ESTIMATES FOR THE SECOND HANKEL–CLIFFORD TRANSFORM AND TITCHMARSH EQUIVALENCE THEOREM Текст научной статьи по специальности «Математика»

144

Probl. Anal. Issues Anal. Vol. 13(31), No 2, 2024, pp. 144-154

DOI: 10.15393/j3.art.2024.15790

UDC 517.544

S. S. Volosivets

ESTIMATES FOR THE SECOND HANKEL-CLIFFORD TRANSFORM AND TITCHMARSH EQUIVALENCE

THEOREM

Abstract. We obtain estimates of integrals containing the second Hankel-Clifford transforms of functions from Sobolev-Hankel-Clifford spaces. As a corollary, we obtain a new variant of Titch-marsh equivalence theorem for the second Hankel-Clifford transform.

Key words: second Hankel-Clifford transform, Hankel-Clifford translation, Sobolev-Hankel-Clifford spaces, Titchmarsh equivalence theorem

2020 Mathematical Subject Classification: 44A15, 47A10

1. Introduction. Let f: R ^ C be in L1(R). The Fourier transform of f is defined by

f(x) = (2^) —1/2 f f (t)e—itx dt, x e R. Jr

If f e LP(R), 1 < p ^ 2, then Fourier transform f(x) is defined as the b

limit of (2^)—1/2 $ f (x)e—ztx dx in the norm of Lq(R), q = p/(p — 1), as

—a

a,b ^ +8.

From the definition it follows that f e Lq (R). The following Hausdorff-Young inequality

\\f\\q ^ C\\f \\p := C (J\f (t)\p dt)1/P, f e LP(R), 1 < p ^ 2, (1)

R

is valid. For p = q = 2, we have the Plancherel equality instead of (1). More about these results can be found in [14, Ch. III and IV] or [3, Ch. 5].

© Petrozavodsk State University, 2024

In [14, Ch. 4, Theorem 85] the following Titchmarsh equivalence theorem is proved:

Theorem 1. Let 0 < a < 1 and f e L2(R). Then the conditions

(i) ||/(• + h) — f (• - h)\\2 = 0(ha), h > 0, and

(ii) $ \f(x)\2 dx = 0(y-2a), y > 0,

are equivalent.

The norm in L2(R) is translation-invariant and \\f (• + h) — f (• — h)\\2 = = \\f (• + 2h) — f (•)|2, h > 0, so the condition (i) may be substituted by \\f(• + 2h) — f(•)\2 = 0(ha), h > 0. Lorentz [8] proved

Theorem 2. If 1 ^ p ^ 2, 1 ^ a > 1/p—1/2, and a 2n-periodic function f e L1 [0, 2^] with trigonometrical Fourier coefficients an, bn belongs to Lip(a) (i.e., \f (x) — f (y)\ ^ C\x — y\a for all x,y e R), then

8

2 (\ak\ + \hk\P) < Cn-ap-p/2+1, n e N.

k=n

Since the proof of Theorem 2 uses the Parseval equality, the condition f e Lip(a) may be replaced by the condition f e Lip(a, 2) (i.e., f is

2tT

2^-periodic, f e L2[0, 2n], and $ \ f (x + h) — f (x)\2 dx = 0(h2a), h > 0.)

0

The aim of this paper is to obtain analogues and generalizations of Theorems 1 and 2 for the second Hankel-Clifford transform. Note that an analogue of Theorem 1 was obtained for the first Hankel-Clifford transform by El Hamma, Daher, and Mahfoud [4], while estimates of this transform in terms of corresponding differential operator were proved by Lahmadi and El Hamma [7], but there are doubts in the last result. A more elementary estimate for the first Hankel-Clifford transform was obtained by the author [15, Theorem 3]. Some close results and facts about the second Hankel-Clifford transform can be found in [16].

2. Definitions. Let 1 ^ p <8, ^ ^ 0, R+ = [0, + 8), and L^(R+) be the space of all real-valued measurable functions, such that

/8 \i/p \\f \\lp = ( 5 \f (x)\Px^ dxJ < 8. If xe is the indicator of a set E c R+

and fXE e LP(R+), then f e LP(E).

The Bessel-Clifford function of the first kind of order ¡i ^ 0 (see, e.g., [5]) is defined by

+8 (—i)kxk m» + k +1) ■ * s 0'

where by r(a) we have denoted the Euler gamma function. It is known that c^(x) is a solution of the differential equation xy" + (i + 1)y' + y " 0.

If jv (x) is the normalized Bessel function of the first kind and order v > —1/2, given by

8 (—1\n »{X)" r( "+ ^ Wn+ l + 1)(l/2)2"'

n=0 x '

then c^ and are connected by

c^(x) " r—1(i + 1) j„(2?x), x > 0. (2)

Hayek [6] introduced the second Hankel-Clifford transform for feL}l(R+) by

+ 8

hvAf)(y) " $ (yx) f(x)x^ dx. 0

By Lemma 2 below and (2), we have \c^(x)\ ^ r—1 (i + 1) on R+. As a corollary, we obtain

\\h2Af)\\l8 ^ r—1(i + 1)\\f\\L., f e Ljl(R+). (3)

For i ^ 0, the transform h2,^ extends from Ljl(R+) x Lfl(R+) onto

Ll(R+) and

\ \ h2,,(f)\\l2 "\ \f\\Ll, feLl(R+). (4)

This Plancherel-type equality can be found in [6] or in [9]. Using Riesz-Thorin interpolation theorem (see [2, Ch. 1, Theorem 1.1.1]), we obtain a Hausdorff-Young type inequality

\ \ h2,,( f)\\Ll <C\\f\\lI, feLl(R+), (5)

where 1 < p ^ 2 and q " p/(p — 1) as in (1).

Let A(x, y, z) " (p(p — x)(p — y)(p — z))1{2, where p " (x + y + z)/2, be the area of the triangle with sides x, y, z. For ¡i ^ 0, set

A2^ 1(x,y,z)

^(x,y,z) 22»(xyz yr(p + 1 ^

when the triangle with sides x, y, z exists, and D^(x,y,z) = 0 in other cases. Then D^(x,y,z) is non-negative and symmetric in x, y, z. In [12], Prasad, Singh, and Dixit suggested the generalized Hankel-Clifford translation of f e Ljl(R+) as follows:

Tx(f)(y) = J f(z)Dß(x,y,z)zßdz , 0 < x,y <8. 0

Using Lemma 1.3 from [12], we have, for f e Lji(R+):

h2,,(Tx(f))(y) = c,(xy)h2M)(y), y > 0. (6)

By Lemma 2.3 in [16], this result is also valid for f e L^(R+), 1 < p ^ 2, a.e. on R+.

Now we introduce the difference of order m e N with step t > 0 by

(x) = (i - r(ß + rnrf (x) ^(-i)^ 7 )r> +

i-0

m / \

y

where I is the identical operator, and the modulus of smoothness of order m in L^(R+), 1 ^ p < 8, by

(f,6)p^hc = sup \\&™ß>hJ\\Lp.

Due to Lemma 1, for 1 ^ p < 8, ß ^ 0, and ö ^ 0, we have wm(f., ¿>)I,,ß,hc ^ c\\f\\LP.

Let S(0, +8) be the set of all infinitely differentiable functions ip(x) defined on (0, +8), such that

pm.k(1) = sup \xm^pk)(x)\ < 8

0<X<8

for all m,k e Z+. In [9] it is proved that h2,ß is an automorphism of S(0, +8). Also, in [9, Proposition 6] it is established that for differential operator Bß(1) = xip" + (ß + 1)1 and 1 e S(0, + 8) the equality

h^B(ip))(y) = (-y)ih2,ß(ip)(y), y> 0, ze N, (7)

holds (see also (1.14) in [12]). For i ^ 0, 1 ^ p <8, and m e N, we define the Sobolev space W^(R+) consisting of f e LP^(R+), such that f, f, ..., f(2m^1q are absolutely continuous on each segment from (0, +8) and B* (f) e L?(R+), i = 1, 2,...,m.

Also, we can consider the space S,e(R+) as the space of pX[0,+8q, where p are even Schwartz functions. Then ) c 5(0, 8) and S,e(R+) is

dense in all L^(R+), 1 ^ p <8. Using the usual density arguments, we state that (7) is valid for ^ e W^(R+) and i = 1, 2,... ,m.

Denote by $ the set of continuous and increasing on R+ = [0, 8)

functions u, such that u(0) = 0. If u e $ and $ t^1u(t) dt = 0(u(8)),

0

8

Xm f +—m—1,

8^ 0, then u belongs to the Bary class B; if u e $ and 8m § t~m~1u(t) dt =

= 0(u(8)) for some m > 0 and all 8 > 0, then u belongs to the Bary-Stechkin class Bm (see [1]). We say that u e $ satisfies the A2-condition (u e A2), if u(2x) ^ Cu(x), x e R+.

3. Auxiliary propositions. Lemma 1. Let 1 ^p < 8, i ^ 0, f e LP(R+). Then

\\T(p + 1)Ttf\\Lfi \ f\\Ll.

The proof of Lemma 1 belongs to Prasad and Singh [13, Lemma 1.1]. Lemma 2. Let ß ^ 0. Then

(i) \ jß(x)\ ^ 1 for x ^ 0 and jß(x) < 1 for x > 0;

(ii) 1 - jß(x) ^ C > 0 forx ^ 1;

(iii) the double inequality C\x2 ^ 1 — jß(x) ^ C2x2 is valid for some C2 > C1 > 0 and all x e [0,1].

Proof. For (i) and (ii), see papers by Platonov [11] and [10, Lemma 3.3]. The assertion of (iii) see, e.g., in [17]. □

From (6), (7), (2), and using induction, we deduce

Lemma 3. Let 1 ^ p ^ 2, ß ^ 0, f e LP(R+), m e N, t^ 0. Then

f)(y) = (1 — J,(2?yt)mh2A f)(y) for a.e. ye R+.

For k e N and f e Wkß(R+), we have:

h^A^Bj(f))(v) = (1 — J»(2Vyt)m(—y)kh2Af)(y) for a.e. y e R+.

Lemma 4 can be found in [16].

Lemma 4. Let ß ^ 0, m > 0, u p Bm, and Git) be a non-negative

measurable function on R+, such that

>

G(t)rdt = 0(u(l/y)), y > 0.

Then tmG(t) is integrable on each segment \a, b] c R+ and i

tmG(t)r dt = 0(ymu(l/y)), y > 0.

Lemma 5 is proved in [1].

Lemma 5. Let u e $ and m e N. Then the conditions (i) u e Bm; and (ii) there exists a e (0,m), such that for all 0 < u ^ v <8 the inequality u(v)/vm~a ^ Cu(u)/um~a holds; are equivalent. In particular, if u e Bm, then u satisfies the A2-condition.

4. Main results. Theorem 3 is an analogue and an extension of Theorem 2.

Theorem 3. Let p > 0, 1 < p ^ 2, \/p + 1/q = 1, m e N, k e Z+, f e L?(R+) fork = 0 or f e W^(R+) for k e N. If 0 <r ^q and a e R, then for all N > 0 we have:

ya\h2,,( f)(y)\ry»dy^C J r-^+^u^ B (f), t-1)p^dt.

N N/2

Proof. By Lemma 3 and Hausdorff-Young inequality (5), we have

\h2,ß(f)(y)\qykq(1 - jß{2Vyt))mqyßdy ^

^ c-iW^B(f)\\* ^ (Bkß(f),tu,hc.

ly , Pi+1 ó ^ ^ +. — 0-iAT-1

Lemma 2 (ii), we find that

Let N > 0 and A = [2lN, 2l+1N), i e Z+, U = 2~lNThen, by

\h2,ß( f)(y)Wdy ^ C2(2iN)-kq<4> B (f), UU,hc

Di

+

By the Holder inequality, for 0 < r < q we obtain:

ya\h2A f)(y)\r ykd y^

Di

(J yaq/{q-r)+k dy) x~r,q (J \ h2,,( f)(y)\q yk dy) r'q ^

Di Di

^ C3(2iNy+^+m-rM^N)-krumB(f), 2-N-1)p^hc ^

2lN

j urm (f, t-1)Pi,hhcta-kr-Pk+1)r/q+kdt. (8)

2i-1N

For = , we see that

ya\h2Af)(y)\qykdy ^ C5(2Nr-k«um(Bk(f), 2-iN-1)p^hc ^

Di

2lN

B k

ul (Bk (f), t-1)p,,Mta-kq-1 dt. (9)

2i-1N

Summing up (9) or (8) over i = 0,1,..., we obtain

ya\h2,k (f)(y)\rykdy^C7 ta-kr-Pk+1)r/iul (Bk (f), t-1)PkhJk dt.

l k

N N/2

□

Corollary 1. Let 1 < p ^ 2, q = p/(p — 1), u e A2, m,k e N, f e WpkAR+) and ui(Bk(f),5)p^hc = 0(u(5)), 5 ^ 0. Then

8

\h2,k(f)(y)\qykdy^Cu(N-1, N > 0. (10)

N

Proof. By Theorem 3, we have for a = 0 and r = q:

\h2,k( f)(y)\qykdy^C1u1 (Bk (f),2/N )p,k,hc t-k-1dt^

i\ kv

N N/2

tC^N-l, N > 0,

^ 2 Nki

due to the condition u e A2. □

Remark. It is interesting to compare Corollary 1 with Theorem 2.1 in [7], where a similar to (10) estimate with instead of h2, ^ is obtained. It seems that a factor N-2kq in the analogue of (10) in [7] is not proper.

Now we can obtain a variant of Theorem 1 (or Titchmarsh equivalence theorem).

Theorem 4. Let p > 0, f e Lfl(R), m e N and u2 e B x B2m. Then the conditions (i) um(f,S)p,^,hc = 0(u(6)), 6 ^ 0;

8

(ii) $\h2Af)(y)Wdy = 0(u2(N-1)), N > 0,

N

and

2 N

(iii) f \h2,,(f)(y)Wdy = 0(u2(N-1)), N > 0,

are equivalent.

N

Proof. Let (i) be valid. By Theorem 3 in the case a = k = 0, r = p = 2, we obtain

\h2Af)(y)Udy^C\ J t-»-1 u2(t-1)t»dt =

N N/2

2 { N

jj ~t

0

22

= Ci\ j2P)dt^CiJ2(2/N) ^C2J2(N-1),

since u e B and, by Lemma 5, u satisfies A2-condition. Thus, we prove (i) ^ (ii) ^ (iii).

Conversely, let (iii) be true. Then

2i+1N

8

\h2A f)(y)Wdy = 2 \h2A f)(y)\Ydy^

N 2iN

1/(2i~1N )

8 8 r

Yi^PP^N )-1) = C4u2(N-1) + CA Y, 2N u2(t)dt^

" " mm)

1/(2i~1N ) 8 r 2(+\

^ 2C^u2(N-1) + y ^dt^CU (N-1), N > 0,

%"1 1/{2iN )

by the condition u2 e B, i.e., (iii) ^ (ii).

By Lemma 3 and Plancherel-type equality (4), we have:

I I A^c f IIh = f \h2Af)(y)\2(l - Jß(2?yt))2my^dy =

R+

1/{4t) 8

" (J ^ )\h2Af)(y)\2(l - Jß(2?yt))2my»dy = h(t) + h(t).

° i/№

By Lemma 2 (i), Lemma 5, and condition (ii) of the Theorem, we obtain:

8

h(t) < 22m J \h2jf)(y)\Ydy^C6u2(4t) ^C7u2(t). (11)

1/(4t)

On the other hand, by Lemma 2 (iii):

1/(4t)

h(t) <C>j \h2M)(y)\2(yt)2myßdy. °

But by (11), the condition u2 e B2m, and Lemma 4, we find that

1/(4t)

y2m\h2Af)(y)\Ydy ^ C8(1/(41))2mu2(4t)

and I\{t) ^ C9u(t), t > 0, by Lemma 5. From the last inequality and (11), we deduce that | | A™>ÄCf||L2 ^ (C7 + Cg)1{2u(t), t > 0, and (ii) ^ (i) follows. Theorem 4 is proved. □

Acknowledgments. This work was supported by the Program of development of Regional Scientific and Educational Mathematical Center "Mathematics of Future Technologies" (project no. 075-02-2023-949). The author thanks the anonymous referee for valuable remarks.

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Received February 27, 2024. In revised form, May 10 , 2024. Accepted May 31, 2024. Published online June 14, 2024.

S. S. Volosivets Saratov State University 83 Astrakhanskaya St., Saratov 410012, Russia E-mail: VolosivetsSS@mail.ru