CC BY
5DOI: 10.15393/j3.art.2024.14330
UDC 517.544
Yu. I. Krotüva
INTEGRABILITY OF q-BESSEL FOURIER TRANSFORMS WITH GOGOLADZE-MESKHIA TYPE WEIGHTS
Abstract. In the paper, we consider the g-integrability of functions \(t)\Fq>v(f)(t)\r, where A(i) is a Gogoladze-Meskhia-Moricz type weight and FqyV(f)(t) is the g-Bessel Fourier transforms of a function f from generalized integral Lipschitz classes. There are some corollaries for power type and constant weights, which are analogues of classical results of Titchmarsh et al. Also, a ^-analogue of the famous Herz theorem is proved.
Key words: q-Bessel Fourier transform, q-Bessel translation, modulus of smoothness, weights of Gogoladze-Meskhia type, q-Besov space
2020 Mathematical Subject Classification: 44Á15, 47Á10
1. Introduction. Let f: R ñ C be an integrable function in Lebesgue's sense over R (f e L1(R)). Then the Fourier transform of f is defined by
by
f (x) = f (t)e~itx dt, x e R.
R
In the case 1 < p ^ 2, we define f (x) as a limit in Lq(R), 1/p + 1/q = 1,
f(x) = (Lq) lim (2n)-1/2 f (x)e-Ux dt. «ñ — 8,
ÍÑ + 8
In particular, f e Lq(R) and the following Hausdorff-Young type inequality proved by Titchmarsh (see [16, Ch. IV, Theorem 74])
||/||q ^ C\\f \\p := C ( i\f (t)\p dt)1/P, f e LP(R), 1 < p ^ 2, (1)
© Petrozavodsk State University, 2024
holds. For p = 2 the inequality in (1) is substituted by the Plancherel equality. More about these results can be found in [16, Ch. III and IV] or [3, Ch. 5].
For f e LP(R), 1 ^ p < 8, we consider the modulus of smoothness of order k e N:
u;k(t, S)p = sup \\Akhf \\p, Akhf (x) = f(-1)^k)f (* + {k — 2j)h/2).
•> = 0 VJ/
_ .. \\Vi . . ,_. . . ,
The following result of Titchmarsh is well known (see [16, Ch. 4, Theorem 84]):
Theorem 1. Let 1 < p ^ 2, 0 < a ^ 1, f e Lip(a,p). Then f (t) e L13(R) for all ft satisfying the inequality
P ^ a ^ P < ft < Q =
p + ap — 1 p — 1
Unfortunately, in [16] and in many papers where Theorem 1 is presented there is no information that this theorem is an analogue of Szasz's results for trigonometric series (see, e.g., [15] and [12]- [14] in literature from this paper).
We will write that a non-negative measurable function X(t) e Ljoc(R+) belongs to the class A7, j ^ 1, if there exists C(j) ^ 1, such that
2 i+l 2i ( J A7(t)dt^1/7 ^ C(T)2*(1-7)/7 J \(t)dt, i e Z. (2)
2 i 2i~1
It is clear that a measurable function X(t) ^ 0 with the property
sup{A(i): 2* ^ t < 2i+1} ^ cinf{\(t): 2i_1 ^ t < 2*}, i e Z.
is contained in all classes A7, j ^ 1. From the last assertion, we deduce that X(t) = ta, a e R, belongs to all classes A7, j ^ 1. Further we assume that \(t) = X(—t) for t > 0.
An analogue of (2) for sequences was introduced by Gogoladze and Meskhia [9]. The condition (2) was suggested by Moricz [13] who proved the following result:
Theorem 2. Let 1 < p ^ 2 and f e LP(R). If 1/p + 1/q = 1, 0 < r < q,
and X e Ap/pp—rp+rq, then
8
m\rn\r dt ^ J xaqt-rfi^ (f, n/tqp dt. \t\>2 1
The aim of the present paper is to obtain an analogue and a generalization of Theorem 2 for the g-Bessel Fourier transform. Some analogues of Theorem 1 for Fourier-Bessel (or Hankel) transform proved by Platonov [14]. An analogue of Theorem 1 for g-Fourier-Dunkl transforms can be found in [5]. Analogues and extensions of Theorem 2 for Fourier-Dunkl transforms are proved by Volosivets [17], while for the first Hankel-Clifford transform see [18]. The main result of the present paper and its corollaries is similar to that of [17] and [18], but the subject of the present paper is discrete and methods are different from used in the cited papers. Also, we obtain an analogue of the famous Herz theorem (see Corollary 4).
2. Definitions and lemmas. Let 0 < q < 1, v >—1, and R+ = {qn: n e Z}. For a e C and n e N = {1, 2,... }, set
n—1 8
(a; q)o = (a; q)n = — aql), (a; ^ = Y\(1 — a<i%).
Let us introduce the ^-integral of Jackson for f defined on R+ on intervals from 0 to a e R+ an from 0 to 8, as follows:
n. 8
8
f (x) dqx " (1 — q)a 2 f (aqn)qn; f (x) dqx = (1 — q) ^ qnf (qn).
o n-0 0 neZ
Then, for 0 < a < b, a,b e R+, set
b b a
f (x) dqX = J f (x) dqX — J f (x) dqX.
a 0 0
For such integral, there is the following simple change of variables formula:
b b/r
jh(x/r)x2u+1dqx = r2u+2 J h(t)t2v+1dgt, r e R+, U > — 1, (3)
a a/
(see [1]). A more general variant is in [11, (19.14)].
The third Jackson g-Bessel function Jv (also called Hahn-Exton g-Bessel function) is defined by
(qn+i. 8 qn(n+1)/2
JV{X] ^ " l^f (q»+1, q)n(q, q)n^
n=0
We consider also its normalized form:
(q, q)8 _„t, , 8, ^ qn[n+1)/2
j" q) Pr+1 ,q)8X~VJv q) 1)ra q)n(q; q)^
n=0
These functions satisfy the orthogonality condition:
8
r q-2n(u+1)
cl,v jv(<fx; Q2)jv(Qmx; q2)x2u+1 dqx = -5nm,
0
where 5nm is the Kronecker symbol,
cq,„ = ((1 — q)(q2; q2)8)-1(q2(v+1); q2)»
(see [12]). Further we write dfiq,v(x) instead of x2v+1 dqx. Let Aq,v f (x) be the g-Bessel operator defined by
f (x) = x-2 [f(q-1x) — (1 + q2v )f(x) + q2v f (qx)].
The function jv (Ax; q) is a solution of the following difference equation:
aq,uf (x) = -\2 f (x).
For 1 ^ p <8, denote by Lpq v the space of all real-valued functions f defined on R+ with finite norm
f\\p,q,v = ( \f(x)\P dßq,u (x))
1/p
If xe is the indicator of a set E and ¡(x)xe(%) e Lpqv, then we write f e Lpq v (E). The space consists of all bounded on R+ functions and is supplied by the usual sup-norm.
Define the g-Bessel Fourier transform Tq>v(f) for f e Lp v, p ^ 1, by
8
(f)(x) cq, v f(t)jv (xt, q2) dßq,v (t). o
Also, the g-Bessel translation operator is introduced by
8
Tq,xif )(y) = Cy,v jFq,V{f ){t)jv{xt,q2)jv(yt, Cj2) d^q,v(t). 0
It is known that for f e L^ the equality
8
)(y) dq,v(y) = J f (y) dq,u(y) (4)
0
holds (see [6, Proposition 5.2]). In [8], the problem of positivity of the operator Tqx was discussed by Fitouhi and Dhaouadi. If from f ^ 0 on R+ it follows that Tqqn f ^ 0 for all n e Z, then T^x is called positive. For v ^ 0 and all q e (0,1), operators T^q„, n e Z, are positive, while for — 1 < v < 0 the situation is more complicated. By virtue of this fact, we consider v ^ 0 further.
In [8], it is proved that \\T^xf ^ \\f }i,q,v for a positive operator Tq x and f e Lq v. On the other hand, Dhaouadi [7] proved that
\\Tq,xf }8,q,v ^ \\f }8,q,v , f e ^ q,v ,
and, by the Riesz-Thorin interpolation theorem, concluded that (see [7, Theorem 4])
\\TqJ\\p,q,u < \\f \\p,q,u, 1 ^ P < 8, f e L^. (5)
Note that
\jv(x,q2)\ ^ 1, x e R+, u > 0, (6)
(see [7, Remark 1]). In [1] it is proved that the inequality (6) is strong for x e R+. From (6), the inequality
\ \ —q,v (f ^\q,q,v ^ Cq,v\\f\\\,q,u (7)
easily follows for f e Lqqv.
For a > 0, m > 0, 6 ^ 1, and p ^ 1, consider the g-Besov space B^'^™ consisting of all f e v, such that
q/9
\ f \\b = \\ f \\P'q,u + ( ka^^(f,(ikW)6) < 8.
keZ
Let Mk = [(1/q)k, (1/q)k+l\ The Herz space K(a,9,p,q, u) contains all function f on R+, such that jXMk p Lp^ v for all k p Z and
( Y^ k 0 \1/d
\I\\k = \\f\\K(a,e,p,q, u) = \\fXMk\\p,q,v) < 8
keZ
Here XE is the indicator of a set E. Lemma 1.
(i) Let f e LptV, p > 1. Then T2q>v(f)(x) = Tq,v(Tg,vf)(x) = f(x), x e R+;
(ii) If f e Lpqv, then Tq)V(f) e and \\Fq,u(f)\\ 2,q,v \\J \\2,q,v;
(iii) For any
e Lpq v, p — 1, 2, the equality holds:
(TvqJ)(y)= JV(yx, q2)Tq,v(f)(y), y,x e R+.
Proof. The statement (i) of Lemma 1 is proved in [6, Theorem 3.2], while (ii) is established in [7, Theorem 3]. The part (iii) is proved in [1] in the case p = 2, but in other cases the proof is the same. □
From (7) and Lemma 1 (ii) by Riesz-Thorin, a theorem follows:
Lemma 2. Let 1 ^ p ^ 2, 1/p + 1/p' = 1, f e L%tV. Then Tq)V(f) e Lpq u and
\\FI,v(f)\\p',g,v ^ C\\f\\p,q,v. (8)
Since in (5) the constant in the right-hand side is equal to 1, for m > 0 the difference of order m with step h may be defined by
j-o v ^
K^n = (!- Tlh)m " D(-1)'' W
where I is an identical operator and
/m\ m(m — 1)... (m — j + 1) /m\
J -m—. -jeN- U) -1
It is known that Xlj" I^j) I <8 for m > 0 (see, e.g., [4]), therefore, by (5) for f eUq v one has A™^/ e Uq v and
f 1, (m)m
From Lemma 1 (iii), it follows that for f e U^ v, p = 1, 2,
)(x) = (1 — Ju(yx,q2))mTq(f)(x), x e R+. (10)
The next Lemma can be found in [1]: Lemma 3. There exist there exist > 0, such that
a ^ \jv(t,q2) — 1\, t e R+ x [1, +8),
Vt2 < \jv(t,q2) — 1| ^ Pt2, t e R+ x (0,1]. We define the modulus of smootheness of order m > 0 for f e Lp by
^m(f,^)p,q,u — sup \\Aq,v,hf\\p,q,v 0<h<S
We will write X(t) e A1t gtV, 7 ^ 1, if for i e Z the inequality
(i/#+1 pi/qy
( I A1(t) (t))1/1 ^ Cq-(2v+2)(1/l-1) I X(t) (t)
(1/g)* (1/q)i
1
holds. In Lemma 4 below, it is provided that \(t) = ta, a e R, satisfies this condition.
3. Main results.
Theorem 3. Let 1 < p ^ 2, 1 + -1 = 1, v ^ 0, f e Lp v, m > 0. If
Jr ^ ? p p' ' ^ ' J q,ui
X e Ap/(p_pr+r),q,u = Ap'/(p>-r),q,u for some r e (0/p'j, X e Lp'/(p'-r)[0,1] and the integral
x(t)t-(2v+2)/p' (f,t-1)p, q,u (t)
converges, then X(t)\TqiV(f)(t)\r e Lqu.
Proof. Let Mi = [(1/qf, (1/q)m], i e Z+. By Lemma 3, (10) and Hausolorf-Young type inequality (8), we have
C1 J (f )(y)\p dfr.v (V) <
Mi
^ J (f)(y)\p'(1 -3u(qly,q2))mp'diq,v(y) ^
Mi
8
,u (f)(y)\P' (1 - jv (q1 V, Q2))mp'd^ (y) = \\Tq ,v (A^, ¿i f)\^
0
^ \A£„,(f)\^ (f,q%tqiV.
By the Holder inequality and the condition À p Ap/pp-r),q,v, we have for 0 < r < p':
À(t)\Tq,v(f)(t)\rdnq,v(t) ^
,v (J )(°)\ UjHJ q, v
Mi
^ (J \À(t)\P'/PP'd»q,v (t))l~r'P' X (J \Tq,v (f)(t)\p'dl,q,v (t)) r/P ^
Mi M,
Vi-1
^Csurm ( f,q%,q,v q-^+2)/p' | À(t)diq, v(t). (11)
By definition,
L
8 8 g(t) diq,v(t) = (1 -q)(qn-i-1 £ g(qn--1) - qn- £ ^-))
n—0 n—0
q-
88
"(1 -l){ H - E )qng(qn) = (1 -q)q-l-19(q-1-1). (12)
n - - 1 n -
Thus, (11) may be rewritten as
À(t)um( f, 1/t)
À(t)\Tq,v(f)(t)\r dig,v(t) ^ j ^ (t). (13)
Mi Mi-1
Summing up (13) over i = 0,1, 2,..., we obtain
8 8
$À(t)\Tq,vumrd^v(t) ^ c4j À(t)u$i ,q,v di9tV(t). (14) 11{
If À e LpqlvV r)[0,1], then \{t) e L1[l/q, 1], and the integral in the right-hand side of (14) is finite, since t-1 and ojm(t, 1/t) are bounded on [1/q, 1]. Finally, by the condition A e LPq[*q'-r)[0,1] and (8):
1 8
if / \ r!p' q,v )p,)}r d^* (t) ^ \Tg * (f ){t)\p d^* (t)\ x
(f)(t)\r d^,* (t) ^ (J ^ umf d»q,v pt)) 0
1
i j> (t)\pf/(pf-r) dN,v (tj)
X I I (t)f/(p' r) dHgiV(t)] < 8.
0
The proof is completed. □
We give two auxiliary statements.
Lemma 4. A function Xa(t) = ta, a e R, belongs to any class A7, 1.
Proof. By (12), we have h = ( [ t^d^(t))lh = ((1 -q)q--l(q--l)»i+2v+l)l/i =
"(1 -q)l/lq -
q,*{ Mi
1/l n-(i+1)('*+(2v+2)H
while
h = j tad^,v(t) = (1 — q)q-V+^v+V,
Mi-!
and we obtain h ^ C(q,u, j)q-(2v+mh-l)i2. □
Lemma 5. A function \a(t) = ta, a e R, belongs to Lpq[i(p'-r)[0,1], 0 < r < p', if and only if a > -(1 — r/p')(2u + 2).
Proof. By definition, we obtain that the integral
1 1
>rl" *{b) ~ I ° ^q,*
tap//pp'-)dfIq* (t) = J t»p,/{p'-r) + 2*+1d^v {t) = 0
8
" (1 _ l) Y (ln{ap'/Pp-T)+2*+2)
n=0
converges if and only if ap'/(p' — r) + 2u + 2 > 0 or a > —(2u + 2)(1 — r/p'). □
Now we obtain some consequences of the main result.
Corollary 1. Let 1 < p ^ 2, m > 0, 1/p + 1/p' = 1, v ^ 0, a e R, f e LP , and r e (0,p'). If a > (r/p' — 1)(2u + 2) and the integral
ta-r(2"+2)/p'^rm(f, t-1)p,q,vdliqv(t) (15)
converges, then talTq,v(f)(t)Ir e L1q^.
Corollary 2. Let p, p', m, v and r be as in Corollary 1, and um( f, S)p,q,v = 0(6 ?), ft > 0. If a > (r/p' — 1)(2u + 2) and
1 p'(a + 2 !/ + 2)
p > r > —-, (16)
1 2 v + 2 +p'ft ' v '
then talTq>v(f)(t)YeL\tV.
Proof. Under conditions of Corollary 2, the integral (15) converges if a — r(2u + 2)/p' — rft + 2u + 1 < —1 and this inequality is equivalent to (16). □
If a = 0, then the case r = q' is also admissible. Corollary 3 is an analogue of the Titchmarsh result.
Corollary 3. Let p, p', m, u, and r be as in Corollary 1 and
m(f, 5)p>q>v = 0(8^),ft > 0. If
(2 v + 2)p' (2 v + 2)p
p ^ r> -
2 v + 2 +p'ft p(2u + 2 + ft ) — (2u + 2)
then (f) e Lrq v.
Now we state some estimates for the -Bessel Fourier transform from -Besov space.
Theorem 4. Let a > 0, 1 <p ^ 2, m > 0, and 9 ^ 1. If f e B£Q;™, then
8
8 C Q! 1
2(i-kQa{2 J i^(f)(y)ip') P <8. (17)
kpPZ i^^k ^^^
Proof. In the proof of Theorem 3, it is established that for i p Z
(f)(y)\p' (1 - 3u (Qi y, Q2 ))mp'd^ (y) ^ ( f,f)
0
Therefore,
I\\% (Q-kaum( f, qk)p,q,u) >
keZ
8 d/p' C-e £ (\Tqv (f)(y)f (i - jv (qky, q2))mP' d/lqu (y)\ * >
k<= z V J /
d c-kCa\ | \T t f\(„,\\p' keZ o
8 8/p' >C- 2 Q-kea( IV* u (f)(y)\p> (1 - ju (qky, q2))mp'd^,v (y)\ * .
keZ
But by Lemma 3 for y ^ q k the inequality 1 — ju (qky, q ) ^ C2 > 0 holds and one has
8
/ r , \ e/p'
\\ f \\% q-kda[ \Tq,v(f)(y)\p d^u(y)j , (18)
keZ
q k
that is equivalent to (17). □
Corollary 4 is an analogue of the famous theorem of Herz for the classical Fourier transform (see [10]). The proof in our case is simpler.
Corollary 4. Let a > 0, 1 <p ^ 2, m > 0, and 0 ^ 1. If f p B^JJ1, then TqtV(f) pK(a,6,p,q, v) and \\Tq,v(f)\\K(a,e,p,q,u) ^ C\\f \\b.
Proof. The result follows from (18) and the obvious inequality
q-k-1
\Fq,v(f)(y)\p' dnqtu(y) > j \Fq,v(f)(y)\p> d/iq,u(y).
rf k ff k
The proof is completed. □
8
q k
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Received August 19, 2023. In revised form, November 30, 2023. Accepted February 15, 2024. Published online February 23, 2024.
Saratov State University
83 Astrakhanskaya St., Saratov 410012, Russia
E-mail: julia.krotova.sgu@gmail.com
