GENERALIZATION OF TITCHMARSH’ S THEOREM FOR THE FIRST HANKEL-CLIFFORD TRANSFORM IN THE SPACE 𝑳ᵖ 𝝁((𝟎, +∞)) Текст научной статьи по специальности «Математика»

56

Probl. Anal. Issues Anal. Vol. 11 (29), No3, 2022, pp. 56-65

DOI: 10.15393/j3.art.2022.11851

UDC 517.98

M. El Hamma, A. Mahfoud

GENERALIZATION OF TITCHMARSH' S THEOREM FOR THE FIRST HANKEL-CLIFFORD TRANSFORM IN THE

SPACE LJ((0, + ra))

Abstract. Using a generalized translation operator, we intend to establish generalizations of the Titchmarsh theorem ( [14], theorem 84) for the first Hankel-Clifford transform for certain classes of functions in the space 0, + x>)), where 1 < p ^ 2.

Key words: first Hankel-Clifford transform, generalized translation operator, Clifford-Lipschitz class, Dini-Clifford-Lipschitz class.

2020 Mathematical Subject Classification: 47G30

1. Introduction. Titchmarsh ( [14], Theorem 84) characterized the set of functions in Lp(R), 1 < p ^ 2, satisfying the Lipschitz condition, by means of an asymptotic estimate growth of the norm of their Fourier transform; namely, we have:

Theorem 1. Let f belong to LP(R), 1 <p ^ 2, such that

If (x + h) - f (x - h)lpdx = 0(hap), 0 <a ^ 1, ash —► 0.

Then its Fourier transform T(f) belongs to L3 (R) for

P </3 ^ p

p + ap — 1 p — 1

On the other hand, Younis in ( [15], Theorem 3.3) studied the same phenomena for the wider Dini-Lipschitz class, as well as for some other allied classes of functions. More precisely,

© Petrozavodsk State University, 2022

Theorem 2. Let f e LP{E) with 1 <p ^ 2, such that ( J \ f {x + h) — f {x)\pdxsj " = o( h"1 ), h 0, 0 <a ^ 1, 7 > 0. Then T{f) e L13 {E) for

P ^ a / ' P

^ P < P

p + ap — 1 p — 1

and 1 < ^, where T{f) stands for the Fourier transform of f.

There are many analogues of these theorems: for the Bessel transform on E+, for the Dunkl transform on Ed, for the q-Dunkl transform on Eg, etc (for example, see [2], [3], [4], [5], [10]).

The aim of this paper is to provide generalizations of Theorems 1 and 2 for the first Hankel-Clifford transform. For this purpose, we use the generalized translation operator.

2. Preliminaries. Let us we briefly collect the pertinent definitions and facts relevant for first Hankel-Clifford analysis, which can be founded in [11], [12], [13], [16].

Assume that L1^ = L1^{{0, + ro)), 1 ^ p < ro and ^ ^ 0, is the space of all real-valued measurable functions f on {0, +ro), such that

11/iu = (/ \f {x)\px^dx) P < ro.

0

Let be the Bessel-Clifford function of the first kind defined by (see [6])

c,{x)=^ {1) x

k=0 fcirfa + k + 1)

which satisfies the differential equation

xy" + {^ + 1)y' + y = 0.

For y ^ — 1, we introduce the normalized spherical Bessel function of index defined by

+: {_1)k /x \ 2k U*) = r{p+DE Hrohiri) (2) *e c (2)

where r(x) is the gamma-function. Moreover, from (2) we see that

t ^(x) — 1 / n lim —2— =0;

x^0 x2

by consequence, there exist C > 0 and rq > 0 satisfying

M ^ V —1| ^ CM2. (3)

The function j^(x) is infinitely differentiable, even, and, moreover, entire analytic.

From [1], we have the following lemma:

Lemma 1. Let ^ ^ — 2. The following inequalities are fulfilled:

1) lfc(aoi ^ 1;

2) 1 — j^(x) = 0(x2), 0 ^ x ^ 1;

3) 1 — j^(x) = 0(1), x ^ 1.

By formulas (1) and (2), we have the following relation, which connect the Bessel-Clifford function and the normalized spherical Bessel function:

1

r(^+1)'

Definition 1. [8], [9] For ^ ^ 0, the first Hankel-Clifford transform for a function f E L* is defined by

hiAf)(X) = J c»(Xx)f(x)dx-0

Proposition 1. If f E L* and h1} ^(f) e L1^, then

f (x)= x^ j c^(Xx)hiM)(X)dX, Vx E (0,

0

For p > 0, let F(A) = hitll(f)(X) and G(X) = hiM(X) denote the first Hankel-Clifford transform of order ^ of f (x) and g(x), respectively. Mendez et al. [9] established the following Parseval relation:

i F(X)G(X)X^dX = f f (x)g(x)x»dx.

c^(x) = w , 1 Jn(2Vx) (4)

Then the first Hankel-Clifford transform h1t^ : f(x) —> h1,,(f)(A) is a linear isomorphism of the space L^ into itself, and for any function f G L^ we have the Parseval identity

llA-^ (/)(A)|2,m = ||x-"/(x)|k,.

Parseval's identity and the Marcinkiewicz interpolation theorem (see [14]) are true for f G L^ with 1 < p ^ 2 and p', such that 1 + 1 = 1

|A-^hi,,(/)(A)|^,M (x)||p,,. (5)

Let A = A(x,y, z) be area of the triangle with sides x, y, z (see [7], [16]). For ^ ^ 0, set

A2M+1

Da(x,y, Z) = -——-:-:-i--—

if A exists, and zero otherwise. Note that D,(x,y, z) ^ 0 and it is symmetric in x, , .

From [12], we define the generalized translation operator by the relation

Th( f)(x)= f(z)D^( h,x, z) z^dz , 0 <x, h< x>.

Assume that ^ ^ 0. Let M be the map of L2^ defined by

Mf (x) = x^f (x) (6)

Prasad et al proved the following well-known proposition: Proposition 2. [12] Let f G L2^ and fix h > 0. Then rh(f)(x) g L2 and

hi,,(MThf(-))(A) = Cfl(Ah)h1>fl(Mf0)(A), A G (0, +œ).

3. Main results. Before giving our first main result, we define the Clifford-Lipschitz class.

Definition 2. Let 0 <8 ^ 1. A function f e L^, 1 < p ^ 2, is said to be in the Clifford-Lipschitz class, denoted Lipc{8,p,^), if

+ 1)Thf {x) — f{x)IIP>fl = 0{hs) ash 0

Theorem 3. Let f belong to the Clifford-Lipschitz class Lipc(8,p, y), 0 <5 ^ 1 and 1 < p ^ 2. Then hltfl(Mf) E L3((0, +m)) for all ft satisfying

№ + P ^ n ^ ' P

< P ^ P

p — ^ + Sp — 1 p — 1

Proof. Assume that f E Lipc(8,p, y); then we have

+ 1)rhf (x) — f (x)\\p^ = 0(h&) ash —^ 0. Using the formula (6), we have

\\r(^ + 1)Thf (x) — f (x) IU = \\x-» № + 1)x»Thf (x) — x»f (x)) =

= \\x-» (r(/1 + 1)M (Thf (x)) — M (f (X))) HP,,.

From proposition 2 and formula (4), we get

^ № + 1)M (Thf (x)) — M (f (x))) (X) =

= (j,(2VXh) — 1) hi,,(Mf (x))(X).

By the Hausdorff-Young formula (5), we have

J X-№'|1 — j,(2VXh)lp/Ihi^Mf)(A)|p'\"dX ^ 0

^ Cf \\x-> (r(jx + 1)M (Thf (x)) — M (f (x)))\\p^ ^

^ Cf' Hrfa + 1)rhf (x) — f (x)\\p^ < Cih&p'.

Hence,

J |1 — U2VXh)f Ihi^Mf)(X)lfX(i-p'^dX ^ CihSp'. 0

If 0 < A < , then 0 < 2^Xh < 'q and inequality (3) implies

11 — j,(2^Xh)l > 4CXh.

From this, we get

4 h

J \ Ah\p'\h1>tl{M/){ A)\P'A(1-P'^dA ^ 0

v2

4 h

< J \1 — J,{2vW\hl^{Mf){A)\p'A(l-pr)^dA ^ 0

^ {¿^ / \1 — 3»{2^Ah)\P'\hiAMf){ A)\^A(1-^»dA = 0{hSp'). 0

So that

v2

4 h

\AhiA Mf){ A)\P'A(1-P' ^dA = 0{h(s-1)p').

0

Thus,

t

\AhiA Mf){ A)\P'A(1-P' ^dA = 0{t(1-s)p').

0

Let

t

1 1-fi/p'

m = J \Ahltfl{Mf){A)f A(1-p'^i/p'd\. 1

Now, if 3 ^ p', by the Holder inequality we obtain

t t № ^y\Ahh,{Mf ){A)\P'A(1-P' J dA^j

11

= 0{t (1-S)p' xl3/p' f1-^/p') = 0{t(1-&)^t 1-<3/p') = 0{t

Therefore,

t t

i\h^{M/){ A)fA^dA = i A-li-(1-p')^/p'^'{A)A'1dA =

+ (f + (1 - p')yf/p' -y) i X-/-(1-P')w//p'+W-1^(X)dX

1

t

,-/3-(1-p

1

= 0(£-//-(1-P,)M/3/p'+M+1—/3+/3/p)

and the right-hand side of this estimate is bounded as t —> if - f - (1 - p') yf/p' + y + 1 - 8f + /3/p< 0.

That is,

f > + p

p - y + 8p - 1 Thus, the proof is finished. □

In the rest of this paper, we give our second main result, which is a generalization of Theorem 2. For this objective, we need to define the Dini-Clifford Lipschitz class.

Definition 3. Let 0 < 5 ^ 1, ^ > 0. A function f e LP, 1 < p ^ 2, is said to be in the Dini-Clifford-Lipschitz class, denoted D-Lipc(S,j,p,y), if

№ + 1)rhf (x) - f(x)\\p,w = O ((y^gr^) ash 0.

Theorem 4. Let f e L^, 1 < p ^ 2. If f belongs to D-Lipc(8,ry,p,y), then h1,/1(Mf) belongs to L3((0, +<x>)), such that

№ +P / « ^ ' P A Ft ^ 1 < f ^ p =-- and f > —.

p - y + 8p - 1 p - 1 7

Proof. Similary to the proof of theorem 3, we can establish the following result:

4h (^ 1) f

J\\fnAM f)(X)\P'X(1-P'^dX = O ((hg-)^) .

Thus,

t

( t(1-$)p'

\AhUM f){ A)\p A(1-p ^dA = 0 -IOg^J .

0

Let us consider again the function defined by

t

i>{t) = j\Ahltfl{M f){A)f A(1-p' ^^ dX. 1

Then, if 3 ^ p', using the Holder inequality we obtain

^ = to-»}

Hence

t

J \hltll{Mf){ A)fA^dA = J A-l3-(1-p')ftf{A)A»dA 11

t

rV-(1-P>WW^{t)+{3+{1— p')^i.—p)!A-P-(1-p'^/p'^{A)dA

1

t

,1-Sp+p/p N . r \1-5p+p/P N

0(t-^-(1-p'^/p' L-—) +0(\A-(i-(1-p'^i/p'^-1 A—— dX)

V {logi)^/ \J {log A)^ /

1

0

-

-¡3-(1-p>) ^ +,+1-&i3+i3/p

{logi)^

and this is bounded as t —> if

—3 — {1 — p')n3/p' + p + 1 — s3 + 3/p < 0 and — 73< —1,

which gives

33 > ^ +/ and 3 > 1 p — p + o p — 1 7

And this ends the proof. □

Acknowledgment. The authors would like to thank the referee for his valuable comments and suggestions.

t

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Received May, 21, 2022. In revised form, August 30, 2022. Accepted September 2, 2022. Published online October 3, 2022.

Laboratoire Mathématiques Fondamentales et appliquées Faculté des Sciences Aïn Chock Universite Hassan II, Casablanca, Maroc

Mohamed El Hamma E-mail: [email protected]

Ayoub Mahfoud

E-mail: [email protected]