Convolution in <span class="mathjax-formula">$(W^p_{M, \ a})^{\prime }$</span>-space

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

R. S. P ATHAK

S. K. U PADHYAY

Convolution in (W

_M,^p _a

)

⁰

-space

Rendiconti del Seminario Matematico della Università di Padova, tome 98 (1997), p. 57-65

<http://www.numdam.org/item?id=RSMUP_1997__98__57_0>

© Rendiconti del Seminario Matematico della Università di Padova, 1997, tous droits réservés.

L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions).

Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

R. S. PATHAK - S. K.

UPADHYAY(*)

ABSTRACT - A characterization of convolutors in

(WL, a)’ -space

^is

given using

the

properties

of the translate _{r :}

WL, a

^-

WL, a- ^Using

^the

^theory

^{of Fouri-}

er transform in these _spaces, the Fourier transform of convolution is studied.

1. - Introduction.

A characterization of convolution

operators

^{on the} space ^was

given by

^Swartz

^[9] generalizing

the characterizations of the _space

0~

of Schwartz

[8]

and of convolutors on the _spaces of distributions of ex-

ponential growth by

^Hausmi

^[5].

This chracterization

naturally yields

^a

characterization for

WM, a-space,

^{which is a}

special

^case space.

A similar characterization of convolution

operators

ⁱⁿ

Kp

^was

given by Sampson

^and

Zielezny [10].

All these results are related to L °°-

norms.

In terms of L p norms the _spaces

WD, P, WD, b, p

^{were de-}

fined and their Fourier transforms were studied in

[6].

^We recall the definition of the _spaces

WL, WL, a’

^Let

^p(~)

^{be a contin-}

uous

increasing

^{function on}

[0, oo ]

^{such that}

/~(0)==0,~(oo)= oo~

^and

for x ~ 0 define an

increasing

^convex continuous function M

by

(*) ^Indirizzo

degli

^AA.:

Department

^of Mathematics, Banaras Hindu Univer-

sity,

^Varanasi 221005, ^India.

MSC (1991): ^46F12.

58

Then

M( 0 )

⁼

0, M( OJ ) = OJ ,

^and

Now the space is defined as the set of all

infinitely

^differen-

tiable functions

lP(x) (- 00

^x

oo) satisfying

for each

non-negative integer

k where the

positive

constants a and

Ck, p depend

upon 0.

Clearly WM

^{is a linear space. The space}

WM

^{can be re-}

garded

^as the union of

countably

^normed spaces

WM, a

^{of all}

complex

valued Coo-functions O which for

any 6

&#x3E; 0

satisfy

Let Q be another

increasing, continuous,

^{convex function}

possessing properties

^{similar to} those of M. Then

WD, P

^is defined to be the set of all entire

analytic

^functions

^{Ø(z) ( z}

^{= x +}

iy ) satisfying

^the

inequali-

ties

The _space defined to be set of all those functions in

WQ, P

which

satisfy

^the

inequalities

In this _paper the translate _{T h:}

WM, a

^~

WM, a

^defined

by

^{T h}

= + is shown to be

continuous,

bounded and differentiable.

A characterization of convolutors in

WIt, a

^is

given. Furthermore, by using

^the

theory

of Fourier transform

of f E

^E

1 /p

⁺

+ ⁼

1,

^{we show that}

2. - Characterization theorems.

THEOREM 2.1.

(i)

^{For each h E R the} 7: h ø is continuous

from

WM,

^{a into}

Wlt, a.

(ii)

^{For a bounded subset A}

of WM,

^{a and E} &#x3E;

0,

^{the set}

~ ih ~: ~

^E,

0 E=- A I

^is bounded in

WM, a .

PROOF. For 0 E

WM,

^{a and we have}

Now, using

^the

convexity property (1),

^we

get

so that

(i)

^and

(ii)

follow from

inequality ^(6).

THEOREM 2.2. For each 0 ^E

WM, a

the translate

rh (P is differen-

tiable in 1.

PROOF. From

[6, p. 734]

^{we know that a} function 0 E

WM, a

^{is dif-}

ferentiable in

WM, a

^{space. Since E} it follows that is dif- ferentiable in

WM,

^{a .}

Now,

^we recall the definition of a convolute

[3, p. 137].

60

DEFINITION 2.3. Let V be _{any test} function _{space and} V be its dual. A

generalized

^{function is said to be a} convolute if for each

~ E E

V, and 0,

^{-~ 0}

implies

^{- 0 in the}

topology

^{of V.}

If

f

^{is a} convolute and _g E

V,

the convolution of

f

^and g is

given by

THEOREM 2.4. Let

f E (WM, a)’

^{and ø E then E}

WM,

^{b ,}

where

~,r~l

^and

PROOF. From

[6, p. 734]

^{we have}

Therefore for 0 ^E

WM, a

^{we have}

So that for

1/p

⁺

1/g

⁼

^1,

^{we have}

Therefore,

^for

^{( b}

&#x3E; a &#x3E;

0),

^{we have}

Hence for

In

particular, taking r

= p ^we

have f*

^{ø E}

WIt, b,

^b &#x3E; a. Therefore

f

^{is a}

convolute in

THEOREM 2.5. Assume that b &#x3E; a &#x3E; 0. Then

WM, b

is a dense sub- space

of WM,

^a

^{for 1 ~ ~}

^00.

PROOF. Let 1

.)

^{such that I}

. Defme

Set uv

^u.

Then uv

^E

D(R).

^{It can be}

easily

^{seen that uv - u in}

WL, a.

Therefore D is dense in

WL, a.

^{Since D c}

(WM, ^b),

^it follows that

WM, b

^{is dense in}

WL, a. Consequently

3. - Fourier transform.

PROOF. From

[6, p. 734]

^we

again

^have

and

62

Now,

which is known to be an element in Since

is an element of

( WM, b )’ , ^{( a ~ b). Also,}

^since

(WM, b)’

^is closed with

respect

^to

differentiation,

^{hence the} distributional derivative

]

^{is also an ele-}

ment of

( WM, b)’.

Furthermore

b)’ implies

^that

E

(Wll, lib)’ by

Gel’fand and Shilov

[4].

Now, Then,

DEFINITION 3.1. is said to

belong

¹

THEOREM 3.2.

If f E (0:)’

^{and then}

. F(g)

^{in the sense}

of equality

ⁱⁿ

PROOF. Let O ^E

WQ, 1/a, p,

^{then we have}

Since

belongs

^to

WL,

^b

^by

^Theorem

2.4,

^then

right-hand

^{side is}

meaningful. ^Now, using

(8)

^{we have}

64

where Then the last

expression

equals

Therefore,

Acknowledgement.

^{The work was}

supported by

^{the N.B.H.M.}

Grant No.

48/1/94-R&#x26;D-II.

REFERENCES

[1] ^{R. D.} CARMICHAEL, Generalized

Cauchy

and Poisson

integrals

^{and distri-}

butional

boundary

values, SIAM J. Math. Anal., ⁴ (1) (1973), pp. 198- 219.

[2] ^A. FRIEDMAN, Generalized Functions and Partial

Differential Equations,

Prentice-Hall, ^New

Jersey

^(1963).

[3] ^I. M. GEL’FAND - G. E. SHILOV, Generalized Functions, ^Vol. 2, ^Academic Press, ^{New York} (1968).

[4] ^{I. M.} GEL’FAND - G. E. SHILOV, Generalized Functions, ^Vol. 3, ^Academic Press, ^{New York} (1969).

[5] ^M. HAUSMI, ^{Note on} n-dimensional

tempered

ultradistributions, ^Tôhoku Math. J., (2), ¹³ (1961), pp. 94-104.

[6] R. S. PATHAK - S. K. UPADHYAY, ^Wp spaces and Fourier

transform,

^Proc.

Amer. Math. Soc., ¹²¹ (3) (1994), pp. 733-738.

[7] S. PILIPOVIC,

Multipliers,

convolutors and

hypoelliptic

convolutors

of

^tem-

pered

ultradistributions, ⁱⁿ

Proceedings of

International

Symposium

^on

Generalized Functions and their

Application

^{held in} Varanasi, ^{Dec. 23-26} (1991), ^edited

by

^{R. S.} PATHAK, Plenum Press, New York (1992), pp.

183-195.

[8] ^L. SCHWARTZ, Théorie des distributions, Hermann, ^Paris (1966).

[9] C.

SWARTZ,

Convolution in

K{Mp}

^spaces,

^Rocky

Mountain J. Math., ² (1972), pp. 259-163.

[10] ^G. SAMPSON - Z. ZIELEZNY,

Hypoelliptic

convolution

equations in K’p ,

p &#x3E; 1, Trans. Amer. Math.

Soc.,

²²³ (1976), pp. 133-154.

[11] ^Z. ZIELEZNY, ^On spaces

of

convolutor operators ⁱⁿ

K’1,

^Studia Math., ³¹ (1968), pp. 111-124.

Manoscritto pervenuto ^{in redazione il 9 novembre 1995.}