2.3. i The statement foows from the observation that if E 1, E 2 Ψ ν ex, then there is a constant A > 0 such that E k 1 φ L 2 X A k E k 2 φ L 2 X 3.2 hods for a k N 0 and a φ C X. In turn, 3.2 foows from the fact that the pseudo-differentia operator E1 k E2 k Ψ 0 ex is bounded on L 2 X with E2 k denoting the parametrix for E2 k, and by the Caderon-Vaiancourt theorem its operator norm can be estimated by A k for some constant A depending ony on finitey many derivatives of symbos of E 1 and E 2. ii The equivaence between 2.12 and 2.10 foows by embedding theorems but we give a short argument for it in order to keep a more precise track of the appearing constants. First we note that 2.12 impies 2.10 with a uniform constant in view of the continuous embedding L X L 2 X. Conversey, suppose we have Let φ Γ {Mk }X. Then using 3.1 we can estimate φ L X = d φ, ke k L X =0 k=1 d φ, k e k L X =0 k=1 C φ0 HS + C d φ, k λ n 1 2ν =1 k=1 d C φ L 2 X + C φ, k λ 2 =1 k=1 1/2 d =0 k=1 λ n 1 ν 2 where we take arge enough so that the very ast sum converges, see 2.2. This impies 1/2 d φ L X C φ L 2 X + C φ, k λ 2 =1 k=1 1/2 C φ L 2 X + E φ L 2 X 3.3 by Panchere s formua. We note that 3.3 foows, in principe, aso from the oca Soboev embedding due to the eipticity of E, however the proof above provides us with a uniform constant. Using 3.3 and M.1 we can estimate E m φ L X C E m φ L 2 X + C E +m φ L 2 X Ch νm M νm + C h ν+m M ν+m C h ν+m hh ν+m 1 M ν+m 1 C h ν+m h ν H B +νm M νm C A νm M νm,

10 10 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY for some A independent of m, yieding iv We note that the proof as in ii aso shows the equivaence of 2.13 and Moreover, once we have condition 2.13, the statement v foows by using M k Ck! and the chain rue. iii Given properties ii and iv, we need to show that 2.10 or 2.12 are equivaent to 2.13 or to Using property i, we can take E = N =1 2. To prove that 2.13 impies 2.12 we use the mutinomia theorem 2, with the notation for muti-indices as in iii. With Y { 1,..., N }, 1, and ν = 2, we can estimate N 2 k φx C =1 =k C =k k! Y 2! 1... Y φx 2 k!! A2 M 2 CA 2k M 2k =k C 1 A 2k M 2k N k 2 k C 2 A 2k 1 M 2k, k!n! with A 1 = 2NA, impying Conversey, we argue that 2.10 impies We write = P E k with P = E k. Here and beow, in order to use precise cacuus of pseudo-differentia operators we may assume that we work on the space L 2 X\H 0. The argument simiar to that of i impies that there is a constant A > 0 such that P φ L 2 X A k φ L 2 X for a νk. Therefore, we get φ L 2 = P E k φ L 2 CA k E k φ L 2 C A k h νk M νk C A νk 1 M νk, where we have used the assumption 2.10, and with C and A 1 = A 1/ν h independent of k and. This competes the proof of iii and of the theorem. Proof of Theorem 2.4. Ony if part. Let φ Γ {Mk }X. By the Panchere formua 2.4 we have E m φ 2 L 2 X = Ê m φ 2 HS = λ m φ 2 HS = λ 2m φ 2 HS. 3.4 Now since E m φ L 2 X Ch νm M νm, from 3.4 we get which impies λ m φ HS Ch νm M νm φ HS Ch νm M νm λ m for a But we use it in the form adapted to the noncommutativity of vector fieds, namey, athough the coefficients are a equa to one in the noncommutative form, the mutinomia coefficient appears once we make a choice for = 1,..., N.

11 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 11 Now, from the definition of Mr it foows that Indeed, this identity foows by writing expmr = exp sup og rνk k M νk inf k N r νk M νk = exp Mr, r > = sup exp og rνk r νk = sup k M νk k M νk and using the identity inf k r k = 1. Setting r = λ1/ν, from 3.5 and 3.6 we sup k r k h can estimate { } h νm φ HS C inf M m 1 λ m νm = C inf m 1 r νm M νm where L = h 1. = C exp M λ 1/ν h If Part. Let φ C X be such that φ HS C exp M Lλ 1/ν hods for a 1. Then by Panchere s formua we have E m φ 2 L 2 X = λ 2m φ 2 HS =0 C =1 Now we observe that λ 2m exp MLλ 1/ν = C exp M Lλ 1/ν, λ 2m exp MLλ 1/ν exp MLλ 1/ν. 3.7 λ 2m sup p N L νp λ p M νp for any p N. Using this with p = 2m + 1, we get λ 2m exp MLλ 1/ν 1 L ν2m+1 λ 2m λ 2m+1 Then, by this and property M.1 of the sequence {M k }, we get λ 2m L νp λ p M νp 3.8 M ν2m+1. E m φ 2 1 λ 2m L 2 X C L ν2m+1 λ 2m+1 M ν2m+1 exp MLλ 1/ν =1 C 1 AH 2νm M 2νm λ 1 exp MLλ 1/ν, 3.9 for some A, H > 0. Now we note that for a 1 we have =1

12 12 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY λ 1 exp MLλ 1/ν λ 1 M νp λ p L νp =1 = M νp L νp 1 λ p+1 In particuar, in view of 2.2, for p such that p + 1 > n/ν we obtain λ 1 exp MLλ 1/ν M νp 1 L νp λ p+1 <. This, M.2 and 3.9 impy = and hence φ Γ {Mk }X. E m φ L 2 X Ã H νm M νm Now we wi check that Seeey s characterisation for anaytic functions in [See69] foows from our theorem. Proof of Coroary 2.5. The first part of the statement is a direct consequence of Theorem 2.4 and 2.16, so we ony have to prove 2.6. Let X be a compact manifod and E an anaytic, eiptic, positive differentia operator of order ν. Let {φ k } and {λ k } be respectivey the eigenfunctions and eigenvaues of E, i.e. Eφ k = λ k φ k. As mentioned in the introduction, Seeey showed in [See69] that a C function f = f φ is anaytic if and ony if there is a constant C > 0 such that for a k 0 we have λ 2k f 2 νk! 2 C 2k+2. By Panchere s formua this means that E k f 2 L 2 X = λ 2k f 2 νk! 2 C 2k+2. For the cass of anaytic functions we can take M k = k! in Definition 2.1, and then by Theorem 2.4 we concude that f is anaytic if and ony if or to f HS C exp Lλ 1/ν f C exp L λ 1/ν, with Mr = sup p og rνp νp! r in view of This impies 2.6 and hence aso Seeey s resut [See69] that f = f φ is anaytic if and ony if the sequence {A λ1/ν f } is bounded for some A > duas In this section we characterise the -dua of the space Γ {Mk }X. This wi be instrumenta in proving the characterisation for spaces of utradistributions in Theorem 2.7.

13 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 13 Definition 4.1. The -dua of the space Γ {Mk }X of utradifferentiabe functions, denoted by [Γ {Mk }X], is defined as { } d v = v N0 : v φ, <, v C d, for a φ Γ {Mk }X. =0 =1 We wi aso write v, = v and v HS = d =1 v, 2 1/2. It wi be usefu to have the definition of the second dua [Γ {Mk }X] as the space of w = w N0, w C d such that d w v < for a v [Γ {Mk }X]. 4.1 =0 =1 We have the foowing characterisations of the -duas. Theorem 4.2. Assume conditions M.0, M.1 and M.2. The foowing statements are equivaent. i v [Γ {Mk }X] ; ii for every L > 0 we have exp MLλ 1/ν v HS < ; =1 iii for every L > 0 there exists K = K L > 0 such that v HS K exp MLλ 1/ν hods for a N. Proof. i = ii. Let v [Γ {Mk }X], L > 0, and et φ C X be such that φ, = e MLλ1/ν. We caim that φ Γ {Mk }X. First, using 2.1, for some q we have φ HS = d 1/2 e MLλ1/ν Cλ q 1 e 2 M Lλ 1/ν e 1 2 M Lλ 1/ν 4.2 for a 1. Estimates 2.15 and 4.2 impy that λ q 1 e 2 M Lλ 1/ν C and hence φ HS C e 1 2 M Lλ 1/ν hods for a 1. The caim woud foow if we can show that e 1 2 M Lλ 1/ν e M L 2 λ 1/ν hods for L 2 = L, 4.3 AH where A and H are constants in the condition M.2. Now, substituting p = 2q, we note that e 1 2 M Lλ 1/ν Mνp 1/2 M 1/2 2νq = inf inf p N L νp/2 λ p/2 q N L νq λ q. 4.4

14 14 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY Using property M.2 we can estimate This and 4.4 impy where L 2 = M 2νq AH 2νk M 2 νq. e 1 2 M Lλ 1/ν M νq L νq 2 λ q, L AH. Taking infimum in q N, we obtain e 1 2 M Lλ 1/ν Therefore, we get the estimate inf q N φ HS C exp M νq L νq 2 λ q = e ML 2λ 1/ν. ML 2 λ 1/ν, which means that φ Γ {Mk }X by Theorem 2.4. Finay, this impies that e MLλ1/ν v HS is finite by property i, impying ii. = d =1 d =1 e MLλ1/ν v φ, v, < ii = i. Let φ Γ {Mk }X. Then by Theorem 2.4 there exists L > 0 such that φ HS C exp MLλ 1/ν. Then we can estimate d v φ, =0 =1 v HS φ HS =0 C =0 exp MLλ 1/ν v HS < is finite by the assumption ii. This impies v [Γ {Mk }X]. ii = iii. We know that for every L > 0 we have exp MLλ 1/ν v HS <. Consequenty, there exists K L such that exp MLλ 1/ν v HS K L hods for a, which impies iii. iii = ii. Let L > 0. Let us define L 2 as in 4.3. If v satisfies iii this means, in particuar, that there exists K = K L2 > 0 such that v HS K exp ML 2 λ 1/ν. 4.5

15 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 15 We aso note that by 4.4 we have exp 1 2 MLλ1/ν M νp 1/2 L νp/2 λ p/2 From this, 4.3 and 4.5 we concude exp MLλ 1/ν v HS =0 =0 =0 K for a p N. exp 1 2 MLλ1/ν exp 1 2 MLλ1/ν v HS exp 1 2 MLλ1/ν exp ML 2 λ 1/ν v HS =0 K + K exp 1 2 MLλ1/ν =1 K + C p =1 M 1/2 νp L νp/2 λ p/2 1 λ p/2 < is finite provided we take p arge enough in view of Utradistributions In this section we prove that the spaces of utradistributions and -duas coincide. Together with Theorem 4.2 this impies Theorem 2.7. Theorem 5.1. Assume conditions M.0, M.1 and M.2. We have v Γ {M k } X if and ony if v [Γ {Mk }X]. Proof. If Part. Let v [Γ {Mk }X]. For any φ Γ {Mk }X et us define d vφ := φ v = φ, v. =0 Given φ Γ {Mk }X, by Theorem 2.4 there exist C > 0 and L > 0 such that Consequenty, by Theorem 4.2, we get =0 =1 φ HS Ce MLλ1/ν. vφ φ HS v HS C e MLλ1/ν v HS <, which means that vφ is a we-defined inear functiona. Next we check that v is continuous. Suppose φ φ in Γ {Mk }X as, which means that sup ɛ M 1 ν sup x X E φ x φx 0 as.

16 16 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY It foows that E φ x φx L X C A M ν, where C 0 as. From the proof of Theorem 2.4 it foows that with C 0. Hence v φ φ as. This impies v Γ {M k } X. φ φ HS C e MLλ1/ν φ φ HS v HS C Ony if Part. Let v Γ {M k }X. By definition, this impies vφ C ɛ sup ɛ M 1 for a φ Γ {Mk }X. In particuar, ν sup x X ve C ɛ sup ɛ M 1 C ɛ sup ɛ M 1 C ɛ sup ν sup x X ν λ x X + n 1 ɛ 2ν λ. M ν E φx e MLλ1/ν v HS 0 E e x sup e x Here in the ast ine we used the estimate 3.1. Consequenty, we get ve C ɛλ n 1 2ν sup ɛ λ M ν C ɛ sup ɛ λ +k M ν 5.1 with k := [ n 1 2ν ] + 1. By property M.1 of the sequence {M k} we can estimate M ν +k AH ν +k 1 M ν +k 1. A 2 H 2ν +k 1 2 M ν +k 2 A νk H νk H fk M ν, for some fk = fν, k independent of. This impies This and 5.1 impy M 1 ν Aνk H fk H νk M 1 ν +k. ve C ɛ ɛ k A νk H fk sup ɛ +k H νk λ +k M ν +k C ɛ,k,a sup ɛ 1/ν H k ν +k λ +k M ν +k C ɛ,k,ae MLλ 1/ν, 5.2

17 EIGENFUNCTION EXPANSIONS OF ULTRADIFFERENTIABLE FUNCTIONS 17 with L = ɛ 1/ν H k. At the same time, it foows from 4.3 that e M Lλ 1/ν e 1 2 M L 3 λ 1/ν This and 2.1 for some q, and then 2.15 impy hods for L = L 3 AH. ve HS Cd 1/2 e MLλ1/ν Cλ q e 1 2 ML 3λ 1/ν Ce ML 3λ 1/ν, that is v [ Γ {Mk }X ] by Theorem 4.2. References [CKP07] R. D. Carmichae, A. Kamiński, and S. Piipović. Boundary vaues and convoution in utradistribution spaces, voume 1 of Series on Anaysis, Appications and Computation. Word Scientific Pubishing Co. Pte. Ltd., Hackensack, NJ, [DHPV04] A. Decroix, M. F. Haser, S. Piipović, and V. Vamorin. Embeddings of utradistributions and periodic hyperfunctions in Coombeau type agebras through sequence spaces. Math. Proc. Cambridge Phios. Soc., 1373: , [DR14a] A. Dasgupta and M. Ruzhansky. Gevrey functions and utradistributions on compact Lie groups and homogeneous spaces. Bu. Sci. Math., 1386: , [DR14b] J. Degado and M. Ruzhansky. Fourier mutipiers, symbos and nucearity on compact manifods. arxiv: , [DR14c] J. Degado and M. Ruzhansky. Schatten casses on compact manifods: kerne conditions. J. Funct. Ana., 2673: , [Gor82] V. I. Gorbachuk. Fourier series of periodic utradistributions. Ukrain. Mat. Zh., 342: , 267, [GR14] C. Garetto and M. Ruzhansky. Wave equation for sums of squares on compact ie groups. arxiv: , [Hör68] L. Hörmander. The spectra function of an eiptic operator. Acta Math., 121: , [Kom73] H. Komatsu. Utradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20:25 105, [Kom77] H. Komatsu. Utradistributions. II. The kerne theorem and utradistributions with support in a submanifod. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 243: , [Kom82] H. Komatsu. Utradistributions. III. Vector-vaued utradistributions and the theory of kernes. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 293: , [Köt69] G. Köthe. Topoogica vector spaces. I. Transated from the German by D. J. H. Garing. Die Grundehren der mathematischen Wissenschaften, Band 159. Springer-Verag New York Inc., New York, [Ney69] M. Neymark. On the Lapace transform of functionas on casses of infinitey differentiabe functions. Ark. Mat., 7: , [Pet79] H.-J. Petzsche. Die Nukearität der Utradistributionsräume und der Satz vom Kern. II. Manuscripta Math., 273: , [PP14] S. Piipović and B. Prangoski. On the convoution of Roumieu utradistributions through the ɛ tensor product. Monatsh. Math., 1731:83 105, [PS01] S. Piipović and D. Scarpaezos. Reguarity properties of distributions and utradistributions. Proc. Amer. Math. Soc., 12912: eectronic, [Rou63] C. Roumieu. Utra-distributions définies sur R n et sur certaines casses de variétés différentiabes. J. Anayse Math., 10: , 1962/1963. [Rud74] W. Rudin. Rea and compex anaysis. McGraw-Hi Book Co., New York-Düssedorf- Johannesburg, second edition, McGraw-Hi Series in Higher Mathematics. [See65] R. T. Seeey. Integro-differentia operators on vector bundes. Trans. Amer. Math. Soc., 117: , 1965.

18 18 APARAJITA DASGUPTA AND MICHAEL RUZHANSKY [See69] [Tag87] R. T. Seeey. Eigenfunction expansions of anaytic functions. Proc. Amer. Math. Soc., 21: , Y. Taguchi. Fourier coefficients of periodic functions of Gevrey casses and utradistributions. Yokohama Math. J., 351-2:51 60, Aparaita Dasgupta: Écoe poytechnique fédérae de Lausanne Facuté des Sciences CH-1015 Lausanne Switzerand E-mai address Michae Ruzhansky: Department of Mathematics Imperia Coege London 180 Queen s Gate, London SW7 2AZ United Kingdom E-mai address m.ruzhansky@imperia.ac.uk