Rend. Sem. Mat. Univ. Poi. Torino Voi. 51, 2 (1993)

JJ . Betancor - I. Marrero

MULTIPLIERS AND CONVOLUTION OPERATORS ON HANKEL-TRANSFORMABLE HILBERT SPACES

Abstract. Multipliers and Hankel convoli]don operators on a chain of Hilbert spaces

where the Hankel transformation is an automorphism are investigateci.

1. Introduction

Let /x e R. The space H^, introduced by A.H. Zemanian [14], consists of ali those infinitely differentiable functions <j> = <l>(x) defined on I =]0,cx)[ such that the quantities

7S;,*W = 8up|*m(x-'1i>)*»-V1/^(x)| (m,*€-N) xei

are finite. When endowed with the topology generated by the family of seminorms { m fc'}m fceN' ^A* becomes a Fréehet space. As customary, H'^ will denote the dual space of H^' A\so in [14], Zemanian defined the space O of ali those 0 e C°°(I) such that for every k e N there exists rif- e N for which (1 -f- x2)~nh(x~lD)k6(x) is bounded on / . In [2], we characterized O as the space of multipliers of H^ and of H'^ and topologized this space by means of the system of seminorms {*y£k : 4> G H^, k e N}, given by

i^m^supix-^ixXx-'D^e^

It is proved in [14] that the Hankel integrai transformation

/•OO

(h^)(y) = / <j>{x){xy)1^J^xy)dx (y G I), Jo

where JM represents the Bessel function of the first kind order ji, is an automorphism of H^, provided that ii> — \. The generalized Hankel transform h'T of T e H'^ is defined by

Then h'^ is an automorphism of H'^ when this space is endowed with either its weak* or its strong topology. The Hankel convolution operators on Hy, and on H' have been investigated by the authors [10]. The space of those operators turns out to be 0^# = /i^(a;M+1/20).

120 J.J. Betancor - I. Marrero

In a previous paper [3], we considered a chain {H^}rez of Hilbert spaces such that h^ (respectively, hjj is an automorphism of Hr^ (respectively, H~r) for ali r e N. Let L2(I) denote the usuai Lebesgue space of square integrable functions on J. If r e N and fi > — , we say that a function ^ e L2(I) lies in HJ, when xmTIAìk4> e L2(J) (m, fc € N,0 < m + fc < r), where T (fc = Np+k-i •. • A^, N/* = x^l/2Dx~^~1/2, and the derivatives are understood in a distributional sense. This space is topologized by the norm

ll*IUr = { E / l*mWO<OI2<M (4>eKp, m+k=0J°

induced by the inner product

m+fc=0''0

which makes of HJ"X a Hilbert space. The dual space of H£ is H~r. From the inequality

(i) ll*ILr<IWL. (*e«J). valid for ali r, s G N with r < s, the chain of inclusions

... D H~ D 7i~ OH^^L (I) D H^ D Tip ...

follows. Moreover, the space B^ (see [15]), hence H^ is dense in H^. We also established the identities H^ = nrGj^7iJ, H^ = UrG^H~r, and the fact that the topology of H» (respectively, the strong topology of H^) is the initial topology (respectively, the final locally convex topology) associated to the family of inclusions {H^ «—• H^}rej^ (respectively,

in~r <-» K)reN). In this work we investigate the space Ov,s of multipliers from H1^ into H^ (Section

2) and the space Ofs of Hankel convolution operators from H~s into H~r (Section 3), with r, s e N. Our study is motivated by a series of papers by J. Kucera ([5], [6], [7], [8]) and J. Kucera and K. McKennon [9]. We prove that OryS (r, s e N) is a Banach space, and describe it by using L2-norms. The union Os = UrGj^Or?s becomes the space of multipliers from H~s

into H^ (s e N). It is shown that O = (\e^j Ur£j^ OrtS holds both in the algebraic and in the topological senses. Since the spaces OriS and Ofa (r, s e N) are related through the /i -transformation, a corresponding identity O' # = flsGN UreN ®% c a n b e established.

In what follows /J, will denote a real number not less that -^, while C will represent an adequate strictly positive Constant (not necessarily the same in each occurrence). The space V{ I) will be understood in the usuai way.

2. Multipliers in H^

Given r, s e N, we denote by OriS = 0^s the space of multipliers from H^ into H*, that is, the linear space of ali those complex-valued functions 0 = 9{x) defined on / such that

Multipliers and convolution operators 121

0(j> e H^ whenever <\> e H^. The next Proposition establishes that 0TìS is actually the space of continuous multipliers from Hr^ into H^.

PROPOSITION 1. Let r, s e N. I/O e OriS then the mapping <j? •—• 9$ is continuous from Hr^ into Ha

h.

Proof. We shall make use of the Closed Graph Theorem. Let {<Au}nGN be a sequence in H* such that the limits

H1

lim <j)n = (j) in Hru, lim 0<j>n = tp in Hs

u

exist. We must prove that #<?!> = V'- Now, as lim <j>n = (f> in W£ implies lim (j>n = (f> in

L2(I), there exists a subsequence {^i,n}nG^ such that

lim fa n(x) — <t>(x) a.e. x e I. r i—•oo '

Consequently,

lim 9{x)<f)1 n(x) = 9(x)<fr(x) a.e. # e . / . n—+00 '

On the other hand, since lim 9<f>\n = ty in H*, also lim 9(\>\n = *l> in £ 2 ( A and n—+00 ' ^ n — • < » '

there exists a subsequence {</>2,n}neN of Wi,n}n€N s u c n m a t

lim 0(x)(f)2n(x) = i/j(x) a.e. x e l .

We conclude that 6(x)(j>{x) = ìp(x) a.e. a; e / .

The space Or,s is topologized by the norm

\mks = \m\^= suP 11 11 (0ea, s). II0II^<1

It follows from Proposition 1 that the bilinear mapping

(0,cj>) v-*9<j>

is continuous. Dually, if 0 e OriS and T e H~s, we define 9T eH~r by the formula

(9TA) = (TM) ( * € H p .

The continuity of the bilinear mapping

is then straightforward.

PROPOSITION 2. (9rjS is a Banach space (r, 5 € N).

122 J.J. Betancor - I. Marnerò

Proof. Fix r, s G N. We only need to show that Gr,s is complete. To this end,

choose a Cauchy sequence {#n}nGN in ^r, s , and take </> e H1^. Since {#n0}nGN i s Cauchy

in H* and this space is complete, there exists QA, G Hs.. such that lim \\0n<j) - 0 J L s = 0. In

particular, {0n(f>}n^ converges to 0$ in L2(I), and hence 0$ is the limit of an a.e. convergent

subsequence of {0n<£}nGN-Now, deflne

. . . ( kx — \ r (a; € / , k € N, fc > 2) ,

where u>fc G £>(/) is chosen so that u>k(x) = 1 [x e

1 1

1 1_" 2 ' 3

x < a/t G Ir? A:2 + 2 ' 2 L

and o;fc(a;) = 0

for A: G N, k > 2. Its is clear that <£fc G ft£, with ^fc(x) = 1

, A; G N, k > 2 ]. Through a diagonal procedure involving the functions ^ , which

we are about to describe, we shall find a subsequence of {#n}nGN, a.e. convergent on I.

Set 0i,n(aO = M ^ ) (n G N). There exists a subsequence {^2,n}nG^ of {0i,n}n€N such that lim 02,n(x)<l>2(x) = 9<t>2(

x) a-e- x e I. Hence, lim 02,n(x) = 0<j>2(x) for x e Bi,

where B<± e ,2 and v \ ^ 0. Next, choose a subsequence {03,n}neN of

{^ .n l^N s u c n ^at l i m ^3,nW^(^) = 0ó3(x) a.e. x G 7 and lim 03 Jx) = ^ 3 ( x )

a.e. x G

of

! •

I I 3 '2

U 2,3

It is clear that 9<t>A{x) = O^x) (x G B2). We denote by £3 the subset,

where {03,n}nGp*j converges. By iterating this process we may construct

1 1 U j-hj , with \Ij\Bj\ = 0 a sequence {Bj}j>2 of sets such that Bj e Ij = ., .

(j G N,j > 2); and a sequence {0j,n},->! nGN 0 f functions satisfying

0) 0i,„ - #n (n G N),

(ii) {6,j+i,n}neN i s a subsequence of {0j,n}nGN (j G N, j > 1), and

(iii) lim 0jyn(x) = 9+ (x) (x G Bj9 j eN,j> 2). n—>oo J

Let us introduce the function 0(x) = ^ ( x ) , (rr G £7, j G N, .7 > 2). As | I \ J B | == 0, where B = U^-^j» 00*0 i s defined a.e. x e I, and may be defìned everywhere on I by assigning to it the value zero (for instance) on I\B. We claim that the diagonal sequence {9j,j}j>i converges to 6(x) a.e. x e I. In fact, if x e B then there exists jo G N such that x G Bj0 implies lim 0jOtn(x) — 0(x). Hence, given e > 0, an no G N can be found so that \0jo,n(x)—0(x)\ < e (n > no). From (ii) we infer that \0jj(x)-9(x)\ < e (j > max{jo, ^0}). In other words, {OJJ}J>I is the a.e. convergent subsequence of {^n}nG^ whose existence we had claimed.

Multipliers and convolution operators 123

Moreover, if (j> e Hra then lim 0n,<£ = 0$ in L2(I). It follows that 0Jx) = 0(x)<f>(x)

a.e. a: G 7, so that 0(f> G H8,, whence 0 e Or,a. Finally, a standard argument allows us to conclude that lim 0n = 0 in 0r s.

The following can be easily derived from (1).

PROPOSITION 3. Lei p, q, r, s e N. There holds

\\O\\qir>\\0\\q,Sì \\0\\qir>\\9\\ptr(OeOqirìr>Sì p > q) .

Hence

< V C C\ s , C \ r C Op,r (r > s, p > q) ,

wM continuous embedding.

The next result will be useful in the sequel.

LEMMA 1. Ifne N then there exist r e N and C > 0 JMC/I r iar

max ||(1 + *2)n(a:-1i))*f l:-'4-1/VWIIoo<C7|HUr ( * € ? £ ) .

Proo/ Let r G N and <£ G H£. Then [3, proof of Theorem 2.2]

(-l)ro!/mTM|fc(V^)(s/) = (-l)fch^+m+k(xkT^m<l>(x)){y) (m, fe e N, 0 < m + fe < r).

If n, A; e N, with 3n < r and 0 < k < n, then

= È f ^*"*^"1/2C-i)*^ik+»(y%.»M)'(y))W-t = 0 W

Hence, for n, A;, r G N, with 0 < k < n and r > 4ra+2v, where t; G N is such that 2v > / / + 1 , by applying well-known properties of the Bessel J^-function [13, pp. 46-48 and p. 199] we find that

sup|(l + a;2)n(ar-1P)fca:-'i-1/2^(a;)r

* E (U) SUP /°0(^)"/i"fc^+^2i(^)2//i+1/2+2fcTAi,2i(/i^)(2/)d2/ g j W mei Jo

n 'n\ f°° y»+1/2

< C max sup|(l + i À " + % j ( M ) ( v ) | • v<3<2nyei

dy

124 J.J. Betancor - I. Marrero

According to [3, Lemma 2.14 and Theorem 2.2],

max sup |(1 + y2)"+kT^(h^)(y)\ < C\\h^\l,r = C\W\„,r .

This completes the proof.

PROPOSITION 4. Let s eN, and let 9 — 9(x) be a complex-valued function defined on I.

(i) If 9 has generalized derivatives such that

(1 + x2)~r/2(x-1D)i9(x) e L°°(I) ( Ì G N , 0 < K s ) ,

with r eN, then 9 e Or+r}S. Moreover,

\\9\\r+sìS < C max ||(1+ z2rr/2(ar1Z})t0(;C)||TO

(ii) If there exists peN such that the generalized derivatives of 9 satisfy

(1 + x2)-p'2xi^+1l'2{x-1Dy9{x) e L°°(I) (j eN,0<j< s),

and ifreN corresponds to [f + s] + 2 as in Lemma 1, then 9 e 0TìS with

\\9\\r,s <C max | | ( 1 + a ; 2 ) - ' , / V + ' 1 + 1 / 2 ( a ; - 1 D ) ^ ( a : ) | | 0 0 . 0< j '< s

(Hi) IfreN and 9 e Or>s+i then there exists m e N such that

(1 + x2)-Tn/2xk+»+1/'1{x-lD)k9(x) e L°°{I) (keN,0<k<s).

Proof. Let <j> e Hr/S, and let

0<i<s " V

There holds

Me= max ||(1 + x2)~rf2{x~lD)i9{x)\l 0<Ì<8

s POO

m-\-k=0J°

m+k=0 jQ

1/2

s k k \ ( f i,^ i K ^ l n ^A-nrr, . , M O , ì '

m+k=0j=0 ^J J° ^M° £ E h {/ \(i+J)r/2x^%,k-M*)\2**}

< CMe\\cf>\\^r+s-

This establishes (i).

Multipliers and convolution operators 125

To obtain (ii) we fix <j> e H1^ and proceed as in the proof of (i). Writing n = [f -f s] +1 , an application of Lemma 1 yields

k

w\\,,s= E £ffi{ria+*2)p/2^^ m+fc=0j=0 W KJ°

1(1 + x 2 ) -P/V*+^+ 1 / 2 (a ; - 1Z))^( a : ) | 2^} 1 / 2

n

<C1£\\(l + x,2r(x-1D)ix-''-^2<l>(x)\\2

t=0

max | | (H-a ; 2 ) - p /V + ^ + 1 / 2 (a ; - 1 W^(a; ) | | 0 0

0 < j ' < s

n+1

^c iKi+ r+H - r - - ^wiioo t=0

max |.|(l+ar2)-»,/V+'4+1/2(.a;-1£>y«(ar)|loo 0<j<s

< CII^II,,,, max ||(1 +x2)-''2x^+1V{x~1Dye(x)\\x. 0<j<s

Finally, we address to the proof of (iii). The case s > ì will be discussed first. Arguing by contradiction, let us assume that some k e N, 0 < k < s, may be found so that (1 + x2)-m/2xk+»+1/2(<x-1D)ke(x) <£ L°°(I) for any m e N, while to every i e N, 0 < i < k, there corresponds ra* € N satisfying (1 + x2)-™i/2xi+tl+1/2(x~1D)i6(x) e L°°(I). Put m = max{mj : % 6 N,0 < % < k}, and take n e N, n > max{m,/i + r + 1}. Then a;A*+-i/2(i +aj2)-n/2 .€ H£. However, if 0 G C?r,s+i then, according to [3, Lemma 2.14], there exists C > 0 such that

C\\x»+1/2(l+x2)-n/2e(x)\\^s+1 > Wx^+^ix-'D^iì +x2)-n'20{x)\\oo

>:||(l..+ a:2)-n/V+ '1 + 1/2(a:-1I>)^(3;) | |0 0

k /A ~J2 ( ) ||sfc+/4+1/2(*~10)i(l + ^2)"n/2(a;-1J9)fc-^(x)||00,

which prevents 0 fromlying in O r s + 1 . This is the expected contradiction.

A similar argument allows to handle the case s = 0. Assume that

(1 + .T2)-m /V i + 1 /20(x) £ L°°(I)

for any m e N, and take n e N, ra > jx + r + 1, so that ( l + x 2 ) - n / 2 ^ + 1 / 2 € ^ r # I f 19 € £>r t

then, again by Lemma 2.14 in [3], there exists C > 0 such that

C||(l + ^ - " / V + ^ ^ I I ^ > | | ( 1 + aPr^af+WoWWoo ,

thus yielding a contradiction which completes the proof.

An immediate consequence of Proposition 4, (i) is the following.

126 7.7. Betancor - I. Marrero

COROLLARY 1. Let r, s e N, and let P = P(x) be a polynomial.

(i) If P has degree m and if r > 2ra, then P{x2) e Or+StS.

(ii) If P does not have any roots in [0,oo[, then . 2, e Or+StS. F[x )

Now we establish an analogue to Proposition 4, replacing L2(I) with L°°(I).

PROPOSITION 5. Let s € N, and let 0 = 0(a?) te A complex-valued function defined on I.

(i) If 9 possesses generalized derivatives which satisfy

(1 + ar2)-r'/2(x~1#)10(z) ^ L2{I) (i e N, 0 < i < s),

with r e N, then 0 e Or -f-s+i)S. Moreover,

\\0\\r+s+lìS<cJ2m+^rr/H^-1Dy0(x)\\2. j=0

(ii) If there exists p € N such that the generalized derivatives of 0 satisfy

(l + x2)-p/2xj+»+l/2(x-1Dy0(x)€L2iI) (jeN,0<j<s),

\P 1 and ifreN corresponds to - + s +1 as in Lemma 1, then 0 e OrìS, with - & -

\\e\\r,s<c^\\(i + x2^

(Hi) .Let.r.eN. If 0 € OryS+1, then 0 € CS(I). If 0 e Or,s> then the distribution (x~1D)s0(x) is regular, and there exists m GN such that

(1 + ^j-m:/2aJ+/*+i/2(ar-ix>)i^(a:) e L2(I) {j e N, 0 < j < s).

Proof To show (i), let </> e HrJ~s+1. By virtue of [3, Lemma 2.14], we may write

1/2

_£_ ( roo . 1/2

< £ { / ^T^ix^ix^dx) m+fc=0 Jo

< E E CÌ) {r km+j^,^^)i2i(x-iD)^(x)i2dx} m + fc = 0.7=0 ^ ^ J°

Multipliers and convolution operators ìli

< C max £ ||(1 + r / ^ ^ T ^ ^ ^ J H o o l K l + x ^ t \ x - ^ 9{x)\\2 ~ T - j=0

< c||^iu,r+s+1 è ||(i + x*)-^(x-iDye(x)\\2. 3=0

Next, let <j) eHr^. Lemma 1 leads to

INU.S E è (fc) { /°° K1 + 2)p/2xm+fc^(x-»z?)fc-^-''-1/V(x)|2

m+k=0j=0 \3/ JO

\(l + xY^xi+^^ix-ìDyOixtfdxy'2

< C max 11(1+ ^2)n(x-1D)-7 'a;-^-1/2^(a;)|| 0<j<n

2 ) n (à ; - 1D)^" ' ' ' - 1 / 3 ,

53 | | ( l+ar 2 )" p / V + " + 1 / 2 (a!"" 1 ^)^(a: ) | |2

Here, rc =

3=0

.2+* + 1. This establishes (ii).

Finally, let us demonstrate (iii). Put (f){x) = x^+1/2e~x2 (x e I). Since 0 e OrìS+1

and <j) e H» C H£ imply $</>'e H^+1 C Cs(7) [3, Lemma 2.14], necessarily 0 e CS(I). Moreover, if 0 G O V then the distribution (x~1 D)s0(x) is regular. In fact, we nave

J'ttx){x-lDye(x) = jy j=o w

where cf>j{x) = a ^ + ^ ^ - i i ) ) ^ - ^ - ! ^ ^ ) e HM G W£. Hence .^(ir)^(a;) € H* (0 <j< s), and

is a regular distribution.

The remaining of (iii) can be shown by proceeding as in the proof of the corresponding assertion in Proposition 4.

Note that (ii) and (iii) in Proposition 5 are converses, while (ii) and (iii) in Proposition 4 are not.

We now proceed to discuss some relationships between the spaces H^ and Ors.

128 J.J. Betancor - I. Marre tv

PROPOSITION 6. Let r,s € N, with r > 2s + p +1. Then H7^ C x/*+1/20SiS, and the mapping <j>(x) !-• x~^~1f2<j){x) is continuous front H1^ into 0SjS.

Proof. Fix (/>eHr^ Then tj> e CS(I). Moreover, by virtue of [3, Theorem 2.2], and according to well-known properties of the Bessel J^-function [16, p. 129], for 0 < i < s and x e I one has

(-ìyix-'DYx-x-^tix) = / y (1 + yr)(h^)(y)(xy)-^iJfl+1(xy)dy. Jo i + y

The boundedness of z'^J^z) (z e I) along with Hòlder's Inequality leads to /•oo l 2 i + / t + l / 2 I

supKx-'Dyx-^^ix)] <c / pV—— l(i + »r)(M)(»)l* x€/ JO | 1 + 3/ I

Uoo / 2i+^+l/2\2 ì 1 / 2 f /.oo- 1 / 2

( i + y r ) dy\ {j0 i(i + 2/r)(M)(2/)i2^} <^IHUr (ìeN,o<i<s).

The proof can be completed by invoking Proposition 4. PROPOSITION 7. tór,s€N,s> /H-l. Foreach 0 e Or?s, the function x*+1/20(x)

(x e I) defines a regular distribution in H~r by /•OO

{x^1'20{x), <f>(x)) = / ^+1/20(x)#c)<fe (4> e UH Jo

and the mapping 0(x) H-> X^+1^20(X) is continuous front Or?s into H~r.

Proof. Given 0 e Or>s, an application of Hòlder's Inequality yields /•OO

\(x»+1/20(x),<t>(x))\< / la^+^l+a;*)—/2| | (1 + x2)s'20{x)cf>{x)\dx Jo

Uoo 2//+1 1 1/2 /-oo j / 2

<C7||^|U.<C||^||r|.|HUr (*€WJ). Thus, a^+1/20(z) e H~r and ||^+1/2 |9(a.)||/x _r < c||0||r |a.

At this point we introduce the space Os = Og = UrGj^Or|B (s e N). The following result can be easily derived from Propositions 4 and 5.

PROPOSITION 8. Let s e N, and let 0 = 0(x) be a complex-valued function defined on I.

(i) If the generalized derivatives of 0 are such that

(1 + x2)-^2x^^^2{x-lDy0{x) e L°°(I) (j e N, 0 < j < s)

for some p e N, then 0 e Os.

Multipliers and convolution operators 129

(ii) 0 e Os if, and only if, there exists p e N such that the generalized derivatives of 6 satisfy

(1 -I- r c 2 ) - p / V + / i + 1 / 2 ( x - 1 i ) ) ^ ( r c ) e L2(I) (j E N, 0 < j < s).

Some other properties of the elements in Os involving L°°-norms are gathered below.

PROPOSITION 9. Let s eN, 0 :1 —• C, and consider the following assertions. (i) 6 e Os+l.

(ii) // <£ G ftM and m, fc e N, 0 < m + A; < s, then xmT^k(0(x)(f>(x)) e L°°(I). (iii) If<t> e H» and m,k eN, 0 < m + k < s, then xm+k(/>(x)(x-1D)k0(x) e L°°(I). (iv) 0eOs.

The implications (i) =*• (ii), (ii) <& (iii), and (iii) =$> (iv), hold.

Proof. (i) =» (ti). Let 9 e Os+ì. There exists r e N such that 9 e OriS+1. Since H» C W£, we find that xrnT^k(9(x)^>{x)) e L°°(I) (<p€H», m,k e N, 0 <m + k < s) [3, Lemma 2.14].

(ii) =$> (iii). If <j> e H^, Leibniz's Formula gives

xm+k(l>(x)(x-1D)k0{x) = ^ ( - l ) - 7 ' ^ ,)a:m+fc+'4+1/2(a;"1I))fc-^(a;-"'4-1/2^(a:)^(a:)) k /k\

= y](-l>M . . ^ V ^ ( x ) ^ W ) (xel, m,keN, 0<m + k<s),

where (pó(x) = x^+^2(x-1D)jx-^-ì/2(l>(x) e H» (j e N, 0 < j < k), and from (ii) we infer that xm+k<p(x)(x-lD)ke(x) e L°°(I) (m,k eN,G <m +k < s).

(iii) => (ti). For every <j> eH^ one has

xmT^k(9(x)ci>(x)) = £ (k)xm+^Tflìk_jcl>(x))(x-1Dy0(x)

= ] T fk^jxm+kcj)j(x)(x-1DY0(x) (m,k e N, 0 < m + k < s), .

where <f>j(x) = x^1^(x~1D)k-3x-fl-^2(i)(x) (j e N, 0 < j < k). Thus, (ii) follows easily

from (iii).

(Hi) =4> (iv). Assume that (iii) holds, and (to reach a contradiction) that (iv) is false. According to Proposition 8 (i), if 0 £ Os then there exists k e N, 0 < k < 8, such that (1 + x2)-m/2xk+^+^2(x-1D)k0(x) <£. L°°(I) for any m e N. By virtue of (iii), we have xk<j)(x)(x~1D)k0(x) e L°°(I) whenever <p e H^.. In particular xk+^+1/2e-x2(x-1D)k0(x) e L°°(I), so that xk+^+1/2(x-1D)k0(x) e L°°]0,T[) (T>0).

130 JJ. Betancor - L Marrero

Now, we may choose a sequence {a^},-G [ of positive real numbers such that XQ > 1, Xj+\ > Xj + 1, and

(2) |x$+ ' l + 1 /V 10)**(*) |_ , | > (1+ *?)J'/2 U € N).

Let a e T>(I) satisfy 0 < a(x) < 1 (x e I), supp a =

to see that the function

ì ì 2 ' 2

, and a(l) = 1. It is not hard

<p(x) = ^ + 1 / 2 ]T) j ( l + x2j)-j/2a{x - Xj + 1) (a; € I)

lies in H^. Moreover, (2) implies

\xfa(xj)(x-1D)kO(x)lx=x.\>j (j e N),

so that xkip(x)(x~1D)k0(x) £ L°°(I). This contradiction completes the proof.

We are about to characterize Os by using L2-norms.

PROPOSITION 10. Let s e N, and let 0 = 6(x) be a complex-valued function defined on I. The following statements are equivalent.

0) 0eOs.

(ii) If^eH^andm.keKOKm-^k^s, then xmT^k(0(x)<i>(x)) e L2(I).

(iii) If(j>en^andm,keN,0<m-{-k<sf then xm+k(j}(x)(x-1D)ke(x) e L2(I).

Proof. That (?') => (ii) => (iii) can be proved as the corresponding implications in Proposition 9.

(iii) => (i). Assume that (iii) holds, and (to reach a contradiction) that (i) does not. By Proposition 8 (ii), if 6 £ Os then there exists k e N, 0 < k < s, such that (1 + x2)-m/2xk+^+1/2(x-1D)ke(x) i L2(I) for any m € N. Moreover, by (iii), and since xn+i/2e-x* € n^ w e find t n a t ^ + a.2)-m/2a.fc+/.+i/2c-«

a(a.-i£))fc^a.) € jr2(jj whenever

m e N, whence (1 +x2)-m/2xk+iL+1/2(x-lD)k0(x) e L2(]0, T[) for every m e N and every T > 0. Thus, a sequence {XJ} .e^ of positive real numbers may be found so that reo = 0, XJ+I > Xj -f 1, and

/ \(l + x2)-i'2xk+»+1/2(x-1D)k0(x)\2dx>j + l ( J G N ) ,

where Ej = [2 ,0^+1] (j e N). For j e N, j > 1, we choose a;- e V(I) satisfying 1 T

0 < ctj(x) <l (x E I), supp a . xù J '^ '+i + 4 , with CXJ(X) = 1 (x e Ej), and such

that sup^g/ \(x 1D)kctj(x)\ does not depend on j . It is easily seen that the function 0 0

rìx)=x»w£a + x ) " ^(rc) (x e I) j=i

Multipliers and convolution operators 131

lies in H^, and that x-^-^2(p(x) > (1 +ar2)~J'/2 (a: € Eó, j e N, j > 1). Consequently /•OO

/ \xk<p(x)(x-1D)k0(x)\2dx Jo

> [ \xk<p(x)(x-xD)k0(x)\2dx JEj

> f |(1 + x2)-j/2xk+IA+1/2(x-1D)k0(x)\2dx > j + 1 (7 € N, j > 1).

This means that xkip(x)(x~1D)k0(x) £ L2(I), which contradicts (iii). The proof is complete.

The next Proposition characterizes Os as the space of multipliers from H^s into H'^.

PROPOSITION 11. Lei s e N. A complex-valued function 0 = 0(x) deflned on I belongs to Os if and only if, the mapping T *-* OT is continuous from H~s into H'^ when Hp is endowed with its strong topology.

Proof Throughout the proof we shall assume that H'^ is equipped with its strong topology.

If 6 e Os then the mapping T »-••• OT is continuous from H~s into H~r, for some r € N. The mapping H~r «-»• H'^ being also continuous [3, Proposition 2.15], we conclude that T *-* OT is continuous from H~s into H'^.

Conversely, assume that the mapping U : H~s —• H'^, deflned by UT = OT, is continuous. As {H'pf = H^ and (H-s)' = H^, the adjoint U* of U, given by U*</> = 0<j> ((f> e H^), maps continuously H^ into H^. Hence xmT^k{0(x)<j)(x)) e L2(I) (<j) e Hfl, m, k e N, 0 < m -f k < s), and from Proposition 10 we deduce that 0 e Os.

It is naturai to endow Os with the final locally convex topology associated to the family of inclusions {OryS <-+ O s } r e ^ . According to Proposition 10, some other topologies can be considered on this space. They are compared below.

PROPOSITION 12. LetseN. (i) The final locally convex topology of Os is fìner than that generated on this space by

the family of seminorms {^s^tzH^ given by

a-2,sAe) = { J2 \xm+k<Kx)(x-1D)ke(x)\2dxj (cj>eH^0eOs).

(il) The topology deflned on Os by {&2,s,(f>}<i>eHh coincides with that generated on this space by the system ofseminormsfa^^^H^where

s />oo . j / 2

T2ìSìm = { E / \xmT»mx)<t>{x))?} & €H„oe oa).

132 7.7. Betancor - L Marrero

Proof. Let 0 e 0TìS for some r e N. If m, k e N, 0 < m + k < s, and if <f> e H^ then

(3) xm+kcj>{x){x-lD)k0{x) = £ ( - j ) ' (k)xm+%ik-3i0(x)<f>3(x)), j=0 W

where ^ ( x ) = a ^ 1 / 2 ^ - 1 / } ) ^ - ^ " 1 / 2 ^ ) G fy, (j € N,0 < j < s). Hence

02,.,+W ^ C ^ X | | ^ | U f l < ( C m a x | | ^ | L r ) l | 0 | | r , a ,

so that the mapping Or,s t-^ Os is continuous when Os is equipped with the topology generated

by {ff2,,,4^w,. This proves (i).

To establish (ii), it suffices to take into account (3) along with the identity

xmT^k{e{x)(j>{x)) = J2 ( f c V m + f c - j ) + J ' ( ^ + 1 ^ ^

valid for 0 e Os, (j> e H^, and m, k € N, with 0 < m + k < s.

PROPOSITION 13. For every s e N, the mapping (0, T) »-> 0T is continuous front Os x H~s into H^, when H^ is endowed with its strong topology.

Proof. Let V be a neighborhood of the origin in H'^. Since H'^ — U r e ^ H ~ r [3, Proposition 2.15], there exists a sequence {«r} r G^ of positive real numbers such that V contains the absolutely convex hull of UrG^Vr, where Vr — {T e H~r : IITH^^-r < ar} (r e N). For r, s e N, set Gr,s = {0 e Or,s : | |0|| r i, < a r } , Us = {T e H~s : ||T||M _a < 1}. The absolutely convex hull Gs of Ur€j^Gr>fl is a neighborhood of the origin in Os, and OT e Vr

whenever 0 € Gr,s and T eUs. Thus, 0T eV provided that 0 e Gs and T e Us.

We write Q = nsGj^Os and endow Q with the initial topology associated to the family of inclusions {Q c—> <9S}S 6N- As we shall now see, the identity Q = O holds both in the algebraic and in the topological senses.

PROPOSITION 14. The linear topological spaces Q and O coincide.

Proof. First of ali, we shall establish the algebraic identity Q, = O. Let 0 e O and s e N. There exist p = p(s) € N, C > 0 such that

max Ux^DYOix^ < C(ì + x2)p (x e I). 0<i<8

Multipliers and convolution operators 133

If <j> € ft£, with r = s + 2p, then

1/2

1/2

0/-OO .

.7=0 ^° 3-

Consequently, ||0^||M,a < C|MUr (<£ € ftp. This means that 0 c f l . Conversely, let 0 e Q, and let <£ e H^ Then 0<£ G W^ for ali s G N. Since

H/x — ns e N ^ 13» Proposition 2.15] and since O is the space of multipliers of H» [2,

Theorem 4.9], we get that lì C O.

Next we aim to prove that the topology of Q, is stronger than that of O. Let <f> e H^ <j) ^ 0, let k G N, and defìne G = {0 e O : ^k(0) < 1}. Since the family of norms (Il • IUr}reN generates on H^ its usuai topology, we may find S e N, C > 0 such that

-#*(*) = s u p l * - " - 1 / 8 ^ ) ^ - ' / ) ) * ^ ) ! «e/

^ E ffcì sup|(a:-1I>)fc^(ar-'-1/^(a.)^(fl.))|

p; w x6/

<tf£iiWiiL* ^ € ° ) '

where ^(ac) - x^^^D)^-»-1/2^) eH^(0< j < k, x e I). Define

k

Gr,S = {9G Or,S : | |%, 5 < [C Y, l l^lUr)"1 }• p=0

The absolutely convex hull Gs of U j^Grjs is a neighborhood of the origin in Os. For every 0 e Gs, there exist m G N and A* > 0 (« G N, 0 < « < m), with ££L0 Ai = 1, such that 0 = £ £ L 0

A ^ ' w n e r e ^ G ^ . s (*' G N, 0 < i < m). U0 e OnGs one may write

k k m

j=0 j==0 i=0

m fc m

< 5>(C|M|r„s£||^||,.,,,) < £ > = L

i=0 j=0 i=0

134 J.J. Betancor - I. Marrero

Having found a neighborhood OnGs of the origin in Q cointained in G, we conclude that Q is continuously embedded in O.

To complete the proof we shall prove that the inclusion O <—• Os is continuous for ali s G N .

Let s G N, let B be a bounded set in O, and let V be a neighborhood of the origin in Os. To each r G N there corresponds a neighborhood K- of the origin in 0TìS such that V contains the absolutely convex closed hull of U£L0K. Given r G N, choose p r > 0 with {6 e OryS : ||0||r>a < pr} C Vr. Since the set {9<f> : 9 e B} is bounded in H* for every 0 G H^, and since the family of norms {pr\\ • IUr}rGN generates the topology of the Fréchet space H^, the Uniform Boundedness Principle yields m e N and er > 0 (r G N, 0 < r < m) such that 110011 ,5 < 1 whenever 9 e B and p r |HU,r < £r (0 < r < m). Now, taking e = mino<r<m£V'1£r we get H^ll/x.s < (^Pm)_1 (0 G J3) provided that <t> e H^ with H IU.m < p^1 . Therefore B c'(epm)"-1Vm C (^Pm)"1^, which proves the boundedness of B in Os. As (9 is bornological, the continuity of the inclusion O <—• Os follows.

REMARK. In connection with Proposition 12 and 14, it should be noted that

the topology of O can be also generated by means of any of the families of seminorms

{a2 | a^}a6N^€W ,-{7ì2,3^}a€Nf0€WM, and also by any one of the families of seminorms

{<Too,a|0}tfeN,*€7v {^'S'^seN^en^ 8 i v e n by

<7oo,sA0) = m a x SUP\xm+k<l>{x){x-1D)k9(x)\ {seN^eH^Oe o), 0<m+fc<s X(zi

Toc,sAe) = ™x sup \xmT^k{0{x)4>{x))\ {s e N, <t> G WM> 0 G 0 ) .

At this point we may give a second proof of an already established property of O (see [2]).

PROPOSITION 15. A complex-valued function 9 e C°°(I) lies in O if and only if the mapping T •-• 9Tis continuous from H'^ into itself when H'^ is endowed with its strong topology.

Proof. If 9 e O then 9 e Os for each s e N, so that T •-»• 9T is continuous from H~s into H'^ with its strong topology, whenever s e N. The strong topology of Hp = US£N'H^S coincides with the final locally convex topology associated to the family of inclusions {H~s c-^ 7^X}S(EN [2, Proposition 2.15], so that T i—• 9T is continuous from H'^ into H'^ when the strong topology is considered on this space.

Conversely, if 9 £ O then there exists s e N such that 9 £ Os. From Proposition 11 we infer that the mapping T i-> 9T is not continuous from H~s into H'^ hence from H^ into H^, when this space is equipped with its strong topology.

In [2, Proposition 5.3] we established that the mapping (9,T) *-• 9T is separately

Multiplìers and convolution operators 135

continuous from O x H'^, when H'^ is endowed with its strong topology. Now we are in a position to improve this result.

PROPOSITION 16. The mapping (0,T) •-> $T is O-hypocontìnuous from O x H^

into H'^ when H'^ is equipped with its strong topology.

Proof. Let V be a strong neighborhood of the origin in H'^, and, for every s G N, define Gs, Us as in the proof of Proposition 13. If B is a bounded subset of O, then for every s e N we may find Xa > 0 such that B e XSGS. Setting Ws — Xj^Ug, it turns out that the absolutely convex hull W of UsG^Ws is a strong neighborhood of the origin in H'^ such that 9 e B and T e W imply 9T G V.

3. Convolution operators in H^r

In this Section we introduce convolution operators in the spaces H~r (r e N). Those operators are closely connected with the multiplication operators discussed in the previous Section.

Let r, s e N. The space 0*s = 0%# consists of ali those Te ^QH~P = W{* s u c h

that y-^-^2(h^T){y) e Or,s. We topologize Ofs by the norm

The main properties of (9#s are gathered below.

PROPOSITION 17. Let r, s G N.

(i) 0#s = h>^^Or,s).

(ii)OfìSisaBanach space.

(Hi) Ifs > /x+1 and r > 2s-f/i-|-1, then H^ C Of^s e H~r, with continuous embeddings.

In particular, Ofs is dense in H^r.

Proof. To show (i) it is enough to observe that h'~l = h^ on H'^, which contains x^+1/2OrìS. Part (ii) follows immediately from (i). As the generalized Hankel transformation ^ is an isometry of H~r [3, Theorem 2.2], the chain of inclusions in (iii) derives from Propositions 3, 6 and 7. Finally, H^ being dense in U~r [3, Corollary 2.13], so is Ofa.

For r, s G N, S G H~ s , and T G 0# 5 , we define the Hankel convolution S # T of 5 andTby

S # r = ^ ( ! / - ^ - 1 / 2 ( ^ r ) ( y ) ( ^ ) ( y ) ) .

Note that S#T e H~r whenever S e H~s and T G 0# a .

Since ( f y j - 1 = h'p on H~ r (r G N), the following holds.

136 J.J. Betancor - I. Marre tv

PROPOSITION 18. Forr.seN, S e H~s, T e Ofs, and x-^-^20(x) e OryS, we have

h^S#T)(y) = y-^'2{KT){y){h^S){y) (y € / ) ,

h>^x-»-VH{x)S{x)) = (h^S)#(h^).

The next properties of the space Ofs derive from the corresponding ones of the space OriS, via Proposition 18.

PROPOSITION 19. The following hold. (i) Let r,s e N. The mapping (5, T) •->• S#T is continuous from H~s x Ofs info H~r. (ii) IfOf = \Jr(-^Ofs (s e N) is endowed with the final locally convex topology associated

to the family of inclusions {Ofa ^ Of }ref$, and if Q^ = nsef$Of is equipped with the initial topology generated by the family of inclusions {Q^># <-+ Of}se^, then the linear topological spaces Q/x,# and O' # coincide.

(Hi) For every s e N the mapping (S,T) *-* S#T is continuous from H~s x Of into H^, if 'ri^ is endowed with its strong topology.

(iv) The mapping {S,T) •-• 5#T is O'^^-hypocontinuous from H'^ x O' # into H\iy ifthe latter is equipped with its strong topology.

REFERENCES

[1] BETANCOR, J.J., MARRERO I., Some linear properties ofthe Zemanian space 7i^, Bull. Soc. Roy. Sci. Liège 61 (1992), 299-314.

[2] BETANCOR J.J., MARRERO I., Muliipliers of Hankel transformable generalized functions, Comment. Math. Univ. Caiolin. 33 (1992), 389-401.

[3] BETANCOR J.J., MARRERO I., A Hilbert-space approach to Hankel-transformable distributions, Appi. Anal. (to appear).

[4] HlRSCHMAN LI. JR . , Variation diminishing Hankel transfonns, J. Analyse Math. 8 (1960/61), 302-336.

[5] KUCERA J., Fourier l?-transfonns of distributions, Czechoslovak Math. J. 19 (94) (1969), 143-153.

[6] KUCERA J., Laplace L2-transfonns of distributions, Czechoslovak Math. J. 19 (94) (1969), 181-189.

[7] KUCERA J., On multipliers of temperate distributions, Czechoslovak Math. J. 21 (96) (1971), 610-618.

[8] KUCERA J., Extension of the L. Schwartz space OM of multipliers of temperate distributions, J. Math. Anal. Appi. 56 (1976), 368-372.

[9] KUCERA J., McKENNON K., Certain topologies on the space of temperate distributions and its multipliers, Indiana Univ. Math. J. 24 (8) (1975), 773-775.

[10] MARRERO I., BETANCOR J.J., Hankel convolution of generalized functions, Preprint (1992). [11] PICARD R.H., A Hilbert space approach to distributions, Proc. Roy. Soc. Edinburgh Sect. A

115 (1990), 275-288. [12] SCHWARTZ L., Théorìe des Distributions, Voi I/II, Hermann, Paris, 1950/51.

Multipliers and convolution operators 137

[13] WATSON G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1956.

[14] ZEMANIAN A.H., A distributional Hankel transform, SIAM J. Appi. Math. 14 (1966), 561-576. [15] ZEMANIAN A.H., The Hankel transfonnation ofcertain distributions of rapid growth, SIAM J.

Appi. Math. 14 (1966), 678-690. [16] ZEMANIAN A.H., Generalized Integrai Transformations, Interscience, New York, 1968.

JJ. BETANCOR, I. MARRERO, Departamento de Anàlisis Matemàtico, Universidad de La Laguna 38271 La Laguna (Tenerife), Canary Islands, Spain.

Lavoro pervenuto in redazione il 17.7.1992 e, informa definitiva, il 15.7.1993.