CC BY
39URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 53-58
DOI: 10.15826/umj.2019.1.006
JACOBI TRANSFORM OF (v, Y,p)-JACOBILIPSCHITZ FUNCTIONS IN THE SPACE Lp(R+, A(a,ß)(t)dt)1
Mohamed El Hamma1^, Hamad Sidi Lafdal11, Nisrine Djellab1,
Chaimaa Khalil1
1 Laboratoire TAGMD, Faculté des Sciences Aïn Chock, Université Hassan II, B.P 5366 Maarif, Casablanca, Marocco
11CRMEF, Laayoune, Morocco
Abstract: Using a generalized translation operator, we obtain an analog of Younis' theorem [Theorem 5.2, Younis M.S. Fourier transforms of Dini-Lipschitz functions, Int. J. Math. Math. Sci., 1986] for the Jacobi transform for functions from the (v, 7,p)-Jacobi—Lipschitz class in the space Lp(R+,
Keywords: Jacobi operator, Jacobi transform, Generalized translation operator.
Younis [8, Theorem 5.2] characterized the set of functions in L2(R) satisfying the Dini-Lipschitz condition by means of an asymptotic estimate of the growth of the norm of their Fourier transforms.
Theorem 1. [8, Theorem 5.2] Let f € L2(R). Then the following conditions are equivalent:
where F stands for the Fourier transform of f.
The main aim of this paper is to establish an analog of Theorem 1 for the Jacobi transform in the space Lp(R+, A(a,^)(t)dt). For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen and Koornwinder [5].
In order to confirm the basic and standard notation, we briefly overview the theory of Jacobi operators and related harmonic analysis. The main references are [1, 4, 6].
Let A € C, a > ft > -1/2, and a = 0. The Jacobi function of order (a,fi) is the unique even C^-solution of the differential equation
1. Introduction and preliminaries
as h ^ 0, 0 < a < 1, ß > 0,
(2) j|A|>r |F(f)(A)|2d\ = O(r-2a(logr)-2ß) as r ^
(Da,ß + A2 + p2)u = 0, u(0) = 1, u'(0) = 0, where p = a + ß + 1, Da,ß is the Jacobi differential operator defined as
1 Dedicated to Professor Radouan Daher for his 61's birthday.
with
A(a>/s)(x) = (2 sinh x)2a+1 (2 cosh x)2fi+1,
and A(a^)(x) is the derivative of A(a ^)(x).
The Jacobi functions can be expressed in terms of Gaussian hypergeometric functions as
4>\(x) = 4>^\ 'l3\x) (^(p — i\), ^(p + 'iA), a + l, —sinh2®
where the Gaussian hypergeometric function is defined as
^ cmm\
with a,b,z € C, c / —N, a0 = 1, and am = a(a + 1) ■ ■ ■ (a + m — 1).
The function z ^ F(a, b, c, z) is the unique solution of the differential equation
z(1 — z)u"(z) + (c — (a + b + 1)z)u' (z) — abu(z) = 0,
which is regular at 0 and equals 1 there.
From [7, Lemmas 3.1-3.3], we obtain the following statement.
Lemma 1. The following inequalities are valid for a Jacobi function <fi\(t) (A,t € R+):
(1) I0a(t)| < 1;
(2) |1 — 0A(t)|< t2(A2 + p2);
(3) there is a constant d> 0 such that
1 — 0A(t) > d for At > 1.
Let L^(R+) = Lp(R+, A(a ^)(t)dt), 1 < p < 2, be the space of p-power integrable functions on R+ endowed with the norm
\ 1/p
|f (x)|PA(«„3)(x)dxJ < ro.
Let L^(R+) = Lp(R+, dp(A)/2n), 1 < p < 2, be the space of measurable functions f on R+ such
l p
that
/ 1 rro \ 1 /p
1 fWMV) '
where dp(A) = |c(A)|-2dA and the c-function c(A) is defined as
2p-iXr(a + 1)r(iA)
c(A) =
r(1/2 ■ (i\ + a + p + 1))r(1/2 ■ (i\ + a - p + 1)) ' Now, we define the Jacobi transform
^ c ^
/(A) = f (x)0A(x)A(«,^)(x)dx,
J 0
for all functions f on R+ and complex numbers A for which the right-hand side is well defined. The Jacobi transform reduces to the Fourier transform when a = p = -1/2. We have the following inversion formula [6].
Theorem 2. If f € L^(R+), then
1 f ~ -
/0*0 = — j /(A)</>a(-*(A).
From [3], we have the Hausdorff-Young inequality
ll/IU < C21|f ||p for all f € La,,
where 1/p + 1/q = 1 and C2 is a positive constant.
The generalized translation operator Th of a function f € La,(R+) is defined as
Thf (x)= / f (z)K (x,h,z)A(a,) (z)dz, Jo
where K is an explicity known kernel function such that
v = 2~2pr(q + I)(cosh x cosh y cosh z)a~l3~1 _ r2^«-i/2 [■hy,«) r(I/2)r(o; + I/2)(sinh.Tsinhj/sinhz)2Q' ;
xF ^a + /3, a — /3, a + ^(1 — for \x — y\ < z < x + y,
and K(x,y,z) =0 elsewhere and
cosh2 x + cosh2 y + cosh2 2 — 1 2 cosh x cosh y cosh z
From [2], we have
(Tf )(A) = 0A(h)fr(A). 2. Main results
In this section, we give the main result of this paper. We need first to define the (v, Y,p)-Jacobi-Lipschitz class.
Definition 1. Let v,7 > 0. ^ function f € La,(R+) is said to be in the (v,Y,p)-Jacobi-Lipschitz class, denoted by Lip(v, Y,p), if
v
l|Tfe/(.T)-/(.T)||p = o((log^//j)7j as h y 0.
Theorem 3. Let f belong to Lip(v, Y,p). Then
/ |/(A)|fd^(A) = O(N-fv(logN)-fY) as N ^ Jn
Proof. Let f € Lip(v, Y,p). Then we have
hv
! as h —>• 0.
Y
Therefore,
/ |1 - ^(h)|fl/(A)|fd^(A) < Cf||Thf(x) - f(x)|p. o
If A € [1/h, 2/h], then Ah > 1 and inequality (3) of Lemma 1 implies that
1 <¿11-^)1**.
Then
r 2/h 1 r 2/h
/ |/(A)|qcfyt(A) < — / |1 - Mh)\qk\f(\)\qdAX) J1 /h J 1/h
1 r+,x , - 1 f hqv
"Wo 11 " <^(/>)n/(A)№(A) < ^C*||Tft/(:r) - f(x)\\l = O (j '
/0 I UV ,1 J - dqk ¿H j J Wllp V(log1/h)qY
Then
/• 2N
/ |f(A)|qdp(A)= O(N-qv(log N)-qY)) as N ^ +to. JN
Thus, there exists C4 such that
f 2N
/ |f(A)|qdp(A) < C4N-qv(log N)-qY.
N
Furthermore, we have
+,
/ |/(A)|q dp(A) =
N
/•2N /-4N /-8N / + / + /
N 2N 4N
|/(A)|q dp(A)
< C4^q"(logN+ C4(2N)"qiy(log2N)"q7 + C4(4N)-qv(log4N)-qY + ...
< C4N-qv(logN)-qY(1 + 2-qv + (2-qv)2 + (2-qv)3 + ...
< C4CkN-qv(log N)-qY),
where Ck = (1 - 2-qv)-1 since 2-qv < 1. This proves that
/ |/(A)|qdp(A) = O(N-qv(logN)-qY)) as N ^ +to,
N
and this completes the proof. □
Definition 2. A function f € L^ ^(R+) is said to be in the (0,p)-Jacobi-Lipschitz class, denoted by Lip(0,p), if
where
(1) 0(t) is a continuous increasing function on [0, to);
(2) 0(0) = 0;
(3) 0(ts) < 0(t)0(s) for all s,t € [0, to).
Theorem 4. Let f € L^ ^(R+), 0 be a fixed function satisfying the conditions of Definition 2, and let f (x) belong to Lip(0,p). Then
r+,
/ |/(A)|qdp(A) = O(0(N-q)(logN)-qY) as r ^ +to.
N
Proof. Let f € Lip(^,p). Then we have
llT^)-/(g)llp = Q((log(l/i)7) as /w0 r+m
/ |1 - 0A(h)|f |/(A)|fd^(A) < Cf ||Thf (x) - f (x)||p. o
If A € [1/h, 2/h], then Ah > 1 and, similarly to the proof of Theorem 3, by inequality (3) of Lemma 1, we obtain
1 <¿11-^)1**.
Then
r2/h 1 r2/h
/ |/(A)№(A) <-j: |1 - MhW'mWdAX)
J1/h J1/h
There exists a positive constant such that
/•2N ^ )
N 5 logN)<n
Thus,
/ |/(A)|q = In
r2N MN /-8N
/ + +
/N ./2N ./4N
l/(A)|q d^(A)
~ 5 (log N)q~f 5 (log 2N)<n 5 (log 4N)<n "
" 5 (log N)<n 5 (logiV)9T + 5 (logiV)97 < gs^^l + '0(2-") + C0(2-«))2 + W(2"*))3 + ...
- (log N)fY'
where K1 = (1 - ^(2-f))-1 since (1) and (3) from Definition 2 imply that ^(2-f) < 1. This proves that
r
/ |/(A)|fd^(A) = O(^(N-f)(logN)-fY) as N JN
and this completes the proof. □
Acknowledgements
The authors would like to thank the referee for his valuable comments and suggestions.
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