RESONANCE CLASSES OF MEASURES

We extend F. Holland's definition of the space of resonant classes of func- tions, on the real line, to the space R(#pq) (I p, q ) of resonant classes of mea- sures, on locally compact abelian groups. We characterize this space in terms of trans- formable measures and establish a realatlonship between R(pq) and the set of positive definite functions for amalgam spaces. As a consequence we answer the conjecture posed by L. Argabright and J. Gil de Lamadrld in their work on Fourier analysis of unbounded measures.


INTRODUCT ION.
F. Holland [I] defined the space R(#q) (I q ) of resonance classes of func- tions, on the real llne, relative to the space of test functions #ooq, and proved that a function belongs to R(q) (2 _ q _ ) iff it is the Fourier transform of an un- bounded measure [I, Theorem 6].He also pointed out that the set P(C c) of positive definte functions in Cooper's sense [2] is included in R(I) [I, I], and proved that every function in R(ool has the same representation in terms of unbounded measures as the functions in P(C c) [I, Theorems 7 and 8], [3, Theorems 4.1 and 4.2] (in fact, as we will prove here, these representations hold for a larger class of functions and they are equivalent).These results of Holland together with Bochner's theorem on po- sltlve definite functions [4]   a function is the Fourier transform of a bounded measure iff it is a linear combination of positive definite functions lead one to speculate that any function in R(oo q) is a linear combination of positive definite functions.
In the present paper we respond to this conjecture in a more general setting.We define the space (#pq) (I p, q ) of resonance classes of measures (on locally compact abellan groups) relative to pq, which includes R(#=o q) as a particular case; we characterize this space in terms of transformable measures [5], and prove that for p , any measure in (pq) (2 q ) is a linear combination of positve de- finite functions for some amagam space (L r, s) [6], and for q < 2, any measure in (pq) can be approximated by linear combinations of positive definite functions for some amalgam (L r s) From these results we conclude that P(C is dense in R(I) and O(C )>, the c c space generated by the set of positive definite measures as defined in [5, 4], is dense in the space of transformable measures.This answers the conjecture posed in [5].
Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group F. For an element in F we write [x,] instead of (x) (x E G).Given two sets A and B we denote by A B the set {x ylx E A, y B}.
For a function f on G we use to denote its involution, i.e. (x) f(-x).The space of contlnuos functions which vanish at infinity, with compact support, will be respectively, We endow C with the inductive limit topology, as denoted by Co, C c, c in [5] By a measure (on G) we will mean an element of the continuous dual of C (G).
We let be the space of measures on G.
A funetlon f belongs to Loc (I =< q =< ) if f restricted to any compact sub- set of G, belongs to L q and f belongs to L q (I < q < ) if f has compact support and belongs to L q.
The amalgam spaces (L p q), (Co, q) (I < p, q < ) and the space of measures M (I < s < ) will be as defined in [7].We will make constant use of the following inclusions and inequalities proven in [7].
< q2 We will assume all the results of duality and convolution product for these spaces, the Holder and Young's inequalities, and the Hausdorff-Young theorem for amalgams as given in [8, I, 2].
The Fourier transform (inverse of the Fourier transform) of a measure on G (on F) will be denoted by ).We will denote by {ca} the approximate identity of the algebra LI(G) consisting of continuous functions with a fixed support and po- sitive Fourier transform in LI(F).
We let {$U U a compact neighbourhood of O} be the family of functions U in (1.10) The duality between a Banach space B and its Banach dual B' will be denoted by <f,F> F B', f B. As in [5] we call a measure on G transformable, if the linear space C 2(G), generated by the set {f*l f Cc(G)}, is included in LI(), and there exists a measure on F such that I f*(x)d(x)= |I12(-)d() for all f C We denote by /_ the space of transformable measures.We follow the definition of positive definite measures given by Dupuis in [6], but using the Segal algebra S0(G) which is equivalent to the space of translation bounded quasimeasures [9].The advantage is that for e S0(G)' its Fourier transform $ be- longs to S0(F)' [I0] and for f e L we have, as proven in [8,2], that We assume all definitions and results about the algebra S0(G) given in [8,2].
From these it is not difficult to see that the Fourier transform of a transformable measure (considered as an element of S0(G)' [10]) corresponds to the measure associated to .
As in [I0], an element o in S0(G)' is positive, o O, if f positive in S0(G) implies o(f) positive.In this sense a function g in S0(G)' is positive iff g(x) 0 almost everywhere.Indeed, let U be the measure gdm and suppose g in S0(G)' positive.For # e C positive, the function *e is a positive element c d of S0(G) and converges to in C [5].So we have that c () lim <*e,> lim <*e,g> O. Hence U is a positive measure and therefore g(x) 0 almost everywhere [II, Chp.III].
DEFINITION 2.1.Let E be a subset of functions of S0(G)'.A measure is a pos- <h,> 0 for all h e E such that O.
We write (E) to denote the set of positive definite measures for E, and P(E) to denote the set of measurable functions in (E).For a set E as in Definition 2.1 we denote by E + the set of functions in E whose Fourier transform is positive, and by <(E)> the linear space generated by (E).
Argabright and Gil de Lamadrid have studied the set (G), of measures U such in connection with the space of transformable mea- that <*$,> _> 0 for all e C c, sures.We use their results in [5] to prove that (G) is equal to (Cc).As in [6, Proposition II] we use the following le to prove eorem 2.5.LE 2.4.Let A be any of the amalgam spaces (L p, Eq), (Co, Es) (I < p, q < , < s < ).If f g A+, then there exists a net {f in A + such that lira f f in A.
n PROOF.Since {@U = C c, we have by [8, Theorem 1.6] that the net {fu*e} is in- cluded in Cc.Thus by (2.1) its Fourier transform (fu*e) (fu) (*U) positive [8, (2.5), (2 6], and since ea belongs to L I, the net {fu*ea} is in- is cluded in A Finally as in [6 Theorem III c)] lim f@u*e f in A (see also [8 Proposition 1.8]).
THEOREM 2.5.Let p, q < .Then P(L p Eq) is equal to (L p q PROOF.If is in (L p', Eq') and f is an element of (Lp, q)+, then there is a net {f as stated in the previous lena, so by [7, Theorem 3.2] we have n that <f,> lm <f D> > O. Therefore g P(L p, Eq).The other inclusion follows from [6, Proposition IV]. S 2.6.From (2.2) and eorem 2.5 we have that if is a positive measure in such that (L p Eq (I p, q < ), then 0 belongs to P(L p Eq) 3. RESONCE CLASSES OF ASUS.
Bertrandias and Dupuis [2] defined the space (I p, q ) of test func- Pq tions on locally compact abelian groups based on Holland's definition of the space (I q ) for the real line.DEFINITION 3.1.Let p, q ! .The space (G) consists of all func- tions in C (G) such that belongs to (C0, Eq)(F) endowed with the norm The space i' used by Bertrandias and Dupuis for their definition of the Fourier transfo is equal (as a set) to A [13] [9] Hence the space C2(G) is included in I(G) [14].We will use this in eorem 4.2.RE 3.2.i) As sets @ for !P, q, r , and C for pq rq pq c p , 2 q by the Hausdorff-Young theorem.
ii) A linear functional T on (G)  and by R(pq) the space of functions in (pq).By (I.I) it is clear that (pq) is included in (# if < s < q < H. Feichtinger has given a more general definition of resonance classes of func- tions relative to the space B' where B is a Banach space of functions containing S0(G) as a dense subspace (private communication).
To prove ii) take in (#pq) and set the map T() <,> on If T is Pq continuous then by Remark 3.2 there exists a measure 9 as stated in the theorem such that I (-)dg() / #(x)d(x).Since C2(G c #=of(G) we conclude that eT and .Conversely if #T and e M ,(r) then for e , e nl(r) [8   q   Theorem 1.4].Hence by [5 Corollary 3ol] and Youngs inequality we have that Therefore I ( ).The proof for II (x)d(x)l I $(-)d()l _< IIllq, ll$11q q p finite is the same.
Part iii) follows from the Hausdorff-Young theorem and the fact that the spaces (L p q and M (2 < q < ) are included in T [15,Remark 6 25] e conclude from Theorem 3.4 [5, Theorem 2.5] that R(pq) (Opq) for p 2 =< q =< ' R (I) LIIoc nT and (*i =T. The following corollary is easily deduced from the previous theorem and the Hausdorff-Young theorem.COROLLARY 3.5.Let !p ! , 2 !q ! .A function, f belongs to R(pq) iff there exists a unique e M ,(F) if p , (L p q )(F) if p < , such that v q f =.
Since q Cc if q => 2, Corollary 3.5 includes the results of Eberlein [16, rem I] (with p q ) and Stewart [7, Theorem 4.4] (with p o) as special cases.
For the remaining cases that is, for 2 <_ p _< =o, 2 < q _< oo; <_ p < 2, 2 _< q _< and q 2, 2 < p < , the inclusions in Proposition 3.6 are proper because the Fourier transforms on (L r, s) (1 < r < 2, =< s =< 2), on Ms (1 =< s =< 2), and on (L 2, s) (1 _< s < 2) are not onto [17,Corollary 6.3].Indeed if 2 _< p _< and 2 < q < then there exists f e (L q, P) such that f + for ali h e (L p', q'), hence f (L p' Eq;) [21, Remark 2.41.So the function g defined by <,,g> l<o,f>l on (L q EP belongs to (L q, P) and clearly to P(L q P ).But g (L p Eq ), otherwise f would be in (L p Eq ).Therefore g R(pq).The remaining cases are s imi lar 4. THE SPACE ( FOR < q < 2. pq We have seen that any measure in (pq) (2 < q < oo) is a linear combination of positive definite functions; we want to prove now that for < q < 2, any measure in ( is approximated by linear combinations of positive definite functions.
PROOF.By Proposition 4.1 we only have to prove that for IIe (1 ), the net {*U belongs to <P(L P')> By Theorem 3.4, the measure is a linear combination of positive measures llj Therefore g ( ). Pq The construction of the function f can be extended to G using the partition of disjoint relatively compact subsets as in [7,3].Probably the same is true for < q < p < 2, but we were unable to decide the matter.
We will prove in this section (Theorem 5.4) that the representation theorems for hold for the space L loc R(pq) (I =< q < 2, =< p _< oo), and they are equlvalent.We first give a remark easily deduce from [5, Theorem 3.3].
REMARK 5.1.Let f e Lq+/-oc R(pq g LI(G), then *f exists and e LI().Therefore for locally almost all x e G we have that f*(x) ii) If the integral on the left is a continuous function of x in a nelghbourhood of 0, then the formula in i) is valid for x 0. Hence under this hypothesis [ f(y)(-y)dy [ ()d().

G r
The next theorem includes [I, Theorem 3] as a particular case.
THEOREM 5.2.Let =< p, q =< .If f L'qloc R( for all e L q' such that $ e (L p q) if p < $ (Co q) if p c PROOF.It is clear that the convolution *f exists for L q' and f e L, q c oc If $ is in either (L p, q) or (C0, q), then e L I().So by our previous remark we have to prove that f* is continuous on a nelghbourhood U of 0. Let E be the support of , and s g U.If < q < 0o, then the map x x where x(y) (x y), is continuous on G.So given e > 0 there exists a nelghbourhood V of 0 such that for all x e V we have that II, x *yl lq, fXu_ E q XU_ E is the characteristic function of U E. So for x e U 8 V we have that If(y) lx(y) s(y) dy < IIfXu_EIIqlIx yllq, < e.
[f*(x) f*(s)[ <_ ]U-E Therefore f* is continuous at s. fXu E is continuous on G, and as before, If q I, then the map x x there exists a neighbourhood of zero V such that for x in U N V we have that f*(x) f*(s) < II lll(fU_E) x (fU_E)sl 11 < .This ends the proof.
We need now to introduce Simon's generalization of Castro summabillty on locally compact abelian groups [22].This consists of a family of functions {U (U being a compact neighbourhood of O) in (C0, I) with the following properties u-->' II%llx__< {U is an approximate identity for L U Cc and lira U() for all F.  (L p q)(F) if p < h e (Co q)(F) if p Furthermore if r 2q/(2q I) and 2 < p < oo, then for alI q, q in (Lr, P') The double integrai exists not necessarily as a Lebesgue in- tegral but as the sum of the convergent series where Va, V 8 are finite union of the sets L, as definedv in [7,3].
If g (Lr, P') then Jig[ [rp' [Iga[ lp')r lip' where ga gXLo" So for #, in (e r, zP') we have that [B(, ) is abso- lutely convergent and the left side of (5.4) exists as stated in the theorem.
Finally, since I ga converges in the norm of (Lr, P') to g we have that [ .B(a,8) f $()() d().The proof for 2 < p < is similar.
THEOREM 5.4.If =< q < 2, =< p < and f e Lloc,q then the following are equi- valent i) f e R(pq) ii) There exists a unique U in M ,(F) if p m, in (L p q )(F) if p < o, such q that for all , in (L r P')(G) where r 2q/(2q-I) the double integral exists as in Theorem 5.3.
iii) There exists a unique measure as in part ii) such that f(x) li, qu(.)[x,] du(.)F where the limit exists as in Remark 4.2.
PROOF.By Theorem 5.3 part i) implies part ii) and by Propsttton 4.1 part tit) t- piles part ).

L 2
Suppose that ii) holds.Since U 8U*SU and 8 U e c for all t in G we have The left side is equal to f(x)8U*U(tx)dx f(X),u(t-x)dx f**u(t), the right side is equal to I U()[t,] du(), and u*f converges to f in the sense of part iii).Hence we conclude that ii) implies iii). 6. RTHER RESETS.
In this last section we want to give a characterization of the set of Fourier mul- tipliers from the space #q (I q ) to L and M I.This will allow us to extend Dupuis' characterization theorem [6,Theorem IIl].
Folling the notation in [23] we denote by M() ((q)) the space of Fourier multipliers from to L (MI). that is M(#) ((#q)) is the space of all func- tions (measures) f on F such that f is in L1 (M I) for all q(G) Since q Cc for 2 =< q =< , we consider the characterization of these spaces for !q 2. EOREM 6.1.Let q 2. Then S(#,) (L q') and (q)__ Mq, PROOF.By the Young's inequality (n I, q') c M(q) and M_,c(# ).
Let {(q) and suppose that lira n in # and lira n h in MI, so for f e C we have that C l<f,$ h'[ l<f,$ n $'1 + [<f'$n < c Ilfllll$ n $ IIq where C is a constant depending on the support of f.Since C is dense in C, we con- C clude by the Closed Graph theorem that the map from #q to M is con- tinuous.Hence by Remark 3.2, e Mq,(F).If f M(q),__ then the measure If[din be- longs to M (F) and therefore f e (L I. q')(F). q' From Theorem 6.1 we see that M(Cc (L 2) as proven in [23] and (C M 2 We write F*q to denote the set EOREH 6.2.Let q 2. Then P(Fq) (L Hence P(Fq) P(C0, q) and (mq) (C0, q).PROOF.By [6.Proposition IV] and eorem 6.1, P(Fmq) (L I, q) and O(Fq) c Mq, O P. Take g M N and $ g (Fq) +.Since q $*e_ g A and ('e_) v > 0 we have that $, lim $*ea, > O. erefore g (F_).Since (L E q is included in M [8, (1.9)] we conclude that -' q q' (L E = P(Fq).The last equality follow from Remark 2.3 and an argument like that of Theorem 2.5.

(
on any compact subset of g {U is an approximate identity of L

PROPOSITION 2 . 2 .
A measure U belongs to (C C) iff <*,U> 0 for all in C c PROOF.The inclusion (C c (G) is clear.Take (G).Since C is inc- c + c luded in T [5, Theorem 2.2] we have that a function in Cc is a continuous po- sitive definite function [5, Theorem 4.1], so <,> 0 by [5, Corollary 4.2] and therefore satisfies condition (D2).M.T. DE SQUIRE REMARK 2.3.It is clear that if El c E2 (El, E2 as in Definition 2.1), then P(EI) c P(E2) so by [5, Theorem 4.1] if C is a subset of E then (E) C Dupuis defined the set of positive definite quasimeasures to be the set of all quasimeasures such that <,o> > 0 for all in A +, and characterized it as the C set {[ M.} [6, Proposition II].
extends Holland's definition of the space of resonance classes of functions[I, 5].DEFINITION  3.3.Let p, q .A measure on G is resonant relative to (resonance classes of measures relative to Pq Pq

U
The following representation theorem is an extension of [I, Theorem 7] (c.f.[3,