Deformed Hankel transform of Dini – Lipschitz functions
UDC 517.5 By using a generalized symmetric difference $\Delta_{h}^{m}$ of order $m$ and step $h>0,$ we obtain an analog of the Titchmarsh theorems [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (Theorems 84 and 85) for the deformed Hankel transform. We also p...
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| author | Elgargati , A. Loualid, M. El Daher, R. Elgargati , A. Loualid, M. El Daher, R. |
| author_facet | Elgargati , A. Loualid, M. El Daher, R. Elgargati , A. Loualid, M. El Daher, R. |
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| description | UDC 517.5
By using a generalized symmetric difference $\Delta_{h}^{m}$ of order $m$ and step $h>0,$ we obtain an analog of the Titchmarsh theorems [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (Theorems 84 and 85) for the deformed Hankel transform. We also provide a further extension of the theorem cited above for functions in $L_k^{p}$ with an abstract deformed Hankel – Dini – Lipschitz condition. |
| doi_str_mv | 10.37863/umzh.v74i8.6134 |
| first_indexed | 2026-03-24T03:26:11Z |
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DOI: 10.37863/umzh.v74i8.6134
UDC 517.5
A. Elgargati (Laboratory Fundamental and Applied Mathematics, Univ. Hassan II, Casablanca, Morocco),
M. El Loualid1 (Chouaib Doukkali Univ. El Jadida, Nat. School Applied Sci., Sci. Engineer Lab. Energy, El Jadida,
Morocco),
R. Daher (Laboratory Fundamental and Applied Mathematics, Univ. Hassan II, Casablanca, Morocco)
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS
ДЕФОРМОВАНЕ ПЕРЕТВОРЕННЯ ГАНКЕЛЯ ФУНКЦIЙ ДIНI – ЛIПШИЦЯ
By using a generalized symmetric difference \Delta m
h of order m and step h > 0, we obtain an analog of the Titchmarsh
theorems [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (Theorems 84 and 85) for the deformed
Hankel transform. We also provide a further extension of the theorem cited above for functions in Lp
k with an abstract
deformed Hankel – Dini – Lipschitz condition.
Використовуючи узагальнену симетричну рiзницю \Delta m
h порядку m i кроку h > 0, отримано аналог теорем Тiтчмар-
ша [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (теореми 84 i 85) для деформованого
перетворення Ганкеля. Крiм того, наведено додаткове розширення вказаної теореми для функцiй у Lp
k з абстрактною
деформованою умовою Ганкеля – Дiнi – Лiпшиця.
1. Introduction. In [8], Titchmarsh gave a Lipschitz condition on a function f \in Lp(\BbbR ) for which
its Fourier transform belongs to L\beta (\BbbR ) for some values of \beta . His result reads as follows.
Theorem 1.1 ([8], Theorem 84). Let f belong to Lp(\BbbR ), 1 < p \leq 2, and\int
\BbbR
\bigm| \bigm| f(x+ h) - f(x)
\bigm| \bigm| pdx = O
\bigl(
h\alpha p
\bigr)
, 0 < \alpha \leq 1, as h \rightarrow 0.
Then \widehat f belong to L\beta (\BbbR ) for
p
p+ p\alpha - 1
< \beta \leq p
p - 1
.
Theorem 1.2 ([8], Theorem 85). Let \alpha \in (0, 1) and assume that f \in L2(\BbbR ). Then the following
statement are equivalents:
\| f(.+ h) - f\| L2(\BbbR ) = O(h\alpha ),\int
| \lambda | \geq r
| \widehat f(f)(\lambda )| 2d\lambda = O
\bigl(
r - 2\alpha
\bigr)
as r \rightarrow \infty ,
where \widehat f stands for the Fourier transform of f .
There are many different analogs of Theorems 1.1 and 1.2: for the Fourier transform on Rie-
mannian symmetric spaces of rank 1 and, in particular, for the Fourier transform on the Lobachevski
plane, for the Fourier – Jacobi transform, for the q-Bessel transform, etc. (see, for example, [3 – 7]).
In the present paper, we obtain a analog of Theorem 1.1 and Theorem 1.2 for the deformed
Hankel transform newly introduced by S. Ben Saı̈d [1, 2]. The importance of this transform lies in
1 Corresponding author, e-mail: mehdi.loualid@gmail.com.
c\bigcirc A. ELGARGATI, M. EL LOUALID, R. DAHER, 2022
1118 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS 1119
the fact that it generalizes many integral transforms. Furthermore, we define the deformed Hankel –
Dini – Lipschitz class Hm,p
\alpha ,\gamma and we obtain an extension of the Theorem 1.1 in this occurence.
This paper is organized as follows. Section 2 is a summary of the main results in the harmonic
analysis associated with the deformed Hankel transform and we prove some auxiliary results required
for the proofs of the main results. In Section 3, first, we define the deformed Hankel – Lipschitz class
Hm,p
\alpha . Next, we prove analogous of the Theorem 1.1. In Section 4, we consider the particular
case, when p = 2, and we provide a characterization of the space Hm,2
\alpha of deformed Hankel –
Lipschitz class functions by means of asymptotic estimate growth of the norm of their deformed
Hankel transform for \alpha \in (0, 1]. In Section 5, we extend the Theorem 1.1 to the deformed Hankel –
Dini – Lipschitz class Hm,p
\alpha ,\gamma .
2. Definitions and auxiliary results. In this section, first, we briefly collect the pertinent
definitions and facts relevant for deformed Hankel transform. Secondly, we prove some auxiliary
results. For more details we refer to [1, 2].
We denote by Lp
k the space of measurable functions f on \BbbR with the finite norm
\| f\| k,p =
\left( \int
\BbbR
| f(x)| pd\mu (x)
\right) 1/p
,
where d\mu (x) = 2 - 1\Gamma (2k) - 1| x| 2k - 1dx.
In [1], the author introduced a new transform \scrF k(f) called deformed Hankel transform which is
a deformation of the Hankel transform by a parameter k > 0. Namely, for k >
1
4
and f \in L1
k, the
integral transform \scrF k(f) is defined by
\scrF k(f)(\lambda ) =
\int
\BbbR
\scrB k(\lambda , x)f(x)d\mu k(x), \lambda \in \BbbR ,
where the kernel \scrB k(\lambda , x), called deformed Hankel kernel, given by
\scrB k(\lambda , x) = j2k - 1
\Bigl(
2
\sqrt{}
| \lambda x|
\Bigr)
- \lambda x
2k(2k + 1)
j2k+1
\Bigl(
2
\sqrt{}
| \lambda x|
\Bigr)
.
Here, j\alpha is the normalized Bessel function of the first kind and order \alpha defined by
j\alpha (z) = \Gamma (\alpha + 1)
\infty \sum
n=0
( - 1)n
\Bigl( z
2
\Bigr) 2n
n!\Gamma (n+ \alpha + 1)
, z \in \BbbC ,
where \Gamma (x) is the gamma-function.
Theorem 2.1. For all k >
1
4
, we have\bigm| \bigm| \scrB k(\lambda , x)
\bigm| \bigm| \leq 1, \lambda , x \in \BbbR ,
\scrB k(\lambda , x) = j2k - 1
\Bigl(
2
\surd
\lambda x
\Bigr)
- \mathrm{s}\mathrm{g}\mathrm{n}(\lambda x)
\Bigl(
j2k
\Bigl(
2
\surd
\lambda x
\Bigr)
- j2k - 1
\Bigl(
2
\surd
\lambda x
\Bigr) \Bigr)
,
\scrF k(f) is an involutional unitary operator on L1(R, d\mu k(x)),
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1120 A. ELGARGATI, M. EL LOUALID, R. DAHER
Parseval’s identity, if f \in L1
k
\bigcap
L2
k, then
\bigm\| \bigm\| \scrF k(f)
\bigm\| \bigm\|
k,2
=
\bigm\| \bigm\| \scrF k(f)
\bigm\| \bigm\|
k,2
,
there exists a unique isometry on L2
k that coincides with \scrF k(f) on f \in L1
k \cap L2
k .
Let f be a suitable function on \BbbR . The translation operator T y
k is defined by
T y
k (f)(x) =
\int
\BbbR
f(z)\scrK k(x, y, z)d\mu k(z),
where
\scrK k(x, y, z) = 2\Gamma (2k)\scrW 2k - 1
\Bigl( \sqrt{}
| x| ,
\sqrt{}
| y| ,
\sqrt{}
| z|
\Bigr)
\nabla k(x, y, z),
and
\scrW k(x, y, z) =
\Gamma (k + 1)
22k - 1\Gamma
\biggl(
k +
1
2
\biggr) \bigl[ \bigl( (x+ y)2 - z2
\bigr) \bigl(
z2 - (x - y)2
\bigr) \bigr] k - 1
2
(xyz)
\bfone ]| x - y| ,x+y[(z).
Here, \bfone A is the characteristic function of the set A and
\nabla k(x, y, z) =
1
4
\biggl[
1 +
\mathrm{s}\mathrm{g}\mathrm{n}(xy)
4k - 1
\Bigl(
4k\Delta
\bigl(
| x| , | y| , | z|
\bigr) 2 - 1
\Bigr) \biggr]
+
+
1
4
\biggl[
1 +
\mathrm{s}\mathrm{g}\mathrm{n}(xz)
4k - 1
\Bigl(
4k\Delta
\bigl(
| z| , | x| , | y|
\bigr) 2 - 1
\Bigr) \biggr]
+
+
1
4
\biggl[
1 +
\mathrm{s}\mathrm{g}\mathrm{n}(yz)
4k - 1
\Bigl(
4k\Delta
\bigl(
| z| , | y| , | x|
\bigr) 2 - 1
\Bigr) \biggr]
,
and
\Delta (a, b, c) =
1
2
\surd
ab
(a+ b - c), a, b, c \in \BbbR \ast
+.
Proposition 2.1. If f \in Lp
k, 1 \leq p \leq 2, and x \in \BbbR , then
\scrF k
\Bigl(
T h
k (f)
\Bigr)
(\lambda ) = \scrB k(\lambda , h)\scrF k(f)(\lambda ). (2.1)
Below, we will define and prove several auxiliary assertions to be used in the proofs of our main
results.
The first and the higher order finite differences of f(x) with step h are defined as follows:
\bigtriangleup 1
hf(x) = T h
k f(x) - f(x)
and
\bigtriangleup k
hf(x) = \bigtriangleup 1
h
\Bigl(
\bigtriangleup k - 1
h f(x)
\Bigr)
=
=
k\sum
i=1
( - 1)k - i
\Biggl(
k
i
\Biggr) \Bigl(
T h
k
\Bigr) i
f(x) for k \in \BbbN \ast ,
where
\Bigl(
T h
k
\Bigr) 0
f(x) = f(x) and
\Bigl(
T h
k
\Bigr) i
= T h
k
\Bigl( \Bigl(
T h
k
\Bigr) i - 1
f(x)
\Bigr)
for i \in \BbbN \ast .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS 1121
Lemma 2.1. There exists C1 > 0, C2 > 0 and \eta > 0 such that
C1| \lambda x| \leq 1 - \scrB k(\lambda , x) \leq C2| \lambda x| for all | \lambda x| < \eta . (2.2)
Proof. By the second statement of Theorem 2.1, we have
\scrB k(\lambda , x) = j2k - 1
\Bigl(
2
\sqrt{}
| \lambda x|
\Bigr)
- \mathrm{s}\mathrm{g}\mathrm{n}(\lambda x)
\Bigl(
j2k
\Bigl(
2
\sqrt{}
| \lambda x|
\Bigr)
- j2k - 1
\Bigl(
2
\sqrt{}
| \lambda x|
\Bigr) \Bigr)
.
Let
\varphi k(x) = j2k - 1
\Bigl(
2
\sqrt{}
| x|
\Bigr)
- \mathrm{s}\mathrm{g}\mathrm{n}(\lambda x)
\Bigl(
j2k
\Bigl(
2
\sqrt{}
| x|
\Bigr)
- j2k - 1
\Bigl(
2
\sqrt{}
| x|
\Bigr) \Bigr)
.
If x < 0, we have
\varphi k(x) = j2k
\bigl(
2
\surd
- x
\bigr)
=
= \Gamma (2k + 1)
+\infty \sum
m=0
xm
\Gamma (2k +m+ 1)
=
= 1 +
x
2k + 1
+ \Gamma (2k + 1)
+\infty \sum
m=2
xm
\Gamma (2k +m+ 1)
.
Then
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0 -
\varphi k(x) - 1
x
=
1
2k + 1
\not = 0. (2.3)
If x \geq 0, we have
\varphi k(x) = 2j2k - 1
\bigl(
2
\surd
x
\bigr)
- j2k
\bigl(
2
\surd
x
\bigr)
=
= 2\Gamma (2k)
+\infty \sum
m=0
( - x)m
\Gamma (2k +m)
- \Gamma (2k + 1)
+\infty \sum
m=0
( - x)m
\Gamma (2k +m+ 1)
=
= 1 - x
k
+
x
2k + 1
+ 2\Gamma (2k)
+\infty \sum
m=2
( - x)m
\Gamma (2k +m+ 1)
- \Gamma (2k + 1)
+\infty \sum
m=2
( - x)m
\Gamma (2k +m+ 1)
.
Then
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0+
\varphi k(x) - 1
x
=
1
2k + 1
- 1
k
\not = 0. (2.4)
Hence, by (2.3) and (2.4), there exist C1 > 0, C2 > 0 and \eta > 0 such that
C1| x| \leq | - \varphi k(x)| \leq C2| x| for all | x| < \eta , (2.5)
and (2.2) follows from (2.5).
Lemma 2.2. For a fixed h > 0, we have
\scrF k(\bigtriangleup m
h f)(x) = (\scrB k(x, h) - 1)m\scrF k(f)(x). (2.6)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1122 A. ELGARGATI, M. EL LOUALID, R. DAHER
Proof. Let h > 0. On the basis of (2.1), we obtain
\scrF k
\Bigl(
T h
k (f)
\Bigr)
(\lambda ) = \scrB k(\lambda , h)\scrF k(f)(\lambda ).
Then, by recurrence on i, we get
\scrF k
\biggl( \Bigl(
T h
k
\Bigr) i
(f)
\biggr)
(\lambda ) = (\scrB k(\lambda , h))
i\scrF k(f)(\lambda ).
Hence,
\scrF k
\bigl(
\bigtriangleup m
h f
\bigr)
(x) =
k\sum
i=1
( - 1)k - i
\Biggl(
k
i
\Biggr) \bigl(
\scrB k(\lambda , h)
\bigr) i\scrF k(f)(x) =
=
\Biggl(
k\sum
i=1
( - 1)k - i
\Biggl(
k
i
\Biggr) \bigl(
\scrB k(\lambda , h)
\bigr) i\Biggr) \scrF k(f)(x).
Using Newton’s formula, we obtain (2.6).
3. Lipschitz conditions in deformed Hankel setting. In this section, we state and prove an
analogous of Titchmarsh’s theorem [8] (Theorem 84) for the deformed Hankel transform. Before, we
need to define the deformed Hankel – Lipschitz class Hm,p
\alpha .
Definition 3.1. A function f : \BbbR \rightarrow \BbbR is said belong to Hm,p
\alpha for \alpha > 0 if\bigm\| \bigm\| \bigtriangleup m
h f(x)
\bigm\| \bigm\|
k,p
= O(h\alpha ) as h \rightarrow 0.
Remark 3.1. The spaces H1,p
\alpha for \alpha > 0 are called the Lipshitz classes \mathrm{l}\mathrm{i}\mathrm{p}p(\alpha ). The spaces
H2,p
\alpha for \alpha > 0 are called the Zygmund classes \mathrm{z}\mathrm{y}\mathrm{g}p(\alpha ).
Theorem 3.1. Let f belong to Lp
k, 1 < p \leq 2. If\int
\BbbR
\bigm| \bigm| \bigtriangleup m
h f(x)
\bigm| \bigm| pd\mu k(x) = O(h\alpha p) as h \rightarrow 0, (3.1)
then \scrF k(f) belong to L\beta
k for
2kp
2kp+ \alpha p - 2k
< \beta \leq p
p - 1
.
Remark 3.2. The statement (3.1) is equivalent to f \in Hm,p
\alpha .
Proof. By virtue of the Lemma 2.2 and Hausdorff – Young inequality, we have
\int
\BbbR
\bigm| \bigm| (1 - \scrB k(x, h))
m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) \leq
\left( \int
\BbbR
\bigm| \bigm| \bigtriangleup m
h f(x)
\bigm| \bigm| pd\mu k(x)
\right)
p\prime
p
.
Thus, using (3.1), we obtain\int
\BbbR
\bigm| \bigm| (1 - \scrB k(x, h))
m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) = O
\bigl(
h\alpha p
\prime \bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS 1123
By the Lemma 2.1, there exist C1 > 0 and \eta > 0 such that 1 - \scrB k(x, h) > C1xh for | x| < \eta
h
.
Then
C1
\eta /h\int
0
\bigm| \bigm| (xh)m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) \leq
\int
\BbbR 2
\bigm| \bigm| (1 - \scrB k(x, h))
m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) = O(h\alpha p
\prime
).
Hence,
\eta /h\int
0
xmp\prime
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) = O
\bigl(
h(\alpha - m)p\prime
\bigr)
.
Let
\phi (y) =
y\int
1
\bigm| \bigm| xm\scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x). (3.2)
Then, if \beta < p\prime , we get
\phi (y) \leq
\left( y\int
1
| xm\scrF k(f)(x)| p
\prime
d\mu k(x)
\right) \beta /p\prime \left( y\int
1
d\mu k(x)
\right) 1 - \beta /p\prime
=
= O
\Bigl(
y(m - \alpha )\beta +2k(1 - \beta /p\prime )
\Bigr)
.
On the basis of (3.2), we obtain
y\int
1
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x) =
y\int
1
x - m\beta \phi \prime (x)dx.
An integrate by parts, we get
y\int
1
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \bigm| \beta d\mu k(x) = y - m\beta \phi (y) +m\beta
y\int
1
x - m\beta - 1\phi (x)dx =
= O
\biggl(
y - m\beta +(m - \alpha )\beta +2k
\bigl(
1 - \beta /p\prime
\bigr) \biggr)
=
= O
\biggl(
y - \alpha \beta +2k
\bigl(
1 - \beta /p\prime
\bigr) \biggr)
,
the quantity is bonded as y \rightarrow +\infty if - \alpha \beta + 2k
\biggl(
1 - \beta
p\prime
\biggr)
< 0, i.e.,
\beta >
2kp
2kp+ \alpha p - 2k
.
Similarly for
\int - 1
- \infty
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x).
The case \beta = p\prime is true by Hausdorff – Young inequality.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1124 A. ELGARGATI, M. EL LOUALID, R. DAHER
4. An equivalence theorem for deformed Hankel – Lipschitz class functions. This section
deals with a particular case, when p = 2 and \alpha \in (0, 1]. We can put the Theorem 3.1 into a form
in which it is reversible. Hence, we give a characterization of the space Hm,2
\alpha of deformed Hankel –
Lipschitz class functions by means of asymptotic estimate growth of the norm of their deformed
Hankel transform.
Theorem 4.1. Let \alpha \in (0, 1] and f \in L2
k . Then the following statements are equivalent:\int
\BbbR
| \bigtriangleup m
h f(x)| 2d\mu k(x) = O
\bigl(
h2\alpha
\bigr)
, (4.1)
\int
\lambda \geq r
\scrF k(f)(\lambda )d\mu k(\lambda ) = O
\bigl(
r - 2\alpha
\bigr)
as r \rightarrow \infty . (4.2)
Proof. By Parseval’s identity, we get\int
\BbbR
\bigm| \bigm| (1 - \scrB k(\lambda , x))
m\scrF k(f)(x)
\bigm| \bigm| 2d\mu k(x) =
\int
\BbbR
\bigm| \bigm| \bigtriangleup m
h f(x)
\bigm| \bigm| 2d\mu k(x). (4.3)
Suppose that (4.1) holds. Then, by virtue of (4.3) yields
\eta
h\int
\eta
2h
| \scrF k(f)(x)| 2d\mu k(x) \leq K
\int
\BbbR
\bigm| \bigm| (1 - \scrB k(\lambda , x))
m\scrF k(f)(x)
\bigm| \bigm| 2d\mu k(x) =
= O(h2\alpha ). (4.4)
Let r > 0 and h =
\eta
2i+1r
for i \in \BbbN . Then, from (4.4), we find
2i+1r\int
2ir
| \scrF k(f)(x)| 2d\mu k(x) =
\eta
h\int
\eta
2h
| \scrF k(f)(x)| 2d\mu k(x) = O(h2\alpha ) = O
\biggl( \Bigl( \eta
2i+1r
\Bigr) 2\alpha \biggr)
=
= O((2ir) - 2\alpha ).
Hence,
+\infty \int
r
| \scrF k(f)(x)| 2d\mu k(x) =
+\infty \sum
i=0
2i+1r\int
2ir
| \scrF k(f)(x)| 2d\mu k(x) =
=
+\infty \sum
i=0
O
\Bigl( \bigl(
2ir
\bigr) - 2\alpha
\Bigr)
=
+\infty \sum
i=0
O
\Bigl( \bigl(
2ir
\bigr) - 2\alpha
\Bigr)
= O
\bigl(
r - 2\alpha
\bigr)
.
Similarly for
\int - r
- \infty
| \scrF k(f)(x)| 2d\mu k(x).
On the other hand, if (4.2), holds, we can write
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS 1125
\phi (r) =
+\infty \int
r
| \scrF k(f)(x)| 2d\mu k(x).
Then, by (4.2), we have
r\int
0
\bigm| \bigm| xm\scrF k(f)(x)
\bigm| \bigm| 2d\mu k(x) =
r\int
0
- x2m\phi \prime (x)dx =
= - r2m\phi (r) + 2m
r\int
0
x2m - 1\phi (x)dx.
Hence,
r\int
0
| xm\scrF k(f)(x)| 2d\mu k(x) = O
\bigl(
r2m - 2\alpha
\bigr)
. (4.5)
By Lemma 2.1, there exist C > 0 and \eta > 0 such that
1 - \scrB k(\lambda , h) \leq C\lambda h for | \lambda h| < \eta . (4.6)
Then, (4.5) and (4.6) gives
\int
\BbbR
\bigl(
1 - \scrB k(x, h)
\bigr) m| \scrF k(f)(x)| 2d\mu k(x) \leq C2h
m
\eta /h\int
- \eta /h
| xm\scrF k(f)(x)| 2d\mu k(x)+
+
- \eta /h\int
- \infty
| \scrF k(f)(x)| 2d\mu k(x) +
+\infty \int
\eta /h
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| 2d\mu k(x) = O(h2\alpha ).
Hence, (4.1) follows from (4.3).
5. Deformed Hankel – Dini – Lipschitz conditions. In this section, we consider a new larger
class of functions Hm,p
\alpha ,\gamma defined by
Definition 5.1. A function f : \BbbR \rightarrow \BbbR is said belong to Hm,p
\alpha ,\gamma for \alpha , \gamma > 0 if
\bigm\| \bigm\| \bigtriangleup m
h f(x)
\bigm\| \bigm\|
k,p
= O
\biggl(
h\alpha \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma \biggr)
.
Remark 5.1. Let \alpha 2 > \alpha 1 > 0, we have
Hm,p
\alpha 2
\subset Hm,p
\alpha 1
and
Hm,p
\alpha 2
\subset Hm,p
\alpha 2,\gamma \subset Hm,p
\alpha 1
.
Now, we are able to establish an extension of Theorem 3.1 to Hm,p
\alpha ,\gamma .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1126 A. ELGARGATI, M. EL LOUALID, R. DAHER
Theorem 5.1. Let f belong to Lp
k, 1 < p \leq 2. If\int
\BbbR
| \bigtriangleup m
h f(x)| pd\mu k(x) = O
\biggl(
h\alpha p \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma p\biggr)
as h \rightarrow 0, (5.1)
then \scrF k(f) belong to L\beta
k for
2kp
2kp+ \alpha p - 2k
< \beta \leq p
p - 1
.
Remark 5.2. The statement (5.1) is equivalent to f \in Hm,p
\alpha ,\gamma .
Proof. By virtue of the Lemma 2.2 and Hausdorff – Young inequality, we have
\int
\BbbR
\bigm| \bigm| (1 - \scrB k(x, h))
m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) \leq
\left( \int
\BbbR
| \bigtriangleup m
h f(x)| pd\mu k(x)
\right)
p\prime
p
.
Thus, using (5.1), we obtain\int
\BbbR
| (1 - \scrB k(x, h))
m\scrF k(f)(x)| p
\prime
d\mu k(x) = O
\Biggl(
h\alpha p
\prime
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma p\prime
\Biggr)
.
By the Lemma 2.1, there exist C1 > 0 and \eta > 0 such that 1 - \scrB k(x, h) > C1xh for | x| < \eta
h
.
Then
C1
\eta /h\int
0
\bigm| \bigm| (xh)m\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) \leq
\int
\BbbR 2
| (1 - \scrB k(x, h))
m\scrF k(f)(x)| p
\prime
d\mu k(x) =
= O
\Biggl(
h\alpha p
\prime
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma p\prime
\Biggr)
.
Hence,
\eta /h\int
0
xmp\prime
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x) = O
\Biggl(
h(\alpha - m)p\prime \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma p\prime
\Biggr)
.
Let
\phi (y) =
y\int
1
\bigm| \bigm| xm\scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x). (5.2)
Then, if \beta < p\prime , we get
\phi (y) \leq
\left( y\int
1
\bigm| \bigm| xm\scrF k(f)(x)
\bigm| \bigm| p\prime d\mu k(x)
\right) \beta /p\prime \left( y\int
1
d\mu k(x)
\right) 1 - \beta /p\prime
=
= O
\Biggl(
y(m - \alpha )\beta +2k
\bigl(
1 - \beta /p\prime
\bigr)
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma \beta
\Biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
DEFORMED HANKEL TRANSFORM OF DINI – LIPSCHITZ FUNCTIONS 1127
On the basis of (5.2), we obtain
y\int
2
| \scrF k(f)(x)| \beta d\mu k(x) =
y\int
2
x - m\beta \phi \prime (x)dx.
An integrate by parts, we get
y\int
2
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x) = y - m\beta \phi (y) +m\beta
y\int
2
x - m\beta - 1\phi (x)dx =
= O
\biggl(
y - \alpha \beta +2k
\bigl(
1 - \beta /p\prime
\bigr) \biggr)
+O
\left( y\int
2
y - \alpha \beta +2k
\bigl(
1 - \beta /p\prime
\bigr)
- 1 \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma \beta
\right) .
The quantity y - \alpha \beta +2k
\bigl(
1 - \beta /p\prime
\bigr)
is bonded as y \rightarrow +\infty if - \alpha \beta + 2k
\biggl(
1 - \beta
p\prime
\biggr)
< 0, i.e.,
\beta >
2kp
2kp+ \alpha p - 2k
.
Since - \gamma \beta < 1, then on the basis of Bertrand rule, the quantity
\int y
2
y - \alpha \beta +2k
\bigl(
1 - \beta /p\prime
\bigr)
- 1 \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
h
\biggr) \gamma \beta
is bonded as y \rightarrow +\infty if \alpha \beta - 2k
\biggl(
1 - \beta
p\prime
\biggr)
+ 1 > 1, i.e.,
\beta >
2kp
2kp+ \alpha p - 2k
.
Hence,
\int y
2
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x) is bonded.
Similarly for
\int - 1
- \infty
\bigm| \bigm| \scrF k(f)(x)
\bigm| \bigm| \beta d\mu k(x).
The case \beta = p\prime is true by Hausdorff – Young inequality. Then, the proof is completed.
References
1. S. Ben Saı̈d, A product formula and convolution structure for a k-Hankel transform on R, J. Math. Anal. and Appl.,
463, № 2, 1132 – 1146 (2018).
2. Salem Ben Sad, Mohamed Amine Boubatra, Mohamed Sifi, On the deformed Besov – Hankel spaces, Opuscula
Math., 40, № 2, 171 – 207 (2020); https://doi.org/10.7494/OpMath.2020.40.2.171.
3. A. Achak, R. Daher, L. Dhaouadi, El Loualid, An analog of Titchmarsh’s theorem for the q-Bessel transform, Ann.
Univ. Ferrara, 65, № 113 (2019); https://doi.org/10.1007/s11565-018-0309-3.
4. R. Daher, M. Hamma, An analog of Titchmarsh’s theorem of Jacobi transform, Int. J. Math. Anal., 6, № 20, 975 – 981
(2012).
5. R. Daher, M. El Hamma, A. El Houasni, Titchmarsh’s theorem for the Bessel transform, Matematika, 28, № 2,
127 – 131 (2012); https://doi.org/10.11113/matematika.v28.n.567.
6. S. Negzaoui, Lipschitz conditions in Laguerre hypergroup, Mediterr. J. Math., 14, № 191 (2017); https://doi.org/
10.1007/s00009-017-0989-4.
7. S. S. Platonov, The Fourier transform of functions satisfying the Lipschitz condition on rank 1 symmetric spaces,
Sib. Mat. J., 46, № 6, 1108 – 1118 (2005).
8. E. S. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948).
Received 24.05.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
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| id | umjimathkievua-article-6134 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:11Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/27/19801de58e0f645a58fb6fa4a7b08827.pdf |
| spelling | umjimathkievua-article-61342022-10-24T17:51:31Z Deformed Hankel transform of Dini – Lipschitz functions Deformed Hankel transform of Dini – Lipschitz functions Elgargati , A. Loualid, M. El Daher, R. Elgargati , A. Loualid, M. El Daher, R. Deformed Hankel transfor Titchmarsh theore Symmetric differenc UDC 517.5 By using a generalized symmetric difference $\Delta_{h}^{m}$ of order $m$ and step $h&gt;0,$ we obtain an analog of the Titchmarsh theorems [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (Theorems 84 and 85) for the deformed Hankel transform. We also provide a further extension of the theorem cited above for functions in $L_k^{p}$ with an abstract deformed Hankel – Dini – Lipschitz condition. УДК 517.5Деформоване перетворення Ганкеля функцiй Дiнi – ЛiпшицяВикористовуючи узагальнену симетричну рiзницю $\Delta_{h}^{m}$ порядку $m$ i кроку $h&gt;0,$, отримано аналог теорем Тiтчмарша [Introduction to the theory of Fourier integrals, Oxford Univ. Press (1948)] (теореми 84 i 85) для деформованого перетворення Ганкеля. Крiм того, наведено додаткове розширення вказаної теореми для функцiй у $L_k^{p}$ з абстрактною деформованою умовою Ганкеля – Дiнi – Лiпшиця. Institute of Mathematics, NAS of Ukraine 2022-10-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6134 10.37863/umzh.v74i8.6134 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 8 (2022); 1118 - 1127 Український математичний журнал; Том 74 № 8 (2022); 1118 - 1127 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6134/9290 Copyright (c) 2022 El mehdi Loualid |
| spellingShingle | Elgargati , A. Loualid, M. El Daher, R. Elgargati , A. Loualid, M. El Daher, R. Deformed Hankel transform of Dini – Lipschitz functions |
| title | Deformed Hankel transform of Dini – Lipschitz functions |
| title_alt | Deformed Hankel transform of Dini – Lipschitz functions |
| title_full | Deformed Hankel transform of Dini – Lipschitz functions |
| title_fullStr | Deformed Hankel transform of Dini – Lipschitz functions |
| title_full_unstemmed | Deformed Hankel transform of Dini – Lipschitz functions |
| title_short | Deformed Hankel transform of Dini – Lipschitz functions |
| title_sort | deformed hankel transform of dini – lipschitz functions |
| topic_facet | Deformed Hankel transfor Titchmarsh theore Symmetric differenc |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6134 |
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