CC BY
2Probl. Anal. Issues Anal. Vol. 13 (31), No3, 2024, pp. 3-22
DOI: 10.15393/j3.art.2024.16330
3
UDC 517.44, 517.983
A. Chana, A. Akhlidj
UNCERTAINTY PRINCIPLES AND CALDERON'S FORMULAS FOR THE DEFORMED HANKEL ¿^-MULTIPLIER OPERATORS
Abstract. The main purpose of this paper is to introduce the deformed Hankel L^-multiplier operators and to give some new results related to these operators as Plancherel's, Calderon's reproducing formulas and Heisenberg's, Donoho-Stark's uncertainty principles. Next, using the theory of reproducing kernels, we give best estimates and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces. Key words: deformed Hankel transform, Calderon's reproducing formulas, extremal functions, Heisenberg's uncertainty principle, Donoho-Stark's uncertainty principle
2020 Mathematical Subject Classification: 42B10, 47G30, 47B10.
1. Introduction. In their seminal papers, Hormander and Mikhlin [5], [12] initiated the study of boundedness of the translation-invariant operators on Rd. The translation-invariant operators on Rd are characterized using the classical Euclidean Fourier transform T(f), therefore they also known as Fourier multipliers. Given a measurable function
m: Rd C,
its Fourier multiplier is the linear map Tm given for all A e Rd by the relation
T(Tm(f ))(A)= m(X)T (f)(A). (1)
The Hoormander-Mikhlin fundamental condition gives a criterion for Lp-boundedness for all 1 < p <8 of Fourier multipliers Tm in terms of derivatives of the symbol m; more precisely, if
" d'
\dyxm(X}\ < |A|~'7' for 0 ^ j7| ^
2
+ 1,
© Petrozavodsk State University, 2024
then Tm can be extended to a bounded linear operator from Lp(Rd) into itself.
The condition (2) imposes m to be a bounded function, smooth over Rd\{0}, and satisfying certain local and asymptotic behavior. Locally, m admits a singularity at 0 with a mild control of derivatives around it up to order [|] + 1. This singularity links to deep concepts in harmonic analysis and justifies the key role of Hormander-Mikhlin theorem in the Fourier multiplier Lp-theory: this condition defines a large class of Fourier multipliers including Riesz transforms and Littelwood-Paley partitions of unity, which are crucial in Fourier summability or Pseudo-differential operator. The boundedness of Fourier multipliers is useful to solve problems in the area of mathematical analysis and Probability theory: see [11], for stochastic processes see [2], and for the study of nonlinear partial differential equations see [8].
The general theory of reproducing kernels has started with Aronszajn's in [1] in 1950, then the authors in [10], [14], [15] applied this theory to study Tikhonov regularization problem and they obtained approximate solutions for bounded linear operator equations on Hilbert spaces with the viewpoint of numerical solutions by computers. This theory has gained considerable interest in various field of mathematical sciences, especially in Engineering and numerical experiments using computers, see [10], [15]. This paper focuses on the generalized Fourier transform associated with the Dunkl-Laplace operator called the Deformed Hankel transform. More precisely, we consider the following differential operator: Aa defined for
a > x by
B2 2a B a
+ T & - (1 -s>•
where S(f)(x) =: f (—x).
The operator Aa is closely connected with the Dunkl theory, see [4], [1], [6], [17], furthermore, the eigenfunctions of this operator are related to Bessel functions and they satisfies a product formula, which permits to develop a new harmonic analysis associated with this operator; see [17] for more information.
The deformed Hankel transform Ta generalizes the usual Fourier transform T and is defined on L^ (R) by
Fa(f)(A) = ¡Ba(\x)f (x)d^a(x), for A 6 R,
where is the measure on R and Ba{\.) is the Bessel kernel given later. Let a be a function in L—(R) and ¡3 > 0 be a positive real number. The deformed Hankel L—-multiplier operators [8] is defined for smooth function on R as
TaJi(f)(x) := T—1 (apTa(f))(x), (3)
where the function ap is given by
ap(A) := ). (4)
These operators are a generalization of the classical multiplier operators given by the relation (1). The remainder of this paper is arranged as follows: in section 2 we recall the main results concerning the harmonic analysis associated with the deformed Hankel transform, in section 3, we introduce the deformed Hankel ¿--multiplier operators Ta, p and we give for them a Plancherel's point- wise reproducing formulas and Heisenberg's, Donoho-Stark's uncertainty principles. The last section of this paper is devoted to giving an application of the general theory of reproducing kernels to Fourier multiplier theory and to give the best estimates and an integral representation of the extremal functions related to the deformed Hankel ¿--multiplier operators on weighted Sobolev spaces.
2. Harmonic Analysis Associated with the Deformed Hankel Transform. In this section, we set some notation and recall some results in harmonic analysis related to the deformed Hankel transform; for more details, see [9], [13], [17], [18].
• For a > , is the weighted Lebesgue measure defined on R by
1 ^,
d^a(x) := 2T(2a) ,
where r is the Gamma function.
• L— (R), 1 ^ p ^ 8 is the space of measurable functions on R, satisfying
i/p
\f\
K \f (x)\pd^a(x)\ <8, 1 ^ p <8,
R J
ess sup \f (x)\ < 8, P = 8.
xeR
In particular, L—(R) is a Hilbert space with inner product given by
(f,g)a = f (x)g(x)d^a (Xq.
2.1 The Eigenfunctions of the Dunkl-Laplace operator Aa. For
A e R, consider the following Cauchy problem:
(S):
Aa(u)(x) = - ^ u(x), u(0) = 1.
x
From [9], [13], [17], the Cauchy problem (S) admits a unique solution Ba(X.) given by
Ba(\x) = j2«-i(2VlA^) - 1qj2a+i(2V|A^), (5)
where ja denotes the normalized Bessel function of order a, see [19] for more information about this function. The function Ba(X.) is infinitely differentiable on R and we have the following important result:
VA,x e R, lBa(Xx)l ^ 1. (6)
Furthermore, from [17], the deformed Hankel kernel (5) is multiplicative on R in the sense
VA e R,x,y e R* Ba(Xx)Ba(Xy) = [Ba(Xz)Ka(x,y, z)d»a(z), (7)
where Ka is the Bessel kernel given explicitly in [17], [18]. The function Ka(x,y,z) is unchanged by permutation of the three variables and there exists a constant Aa independent of x,y, such that
lKa(x,y,z)| dya(z) ^ Aa. (8)
From [17], the product formula (7) permits to define a translation operator, convolution product, and to develop a new harmonic analysis associated to the Dunkl-Laplace operator Aa.
2.2 The Deformed Hankel transform.
Definition 1. The deformed Hankel transform Ta is defined on L^(R) by Mf)(A) = f Ba(Xx)f (x)d^a(x), for X e R.
Some basic properties of this transform are as follows: (find the proofs in [9], [13], [17], [18]).
Proposition 1.
(1) For every f e L^R), we have
\\Mf)\L,»a ^ \\f111,*.. (9)
(2) (Inversion formula). For f e (Lla x L2a)(R), such that Ta(f) e Lla(R), we have
f (x)= f Ba(Xx)Ta(f )(X)d^a(X), a.e. x e R. (10)
(3) (The Parseval formula). For all f,g e L2(R) we have:
(f,g)a =(Ta(f), Ta(g))a , (11)
In particular, we have
\ \ f \ \ 2,^« = \ \ ?y(f ) \ \ 2,/ua . (12)
(4) (The Plancherel theorem). The deformed Hankel transform Ta can be extended to an isometric isomorphism from £2(R) into L2'
2.3 The translation operator associated with the deformed Hankel transform. The product formula (7) permits to define the translation operator as follows:
Definition 2. Let x,y e R and f be a measurable function on R; the translation operator is defined by
Taf (y) = f (z)Ka(x,y,z)d^a(z).
The following proposition summarizes some properties of the deformed Hankel translation operator see [13], [17].
Proposition 2. For all x,y e R, we have:
<f (y) = t* f (x). (13)
JrZf (y)d^q(y) = f f (y)d^q(y). (14)
R R
(iii) for f e Lq(R) with p e [1; +8] r*f e Lq(R) and we have
WTaf }p,na ^ Aa\\f , (15)
where Aa is the constant given in (8).
(iv) For f e Lla(R), r*f e Lla(R) and we have
Tq (Txa f) (X) = Ba(Xx)Ta(f)(X), @X e R. (16)
The relation (16) shows that the translation operator rX is a particular case of the deformed Hankel multiplier operator (3).
By using the translation, we define the generalized convolution product of f, 9 by
(f *a 9) (x,t) ^rx(f)(y)g(y)d^a(y).
K
This convolution is commutative, associative, and satisfies the following properties (see [17]):
Proposition 3.
(i) (Young's inequality). For allp, q,r e [1; +8], such that 1 +1 = 1 +1 and for all f e LX(R),g e LqX(R) the function f *q g belongs to the space Lq(R) and we have
\\f *q 9}r,^a ^ Aq\\f \\p,^a\\gWq. (17)
(ii) For f,g e LX(R), the function f *q g belongs to LX(R) if and only if the function Tq(f )Tq(g) belongs to LX(R), and in this case we have:
Tq (f *q g)= Tq(f)Tq(g). (18)
(iii) For all f,g e LX(R) we have
*q 9(X,t)\2 d^q(x) = J \Tq(f)(X)\2 \Tq(9)(X)\2 d^q(X), (19)
RR
where both integrals are simultaneously finite or infinite.
3. Deformed Hankel .^-multiplier operators. The main purpose of this section is to introduce the deformed Hankel L2-multiplier operators on R and to establish for them some uncertainty principles and Calderon's reproducing formulas.
3.1 Calderon's Reproducing Formulas for the Deformed Hankel .L^-multiplier operators.
Definition 3. Let a e £2(R) and [3 > 0; the deformed Hankel L2a-multiplier operators are defined for smooth functions on R as
(f)(x) : = T-1 (a^Ta(f))(x), (20)
where the function ap is given by
a?(X) := a(p\),
for all \e R.
By a simple change of variables, we find that for ¡3 > 0, a g e L
and
I I aP \ \ 2,^ " 11 o \ \ . (21)
P 2
Remark. According to the relation (18), we find that
%,p(f)(x) = {T-1 (ap)*- f) (x), (22)
where
1 )(*) = V)( £)- (23)
Let us give some properties of the deformed Hankel ¿--multiplier operators.
Proposition 4.
(i) For every a e L-(R) and f e L-(R), the function Ta,g(f ) belongs to
Li (R), and we have
a
Aa
\ I (f) \1 2,u„ ^ n2q-1 11 ° \ \ 2,fJ,a \ \ f \1 1,fla . P 2
(ii) For every a e L8(R), and for every f e L^(R), the function Ta,p(f) belongs to L^(R), and we have
1 1 (f ) \ \ 2 >tia ^ 1 1 a 11 \ \ f \ \ 2,/.« . (24)
(iii) For every a e L«(R), and for every f e L«(R), (f) e L%(R), and we have
%,/s(f )(x, t) ^ a(fi\, m)<pxtm(x, t)Ta(f )(A, m)d^a(X, m), a.e(x, t) e K,
R
(25)
and
Aa
(f )\\8,ua ^ ,02a-1 \\f \\2,^a .
P 2
Proof. (i) By the relations (17), (22), we find that
\\r°M= 1 ) *« f< Al\\f\\L„ 1 ^.
Plancherel's formula (12) and the relation (21) gives the desired result.
(ii) Is a consequence of Plancherel's formula (12).
(iii) Is a consequence of the relations (12), (17), (21), and (22); on the other hand, the relation (25) follows from inversion formula (10). □
In the following result, we give Plancherel's and pointwise reproducing inversion formula for the deformed Hankel L«-multiplier operators.
Theorem 1. Let a e L«(R) satisfy the admissibility condition:
We (A)!2 ^ = 1, A e R. (26)
(i) (Plancherel's formula). For all f in L«(R), we have
8
!f (x)|2(x)= \ \\%Af)\\2>/ta (27)
(ii) (First Calderon's formula). Let f e Ll(R), such that T«(f) e L«(R); then we have
8
f (x) " J {%M) *a K1 (oe)) (x)a.e. x e R. 0
Proof. (i) By using the Fubini theorem and relations (19) and (22), we get
i7^ ( /)iiLa -j -
0
8
I( f)(x)\2 df!a(x)
dj
J
0
8
\Fa 1 (ôp)*a f(x)\ dfla(x)
dj
J
\Ta( f)( A)|2 dfia(X)
\ôp(A)|2 f-
The admissibility condition (26) and Plancherel's formula (12) give the desired result.
(ii) Let f e L«(R), such that Ta(f) e L«(R); by using the Fubini theorem and the relations (11), (16), we find that
Cfc,p(f) 1 (ôp)) (x)-J "
%,P ( f)(y)T.« (Ta 1 (ôp )) (y)d^a(y)
dj
J
\vp ( A)|2 Ta(f)(X)Ba (Xx)d ¡ia(X)
dj J-
Admissibility condition (26) and inversion formula (10) give the desired result. □
To establish the second Calderon reproducing formula for the deformed Hankel ¿«-multiplier operators, we need the following technical result.
Proposition 5. Let ô e L«(R) x L8(R) satisfy the admissibility condition (26); then the function defined by
( x)= | \ôp(x)\2dj
Using the admissibility condition (26), relation (28) follows from the dominated convergence theorem. □
3.2 Uncertainty principles for the Deformed Hankel ¿^-multiplier operators. The main purpose of this subsection is to establish Heisenberg's and Donoho-Stark's uncertainty principles for the deformed Hankel L2-multiplier operators Ta,f.
3.2.1 Heisenberg's uncertainty principle for . In [6], the authors proved the following Heisenberg's inequality for Ta: there exist a positive constant c, such that for all f e L2 (R) we have:
11/III* ^ *IN2/||A|2J^(/)|2,Ma.
We generalize this inequality for 7^,f:
Theorem 3. There exists a constant c > 0, such that for all f e L we have
(29)
I/IIl. ^ c||a|2j;(/)|
2
( /)IL f
Proof. Suppose that | | |A|2^a( /) 11 2 relation (29), we have:
+
5IN2"^ ( /)|2„ f
< 8. Using
17^ ( /)(x)|2^a (x) ^c I |x|2TCT)/3 ( /) I 2M I |A|VFa(/)
Integrating over ]0, + 8[ with respect to measure jf and using Plancherel's
formula (27) and Schwartz's inequality, we get } }
2
1 2,^q
co
^ C
m2^ ( /)|2„q f
||A|4a^( A)|2 |J-«(/)(A)|2^(A)
d/3_
The Fubini theorem and the admissibility condition (26) gives the desired result. □
3.2.2 Donoho-Stark's uncertainty principle for .
Developing the ideas of Donoho and Stark [3], the main purpose of this
2
subsection is to give an uncertainty inequality of concentration type in L^R), where L^R) is the space of measurable functions on ]0, +ro[xR, such that
}
2,0a
1 f(J,.)ll2,na /
Denote by 9a the measure defined on ]0, + ro[xR by
d/3
d0a(/3,x) = d/ia(x) b —.
Definition 4. [3]
(i) Let E be a measurable subset of R. We say that the function f e L2 is £-concentrated on E if
\f - XeÎh,ua ^ e\\f
}2 ,/J.a
(30)
where \e is the indicator function of the set E.
(ii) Let F be a measurable subset of ]0, + ro[xR. We say that the function Ta,/3 ( f) is p-concentrated on F if
\\Ta,p ( f) - XF %,p {f)h,da < p \\Ta,l3 Whs
We have the following result: Theorem 4. Let f e L2a(R) and a e L2a
(31)
x L«(R) satisfy the admissibility condition (26). If f is £-concentrated on E and Ta>p (f) is p-concentrated on F, then we have
1 ô11 (fia(E)) 1
F
dÔa(/3,x) / 4a-2
> 1 -(e + p).
Proof. Let f e L
^a(E) < 8 and (31), we have
and ô L2
dea(p,x)
p4a —2
xL8
satisfy (26). Assume that
< 8. According to the relations (30) and
\%,p ( f) - XF Ta,p (XeÎ )} }2,ea ^ ¡%,p (f) - XF Ta,p (f)} }2,da +
+ 11 XF Ta, p ( f - XE f)} }2,ea ^ p|| Ta,p (f )} }2,0a + ¡Ta,p (f - XE f )} 12,(
LV
C-V
Using Plancherel's relation (27), we get
I I Ta,l3 ( f ) - XF Ta,? (XE/) | ¡2,9a < + p)|| fh,^a .
So, we get
1 1 %,/3 ( f) 11 2,ea ^ 1 1 Ta,/3 ( f) - XF Ta,/3 (XE f) } }2,da + }|XF (XE f )} }2,0a ^
^ (e + p) I I f I I 2,*, + 11 XF (xe f ) 11 2^. (32)
On the other hand, by relations (9), (25), and Holder's inequality, we find that
(E ),
so we arrive at
¡XF%,f} (XE f)h,oa < IIfh^a H i,7a (P(E)) 1
By the relations (32) and (33), we deduce that
F
dda CM)
(33)
I Ta,/3 ( f) 11 2,ea ^ I I f
I 2,^a
(e + p) + 11 a 11 i,7a (fia(E)) 1
F
4a—2
Plancherel's formula (27) for Ta,p gives the desired result. □
4. Extremal functions associated with the deformed Hankel L^-multiplier operators.
In the following, we study the extremal functions associated with the deformed Hankel ¿^-multiplier operators.
Definition 5. Let be a positive function on R satisfying the following conditions:
1 ^
and
^(A) ^ 1, (A) e R. We define the Sobolev-type space (R) by
(34)
= {/GL2(R) : a^j;(f)6L2(R)}
equipped with inner product
(f, 9% = ^(X,m)Ta(f)(X)Ta(g)(X)dtia(X),
and the norm
f I I P = \K fi Dp .
Proposition 6. Let a be a function in L8(R). The deformed Hankel L2a multiplier operators Ta,g are bounded and linear from W^ (R) into L2a\ and we have for all f e W^ (
1 I ^ ( f) 11 ^ I I a 11 8,7a | | f | | ^ . (36)
Proof. Using relations (12), (24), (35), we get the result. □
Definition 6. Let r] > 0 and let a be a function in L8(R). Denote by X f, the inner product defined on the space W^ (R) by
X f, 9>",v = J (#( A) + Wß ( X)|2) Ta( f)( X)Ta(g)( X)dßa{ X),
R
and the norm
| I f | I " ■
In the following results, we show that the norm 11 • | | can be expressed as a function of norm of the Hilbert space H" (R) and the norm of deformed Hankel ¿«-multiplier operators. Moreover, we show the equivalence between the norms 11 • | | and 11 • | | ".
Proposition 7. Let u be a function in L8(R) and f e H"(R)a. Then
(i) the norm 11 • 11satisfies
11 f11" 11 f11 " + | | Ta,ß ( f) 112,ßa ■
(ii) The norms 11 • 11and 11 • 11^ are equivalent and
?11111" ^ | | f11< 11pII■
Proof. The results follow from Plancherel's formula (12) and relation (36). □
Theorem 5. Let a e L8(R), the Sobolev-type space (V(R)), <•, ) be a reproducing kernel Hilbert space with kernel
K*"{x*>-/ „X x+kS)2d""(A)'
R
that is
(i) For all ye R, the function x ^ (x,y) belongs to V^ (R).
(ii) For all / e V^ (R) and y e R, we have the reproducing property:
f(y)=< f, (., (y))>^ . Proof. (i) Let ye R. From relations (6), (34), we see that the function
5(Ay )
9y: A
<#( A) + log ( A)\2
belongs to ¿^(R) x L2 ((R). Hence, the function is well-defined, and by the inversion formula (10), we get
(xy) = gy )(x).
Using Plancherel's theorem for Ta, we find that (•,y) belongs to L2 (R), and we have
(•, y))( A) =
(Ay )
#( A) + \a^ ( A)\2 ' Using relations (6), (34), and (37), we find that
(37)
2
.
This proves that for every ye R the function x ^ (x,y) belongs
to V (R).
(ii) Using the relation (37), we find that for all / e V^ (
<.a ^ ^
= (<#( A) + K( A)|2) Ta(f)( A)J^,,, (.,y)( A)dy«( A) =
= Ba(Xy)Ta (f)(X)dfia(X),
and inversion formula (10) gives the desired result. □
Taking a a null function and r] = 1, we find the following result:
Corollary 1. The Sobolev-type space (Wj(R)) , (•, ) is a reproducing kernel Hilbert space with kernel
Ba(Xx)Ba(Xy)
fcj(x, y) = J -^{X)-dlla(X).
R
The main result of this section can be stated as follows:
Theorem 6. Let a e L8(R) and fi > 0, for any h e L2a (R) and for any r] > 0, there exists a unique function f^,/3,h, where the infimum
inf {,, I i f 11 j + II h - %, ( D 11
(38)
is attained. Moreover, the extremal function f*^ h is given by
fvAh(y) " I h(x)®V,/3(x, y)d^a(x),
where Ov,p is given by
ap (X)Ba(Xx)Ba(Xy) #( X) + \ap (X)|2
n ! \ UP (X)Ba(Xx)Ba(Xy) , /an
®v,!3(x, y) = | d/la(X).
Proof. The existence and the unicity of the extremal function f*j3h satisfying (38) is given in [7], [10], [14]. Furthermore, f%tpth is given by
fvAh(y) = (h, (^J,v ))y^a. Using inversion formula (10) and the relation (37), we get
ap (X)Ba(Xx)Ba(Xy)
%,P (fcj,v (•, y) (x) = J ^ X) + a (X)\2 d^a(X) = eVtp (x, y)
R
and the proof is complete. □
Theorem 7. Let a e L8(R) and h e L2 (R). The function f*^^ satisfies the following properties:
Mf*,^)(A) " ^( A)\2Ta(h)(A)
(39)
and
\\ f ^ W^2^«.
Proof. Let ye R. The function
a^ (A)Ba(Ay)
ky : (A)
<#( A) + a ( A) |2 belongs to L2(R) x L*(R). Using inversion formula (10), we get
®v,/3 (X y) = ^)(x).
Using Plancherel's theorem and Parseval's relation (11), we find that 6^(•,y)eL2(R) and
^ (A)
f!,p,h(y) = J Ta(/)( ( A)dy«( A) = J A) + |a/?( A)|2 J'a(h)( A)c^«( A).
R R
On the other hand, the function
F.x_ . ^(A)Ta(fe)(A)
#( A) + \<Tg ( A)\2
belongs to L^(R) x L8(R). Using inversion formula (10) and Plancherel's theorem, we find that f^^h belongs to L2 (R) and
Ta( /11VX A) = F (A).
On the other hand, we have
I^a(fl,,h)(A)|2 = ( / ^( A)|2 2,2l^a(h)(A)|2 ^ -J—|^(h)(A)|2. (#( A) + |a^ ( A)|2) 2#( A)
By Plancherel's formula (12), we find that
II f 1,i3,h\U ^ W^2^«.
The proof is completed. □
Theorem 8. (Third Calderon's formula). Let a e L8(R) and f e W* The extremal function given by
¡lf},hpy) = J Ta,!3 (f)(x)®r,? (x, y)d|lа(x),
R
satisfies
lim|| f* - f\\ = 0. (40)
*^0 + "I" "2,^a
Moreover, we have f*^ —> f uniformly when rq —> 0+.
Proof. Let f e W* (R)- Put h = Ta/I (f) and f*>/3>h = f*>/3 in relation (39) to find that
-V l(A)Ta(f)(A) rl( A) + \ap ( A)f
Therefore,
Kif^h -f)(A) " ■ V^. (41)
2 f v2 (l(A)
^ -f 1 1 * " I #( A) + ( A)\2 ^ f)(A1 dla(A).
On the other hand, we have
V2 (l(A))3
\Fa(f)(A)\2 ^l(A) \Ta(f)(A)\2. (42)
#( A) + \(j/3 (A)\2
The result (40) follows from (42) and the dominated convergence theorem. Now, for all f e W* (R) we (10) and (41), we find that
f. f(q xf -v m Mf)(A) J*,fi{y,s) nV ) J Vl( A) + \ap( A)f
R
and
^ \Ta( f)( A, m) \. (43)
Now, for all f e W*(R) we have Ta(f) e L2a(R) x L«(R). Using relations
--n m^m))
VP(\) + \as(A)\2 (
Using relation (43) and the dominated convergence theorem, we deduce that
lim+ \f (y)-f(y)\= 0.
This completes the proof of the theorem. □
Acknowledgment. The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article.
References
[1] Aronszajn N. Theory of reproducing kernels. Trans. Am. Math. Soc., 1950, vol. 68(3), pp. 337-404.
DOI: https://doi.org/10.1090/S0002-9947-1950-0051437-7
[2] Banuelos R., Bogdan K. Levy processes and Fourier multipliers. J. Funct. Anal, 2007, vol. 250(1), pp. 197-213.
DOI: https://doi.org/10.1016/jjfa.2007.05.013
[3] Donoho D. L., Stark P. B. Uncertainty principles and signal recovery. SIAM J. Appl. Math, 1989, vol. 49(3), pp. 906-931.
[4] Dunkl C. F. Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc, 1989, vol. 311(1), pp. 167-183.
DOI: https://doi.org/10.1090/S0002-9947-1989-0951883-8
[5] Hormander L. Estimates for translation invariant operators in Lp spaces. Acta Math., 1960, vol. 104(1-2), pp. 93-140.
DOI: https://doi .org/10.1007/BF02547187
[6] Johansen T. R. Weighted inequalities and uncertainty principles for the (k, a)-generalized Fourier transform. Int. J. Math, 2016, vol. 27(03), 1650019. DOI: https://doi.org/10.1142/S0129167X16500191
[7] Kimeldorf G., Wahba G. Some results on Tchebycheffian spline functions. J. Math. Anal. Appl, 1971, vol.33(1), pp. 82-95.
[8] Kumar V., Ruzhansky M. LP — Lq boundedness of (k, a)-Fourier multipliers with applications to nonlinear equations. Int. Math. Res. Not, 2023, no. 2, pp. 1073-1093. DOI: https://doi.org/10.1093/imrn/rnab256
[9] Kumar V., Restrepo J. E., Ruzhansky M. Asymptotic estimates for the growth of deformed Hankel transform by modulus of continuity. Results Math, 2024, vol. 79, article number 22.
DOI: https://doi.org/10.1007/s00025-023-02051-w
[10] Matsuura T., Saitoh S., Trong D. D. Approximate and analytical inversion formulas in heat conduction on multidimensional spaces. Journal of Numerical Mathematics, 2005, vol. 13, no. 5, pp. 479-493. DOI: https://doi.org/10.1515/156939405775297452
[11] McConnell T. R. On Fourier multiplier transformations of Banach-valued functions. Trans. Am. Math. Soc, 1984, vol. 285(2), pp. 739-757.
DOI: https://doi.org/10.1090/S0002-9947-1984-0752501-X
[12] Mikhlin S. G. On the multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR, 1956, vol. 109, pp. 701-703. MR 0080799. (in Russian)
[13] Negzaoui S., Oukili S. Modulus of continuity and modulus of smoothness related to the deformed Hankel transform. Results Math, 2021, vol. 76, article number 164. DOI: https://doi .org/10.1007/s00025-021-01474-7
[14] Saitoh S. Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc, 1983, vol. 89(1), pp. 74-78.
[15] Saitoh S., Matsuura T. Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces. Appl. Anal, 2006, vol. 85(8), pp. 901-915.
DOI: https://doi.org/10.1080/00036810600643662
[16] Said S. B., Kobayashi T., 0rsted, B. Laguerre semigroup and Dunkl operators. Compos. Math, 2012, vol. 148(4), pp. 1265-1336.
DOI: https://doi.org/10.1112/S0010437X11007445
[17] Said S. B. A product formula and a convolution structure for a k-Hankel transform on RR. J. Math. Anal. Appl, 2018, vol. 463(2), pp. 1132-1146. DOI: https://doi.org/10.1016Zj.jmaa.2018.03.073
[18] Said S. B., Boubatra M. A., Sifi M. On the deformed Besov-Hankel spaces. Opusc.Math, 2020, vol. 40, no. 2, pp. 171-207.
DOI: https://doi.org/10.7494/0pMath.2020.40.2.171
[19] Watson G. N. A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, 1922.
Received June 22, 2024. In revised form,, September 09 , 2024. Accepted September 25, 2024. Published online October 23, 2024.
Laboratory of Fundamental and Applied Mathematics,
Department of Mathematics and Informatics, Faculty of Sciences Ain Chock,
University of Hassan II,
B.P 5366 Maarif, Casablanca, Morocco
A. Chana
E-mail: [email protected] A. Akhlidj
E-mail: [email protected]
