Rev. Acad. Canar. Cienc., XIV (Núms. 1-2), 65-77 (2002)
Mittag-Leffier integral transform on Lv.r-spaces
Blanca Bonilla, Margarita Rivero, Luis Rodriguez-Germá, Juan J. Trujillo
Departamento de Análisis Matemático, Universidad de La Laguna,
'18271 La Laguna-Tenerife, Spain
Anatoly A. Kilbas, Natalia G. Klimets
Department of Mathematics and Mechanics, Belarusian State University,
220050 Minsk, Belarus
Dedicated to 80tb birtbday oí Prof. Nacere Hayek
Abstracts
The paper is devoted to the study of the integral transform
(E>.,uf)(x) = fo00 E>.,u(-xt)f(t)dt (x >O)
with positive ,\>O and complex u involving the Mittag-Leffier function E>.,u(z) in the kernel,
on the space Lv,r of Lebesgue measurable functions f on R + = {O, oo) such that
Jo{ "° dt ltv f(tW t < oo {l Sr< oo), ess SUPtER+ Wlf(t)IJ < oo (r = oo)
with real v E R , coinciding with the space Lr{R+) (1 Sr S oo) when v = l/r. It is proved
that the transform E>. uf can be represented by the integral transform with the H-function in
the kernel. Mapping properties such as the boundedness, the representation and the range of
this transform on Lv,r-spaces are established and its inversion formulas are given.
Key words: Mittag-Leffier integral transform, H-transform, Mittag-Leffier function, spaces
of p-summable functions
1991 AMS Subject Classification: 44Al5, 33El2, 47B38, 47G10
l. Introduction
The paper deals with the integral transform of the form
(E>.,uf)(x) = 100 E>.,u(-xt)f(t)dt (x >O) {1.1)
with real positive ,\ > O and complex u E C. The kernel of this transform contains the MittagLeffier
function E>.,u(z) defined for ,\>O and u E C by [7, Section 18.1]
00 k
E>. ,u(z) = L r(>.: +u) (z E C),
k=O
{1.2)
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where f(z) is the Euler gamma-function (6, Section 1). The transform (1.1) is clearly defined
for continuous functions f E Ca with compact support on R+ =(O,=).
It should be noted that the interest to studying integral transforms with special function
kernels was increased recently in connection with their applications in different problems of
pure and applied mathematics, in this connection see the books by Brychkov and Prudnikov [2],
Debnath [3) and Zayed [23). Basically, they have considered integral transforms contain special
bessel or hypergeometric functions in the kernels. The transforms (1.1) with the Mittag-Leffier
function as the ker~el were studied less. We can indicate only the book by Djrbashian [4] who
considered the integral transforms in the form
(1.3)
with positive >. > O and /3 > O and real B. Djrbashian [4, Lemma 4.5) proved that if O< >.::; 2,
3/2 < u < a+ 3/2 and () E [(>.rr)/2, 2rr - (>.rr)/2], then the transform EA,u ;9 is bounded from
L2(R+) into L:2u-1,2 (see (1.4)), established its inversion formula and some abelian theorems;
the results of Djrbashian were also presented in his book [5, Section 1.6) and in the book by
Zayed [23, Sections 16.2-16.4).
We study the Mittag-Leffier transform (1.1) in the spaces L:.,,r of those complex-valued
Lebesgue measurable functions f on R+ such that 11/llv,r < CXJ, where
( ("' di) 1/r
11/llv,r = lo lt" f(t)J' t (1 :S: r < CXJ, V E R = (-=, =)) (1.4)
and
11/llv,oo = ess SUPx>ox"lf(x)I (v E R). (1.5)
We note that, when v = l/r (1 ::; r ::; =), the space L:1¡r,r coincides with the classical Lr(R+)space:
Lr(R+) = L:1¡r,r·
We investigate the mapping properties such as the boundedness, the representation, the
range and the inversion of the Mittag-Leffier transform E>.,u on L:.,,r-spaces.
We show that the Mittag-Leffier transform (1.1) is a special case of the so-called H- transfrom
roo [ (a¡, a¡) , ... , (ap, ap) l (Hf)(x) =lo H;::¡t xt f(t)dt
(b1 , /31), ... , (bq/3q)
( 1.6)
with the H-function as kernel - see, for example, [15, Chapter 2), [16, Section 8.3) and [22,
Chapter l). This transform has the property
(MH!)(s) = 1l(s)(Mf)(I - s), (1.7)
with
= (MHm,n [t p,q
(1.8)
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under certain conditions on the function f. Here m, n, p, q are integers such that O :'S m :'S
q, O :'S n :'S p; a;, bj E C; a;, (Jj E R (1 :'Si :'S p, 1 :'S j :'S q) and ernpty products, ifthey occur,
are taken to be one. M is the Mellin transforrn defined by
(Mf)(s) = l'° f(t)t'- 1dt. (1.9)
We note that for f E C,,,r with 1 :'S r :'S 2 the Mellin transforrn M is defined by
(MJ)(s) = ¡: e<o+it)T f(e7)dr (s = <T + it, u, t E R) , (l.10)
and if f E Cv,r ílC,,,1 and Re(s) = 11, then (1.10) coincides with (1.9), see (17).
Mapping properties such as the boundedness, the representation and the range of the Htransforrn
(1.6) were proved independently in [8)-[9), [ll)-[13) and (1), while the invertibility of
(1.6) in C,, ,r was given in (19); in this connection see also [10, Sections 3 and 4).
In this paper we apply the results in (8) and [19) to investigate such properties of E>. ,0 -
transforrn (1.1). Section 2 deals with sorne results frorn the C,,,r-theory of the H-transfrorn
(1.6). In Section 3 we discuss the Mittag-Lefller tfansforrn as the H-transforrn. Section 4 is
devoted to the boundedness, the range and the representation of the Mittag-Lefller transforrn
E>.,o in the space Cv,r for r = 2 and any r :'.:: l. Section 5 deals with the inversion of the
transforrn E2,o in Cv,r ( r :'.:: 1).
It should be noted that we can investigate the rnapping properties of the Mittag-Lefller
transforrn E>. ,o in C,,,r with O < >. :'S 2, and that the results will be different in the cases
O < >. < 1, >. = 1, 1 < >. < 2 and >. = 2. Moreover, we can construct the inversion of such a
transforrn in the frarne of this space only for the operator E>.,o with >. = 2:
(E2,o/)(x) =fo"° E2,o(-xt)J(t)dt (x >O; <TE C). (1.11)
Therefore the problern to construct C,,,r-theory ofthe Mittag-Lefller transforrn (1.1) for >. > 2
stay open as well as the problern of its inversion for >. f. 2.
We also indicate that recently interest to the Mittag-Lefller function (1.2) and sorne oftheir
rnodifications and generalizations is increased by their applications in various physical and
mathernatical problerns, in particular in the frarnework of fractional differential equations; for
exarnple, see the survey papers [10] and [14). In this connection we hope that the results obtained
in this paper can be applied to the study of sorne problems in pure and applied rnathernatics
by analogy with the investigations carried out by using Hankel-type integral transforrns; for
exarnple, see the books by Sneddon (20)-(21).
2. Auxiliary Results
In this section we give sorne results frorn the theory of the the H-transforrn (1.6) on C,,,rspaces
given in (9) and [19). Following these papers we use the notation
max [-Re(b¡) ... _Re(bm)] (m >O),
f31 ' ' f:Jm
(2.1)
-oo (m =O);
min rl - Re(ai) ... 1-Re(a.)] (n >O),
O:'¡ \ ' ' O'.n
(2.2)
oo (n =O);
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n p m q
a*= L°'i - L a¡+ L.Bi - L ,Bj; (2.3)
i=l i=n+l j=l j=m+l
m p n q
at = L .Bi - L a;, a;= L°'i - L ,Bj; (2.4)
j=l i=n+l i=l j=m+l
q p
~ = L.Bi - L°'i' (2.5)
j=l i=l
q p
μ=L" .,, bj - "L .,, a;+p--q- .
j=l i=l 2
(2.6)
We denote by E11, the exceptional set of the function 1l defined in (1.8) which is the set of real
numbers v such that a < 1-v < .B and 1l(s) has a zero on the line R e(s) = 1-v. We denote by
[X, Y] the collection of bounded linear operators from a Banach space X into a Banach space
Y.
The L:..,,rtheory of the H-transform is given by two following statements.
Theorem A {[9, Theorem 3].) Let a < 1 - v < .B and either a* > O ora* =O, ~(1 -
v) +Re(μ) ~O. Then the following assertions hold:
{a) There is one-to-one transform H E (L:..,,2, L1-v,2] so that the relation (1. 7) holds for
f E L:..,,2 and Re(s) = 1 - v. lf a*= O, ~(1 - v) +Re(μ) = O and v 'f. E11,, then the operator
H maps Lv,2 onto L1-v,2·
(b) For f, g E L:..,,2 the relation of integration by parts holds
L"' f(x)(Hg)(x)dx = fo00 g(x)(Hf)(x)dx .
(c) Let f E L:..,,2, w E C and h > O. lf Re(w) > (1 - v)h - 1, then Hf is given by
(HJ)(x) = xl-(w+l)/h~x(w+l)/h
d:r.
100 [ m,n+l x 0 Hp+l,q+l xt
(-w,h),(a1,a1),· ·· , (ap,ap) l
(b1,.B1) , · · ·, (bq,Bq ), (-w- l ,h)
lf Re(w) < (1 - v)h - 1, then
(HJ)(x) = -xl- (w+L)/h~X(w+L)/h
dx
f(t)dt.
X H m+l,n xt 1 f(t)dt . 100 [ (a1,a1) ,···, (ap ,ap),(-w, h) l p+l,q+l
o (-w - 1, h)(b¡ ,,B¡), · · ·, (bq,Bq)
(2.7)
(2.8)
(2.9)
( d) H is independent on v in the sense if v1 and v2 satisfy the conditions in the first sentence
of this theorem, and if the transforms H¡ and H2 are defin ed in L:..,1 ,2 and L:..,,,2 respective/y by
(1. 7), then H¡f = H2/ for f E L:..,1 ,2 ílL:v,,2·
Theorem B ([9, Theorem 4].) Let a < 1 - v < ,B, f E Lv,2 and either a* > O or
a*= O, ~(1 - v) +Re(μ) < O. Then for x E R+ the transform (HJ)(x) is given by (1.6).
Sorne of the results above can be extended to the space Lv,r with any r E (1, oo). The range.
of the H-transform on L v,r is different in nine cases. We shall use only four cases; a) a* = O
and ~ > O ; b) ai > O and a5 > O; c) ai > O and a5 = O and d) a* > O, ai > O and a5 < O. For
this we need the elementary transform M€ of the forms
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(M€f)(x) = xE f(x) (~E C), (2.10)
the Erdelyi-Kober fractional integration operator l ':!.;o,ry (see (18, Section 18.l]) defined for a, r¡ E
C with Re(a) > O and 6 E R+ by
(!~. J)( x) = __!.._____ (tº - xº)"- 1t 0(l-a-ry)-l f(t)dt (x > O) ,
6 ury ¡oo
,ó,ry f(a) X
(2 .11)
the modified Hankel transform Hk ,ry given for k E R (k f. O) and r¡ E C with Re(r¡) > -1 by
(Hk ,ryf)(x) = 100 (xt) 1fk-lf 2 Jry (1kl)(xt) 11k) f(t)dt (x > O) ;
and the modifi.ed Laplace transform Lk ,€ defined for k E R (k f. O) and ~E C by
(Lk,{/)(x) = L"' (xt)-€ exp ( - lkl(xt) 1fk) f(t) dt (x >O).
Note that if k = 1, the transform (2.12) coincides with the classical Hankel transform
(H1,ry /)(x) ::=: (Hry!)(x) = 1"° (xt) 112 Jry(xt)f(t)dt (x >O),
while for k = 1 and ~=O (2.13) yields the Laplace transform
(L1,o/)(x) ::=: (Lf)(x) = 1"° e-xt¡(t)dt (x > O).
There hold the following assertions.
(2.12)
(2.13)
(2.14)
(2.15)
Theorem e ([9, Theorem 7]). Let é = O, ~ > O, a < 1 - V < (3 , 1 < r < 00 and
~(1 - v) +Re(μ) :::; 1/2 - 7(r), where
¡(r) = max [~, ~1 , ~ + ~ = l.
r r' r r
(2.16)
(a) The H-transfo rm defined on Lv,2 can be extended to L v,r asan element of [Lv,ri L1-v,p]
Jor ali p with r :=:; p < oo such that p' 2: (1/2 - ~(1 - v) - Re(μ)J- 1 .
(b) lf 1 :::; r :::; 2, then H is a one-to-one transform on Lv,r and there holds the equality
(1. 7).
(e) IJ / E Lv,r and g E Lv,p, 1 < p < oo, 1/r + 1/p 2: 1 and ~(1 - v) +Re(μ) :=:;
1/2- max[¡(r) ,¡(p)], then the relation (2. 7) holds.
( d) IJ v <f_ E:'H , then the transform H is one-to-one on Lv,r. Jf we set ~ = -~a - μ - 1,
then Re (~) > -1 and there holds
(2.17)
When v E &'H, then H (.Cv,r) is a subset of the right hand side of (2. 17).
(e) lf f E .Cv,r, w E C , h >O and ~(1-v)+Re(μ):::; l/2- 7(r), then Hf is given in (2.8)
for Re(w) > (1- v)h-1, and by (2. 9) for Re(w) < (1- v)h - l. lf ~(l - v) + Re(v) < O, Hf
is given by (1 .6).
Theorem D ([9, Theorem 9]). Let a* > O, a < 1 - v < f3 and 1 :=:; r :=:; p :=:; oo.
(a) The H -transform defined on L v,2 can be extended to L v,r as an element o/[Lv,r, L 1-v,p].
lf 1 :=:; r :=:; 2, then H is a one-to-one transform from L v,r onto L1 - v,s·
(b) lf f E .Cv,r and g E Lv,p' with ( 1/ p + 1/ p' = 1, then the relation (2. 7) holds.
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Theorem E ([9, Theorem 10]). Let at > O, a5 > O, m > O, n > O, a < 1 - v < f3 and
w =μ+ata - a5f3 + 1 and {et 1 < r < oo.
(a) lf v <t &11, ar if 1 < r :::: 2, then the transfarm H is ane-ta-ane an Cv,r.
(b) l/Re(w) 2: O and v <t &11, then
When v E &11, H (C,,,r) is a subset a/ the right hand side a/ {2.18).
(e) lf Re(w) <O and v <t &11, then
When v E &11 , H (C,,,r) is a subset a/ the right hand side a/ (2.19).
(2.18)
(2.19)
Theorem F ([9, Theorem 11]}. Let at > O, a5 = O, m > O, a < l - v < f3 and
w =μ+ata+ 1/2 and let 1 < r < oo.
(a) IJ v <t &11 , ar if 1 < r :::: 2, then the transfarm H is ane-ta-ane an Cv.r.
(b) lf Re(w) 2: O and v <t &11. then
H (C,,,r) = ( La;,o-w/a;) (C,,,r) ·
When v E &11., H (C,,,r) is a subset a/ the right hand side a/ {2.20).
(e) I/Re(w) <O and v <t &11, then
H (Cv,r) = (C~¡a• -a'oLa;,o) (C,,,r) ·
1 l ' l
When v E &11, H (C,,,r) is a subset a/ the right hand side a/ {2.21).
(2.20)
(2.21)
Theorem G ([9, Theorem 13]). Let a• > O, a1 > O, a5 < O, a < 1 - v < f3 and
1 < r <OO.
(a) lf either v <t E11 ar 1 < r :::: 2, then H is a ane-ta-ane an C,,,r.
(b) Let
• 1
w=a~ - μ--,
2
where μ is given by (2. 6), and let ~, ( E C be chasen as
If v <t E11 , then
a*Re(<) 2: 7(r) + 2a;(v - 1) +Re(μ) , Re(~)> v - l
Re(()< 1- v.
(2.22)
(2.23)
(2.24)
H(Cv,r) = ( M1/2+w/(2a;)H- 2a; ,2a;(+w- 1L-a* ,l/2+{-w/(2a; )) (C3¡2-v+Re(w)/(2a; J.r ). (2.25)
When v E E11, then H(C,,,r) is a subset a/ the right hand side a/ {2. 25).
3. Mittag-Leffier transformas the H-transform
The C,,,r-theory of the Mittag-Leffler transform (1.1) is based on the Mellin transform of
the Mittag-Leffler function (1.2) .
Lemma l. Let .A, <T E R and s E C be such that O < .A :::: 2, 3/2 < <T < .A + 3/2 and
.ARe(s) = <T - 3/2.
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Then the Mellin transform (1.9) of E>.,a(-x) is given by
(M[E (- )])( ) = f(s)f(l - s)
>. ,a x s f(u-..\s). (3.1)
Proof. It is known the relation (4, Lemma 5]:
"' e(•+μ-1)-1 - ¡oo . 7r eie(7r-a)(•+μ-l)
0 E1/e,μ+i(xe )x dx- f(2-s)sin(rre(s+μ-1)] ( Re(s) = D, (3.2)
being held for
11 llrr 7r
e"2-2' -<μ<-+- -<a<2rr--.
2 2 e' 2e - - 2e
(3.3)
Taking a = rr, e = 1/..\, μ=u - 1 and replacing e(s + μ - 1) by s, the conditions in (3.2)
trans[er to the condition of Lemma, and (3.3) is reduced to the relation
100 •-1 7í 1 ( 1) E>.a (-x)x = ( ..\)-.-(-) Re(s..\-u+2)=-2 .
O ' f <T - S Sll1 7íS
(3.4)
In accordance with the reflection formula for the gamma-function (see (6, 1.2(6)])
r(s)f(l - s)
sinl rrs) 7r
and hence (3.4) yields (3.1). Thus lemma is proved.
The relation (3.1) can be extended to more general complex <T and s.
Lemma 2. Let..\ E R , <T E R and s E C and let either of the conditions (a) O < ..\ < 2,
O< Re(s) < 1 or (b) ..\ = 2, O< Re(/3) < 3, O< Re(s) < min[l,Re(u)/2] hold.
Then the Mellin transform (1 .9) of E>. ,a(- x) is given by (3. 1).
Proof. We use analytic continuation to extend (3.1) to the range of complex /3 and s
indicated in (a) and (b). Since f(z) is analytic function of z E C having simple poles at the
points z =O, -1, -2, · · · (6, Section 1], the right hand side of (3.1) is analytic function of s E C
except points s =O, ±1, ±2, · · ·. The left hand side of (3.1) is also analytic of those s E C for
which the integral
(M[E>. ,a(- x)])(s) = ['º E>. ,a(- x)]x'-1dx (3.5)
converges. By (1.2)
E>. ,a(-x)x'-1 =O (x1
1_,) (x-+ +0)
and the integral (3.5) converges at zero for Re(s) > O. According to (4, Lemmas 3.4 and 3.5],
the Mittag-Leffier function (1.2) has different asymptotic behavior at infinity for O < ..\ < 2 and
..\ = 2, O < Re(/3) < 3:
E>. ,a(-x) =O (D (x-+ +=; O<..\< 2)
and
E>. ,a (-x) = x(l-a)/ 2 cos [vx + Í(l- u)]+ O G) (x-+ +=; ..\ = 2) ,
respectively. Therefore
E>.,a (-x)x'- 1 =O (xL') (x-+ +=; O<..\ < 2)
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and
EÁ, 3-l 1 [ r.::: 7r J ( 1 ) 0 (-x)x = x-•+(I+o)/2 cos vx+2(1-u) +O x 2_, (x--+ +oo; A= 2).
and, in accordance with the known convergence theorems of analysis, the integral in left hand
side of (3.5) converges at infinity for O < A < 2, Re(s) < 1 and for A = 2, O < Re(/1) <
3, Re(s) < min[l, Re(u)/2], and these conditions are satisfied by the conditions of Lemma.
Therefore the analytic continuation yields the result in lemma.
Remark l. The relation (3.1) was indicated in (16] but without the condition O< Re(/1) < 3
in the case A = 2.
Using (3.1), it is directly verified for "sufficiently good" function f that the Mellin transform
M of (1.1) is given by
(MHf)(s) = r(s)f(l - s) (Mf)(l - s).
r(u - As)
According to (1.8) the function in the right hand side of (3 .1) is the 1l-function
f(s)r(l - s) _ 1,1 [ (O, 1) 1 ]
f(u - As) - Hr,2 (O, 1), (1 - u, A) 8 .
(3.6)
(3.7)
Therefore (3.6) is the relation of the form (1.7), and hence the EÁ ,o f -transform (1.1) is a
special H-transform (1.6):
(EÁ,of)(x) = l'º Hi,'i [xt 1 (O, l):~l ~ u,A) ] f(t)dt. (3.8)
According to (2.1)-(2.6) we have
1
a = O, /1 = 1, a• = 2 - A, a;- = 1, a; = 1 - A, .:l = A, μ = - - u. (3.9)
2
lf E1i is the exceptional set of the 1l-function in (3.7), then vis not in the exceptional set E1i,
if
S _.,.J... O" +A k (k =O, 1, 2, · · ·) .
4 . .Cv,r-theory of the Mittag-LefBer transform
(3.10)
Using the results in the previous section, we can apply Theorems A-G to construct the
.Cv,r-theory of the Mittag-Leffler transform (1.1). First from Theorems A and B we deduce the
.Cv,rtheory of this transform.
Theorem l. Let O < v < 1 and let a > O and u E C be such that either O < A < 2 ar
A= 2, O< Re( u) < 3, 2v +Re( u) ~ 5/2. Then the following assertions hold:
(a) There is one-to-one transform EÁ,o E [.Cv,2, .C1-v,2] so that the relation (3.6) holds far
f E .Cv,2 and Re(s) = 1 - v . lf A = 2, O< Re( u) < 3, 2v +Re( u) = 5/2 and the condition in
(3.10) is fulfilled, then the operator EÁ,o maps .Cv,2 onto .Ci- v,2·
(b) Far f, g E .Cv,2 the relation o/ integration by parts holds
(c) Let f E .Cv,2, w E C and h > O. lf Re(w) > (1 - v)h-1, then EÁ,of is given by
(EÁ,of)(x) = XI-(w+I)/h ~ :r(w+I)/h
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(4.1)
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X f H!j [xt (-w,h),(O,l) l f(t)dt.
(O, 1), (1 - u,..\), (-w - 1, h)
(4.2)
If Re(w) < (1- 11)h- 1, then
(E.>.,uf)(x) = -xl-(w+l)/h ~ x<w+l)/h
roo [ (0,1),(-w,h) l x Jo Hi;~ xt f (t)dt.
(-w - 1, h), (O, 1), (1 - u,..\)
(4.3)
( d) E.>. ,u is independent on 11 in the sense if 111 and 112 satis/y the conditions in the first
sentence of this theorem, and if the transforms E.>.,u;l and E.>. ,u;2 are defined in Lv1 ,2 and Lv2 ,2
respectively by (3.6}, then E.>.,u;lÍ = E.>.,u ;2Í far f E Lv1,2ílCv2 ,2.
(e) If f E Lv,2 and either O<..\< 2 or ..\ = 2, O< R e( u)< 3, 211 +Re( u)> 5/2, then far
x E R+ the transform (E.>. ,uf)(x) is given by (1.1} and (3.8).
If ..\ = 2, then according to (3.9) a• = O and from Theorem C we deduce Cv,r-theory of the
Mittag-Leffler transform (1.1) for any r > l.
Theorem 2. Let ..\ = 2, O < Re(u) < 3, O < 11 < 1 and 1 < r < oo be such that
211 +Re( u) 2: 2 + l'(r), where !'(r) is given in (2.16).
(a) The E2,0 -transform defined on Lv,2 can be extended to Lv,r asan element of[Cv,r, L1-v,p]
far ali p with r:::; p < oo such that p' 2: [211- Re( u) - 2]- 1 .
(b) If 1 ::=; r ::=; 2, then E2,u is a one-to-one transform on L v,r and there holds the equality
(3.6) with ..\ = 2.
(e) If f E L v,r and g E Lv,p, 1 < p < oo, l/r+ 1/ p 2: 1 and 211+Re(u) 2: 2+max[!'(r) , ')'(p)],
then the relation (4.1) with ..\ = 2 holds.
(d) If the condition in (3.10} with ..\ = 2 is satisfied, then the transform E2,u is one-to-onf"
on Lv,r and and there holds
(4.4)
lf the condition in (3.10} with ..\ = 2 is not satisfied, then E 2,,, (Cv,r) is a subset of the right
hand side of (4 .4).
(e) lf f E Cv,r, w E C , h >O and 211) +Re( u):::; 2 + !'(r), then E2,uf is given in (4.2) far
Re(w) > (l -11)h- 1, and by (4.3) far Re(w) < (1- 11)h- 1 with ..\ = 2. lf 211 +Re( u) > 5/2.
E2,uf is given by (1 .1} and (3.8) with ..\ = 2.
Let now O< ..\ < 2, and hence a* > O by (3.9) . Fr-0m Theorem E we obtain the boundedness
of the Mittag-Leffler transform (1.1) in Lv,r and the relation of the integration by parts.
Theorem 3. Let O < ..\ < 2, O < 11 < 1 and 1 :::; r :::; p :::; oo.
(a) The E.>, ,,, -transform defined on Lv,2 can be extended to Lv,r asan element of[Cv,r . L1-v,p].
IJ 1 ::=; r ::=; 2, then E.>. ,u is a one-to-one transform from Lv,r anta LJ-v,•·
(b) lf f E Lv,r and g E Lv,p' with (l/p+ l/p' = 11 then the relation (4.1} holds.
When O < ..\ < 2, then by (3.9) aj = 1 and a; = 1 - ..\. Theorems E-G give the following
results in which the range of the Mittag-Leffier transform E>. ,u will be different in the cases
O < ..\ < 1, ..\ = 1 and 1 < ..\ < 2 corresponding to the ones in a2 > O, a; = O and a; < O,
respectively.
Theorem 4. Let O<..\< 11 O< 11 < 11 w =u - ..\ - 1/2 and let 1 < r < oo.
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(a) IJ the condition in {3.10) is satisfied, or if 1 < r ::; 2, then the transform E>. ,17 1s
one-to-one on Cv,r.
(b) lf Re( u)::; .X+ 1/2 and the condition in {3.10} is satisfied, then
E>. ,a (Cv,r) = (LL1 ->.,w/(l ->.)) (C1-v,r) · (4.5)
When the condition in {3.10) is not satisfied, E>.,a (C.,,r) is a subset of the right hand side of
{4.5).
(e) IJ Re( u)> .X+ 1/2 and the condition in {3.10} is fulfilled, then
(4.6)
When the condition in {3.10} is not satisfied, E>. ,a (Cv,r) is a subset of the right hand side of
(4.6).
Theorem 5. Let O < v < 1 and 1 < r < oo.
(a) lf the condition in {3.10) with .X= 1 is satisfied, or if 1 < r::; 2, then the transform
E1,17 is one-to-one on Cv,r·
(b) lf u S: 1 and the condition in {3.10) with A= 1 are satisfied, then
(4.7)
When the condition in {3.10} with .X = 1 is not satisfied, Ei,17 (C.,,r) is a subset of the right
hand side of (4. 7).
(e) lf u > 1 and the condition in {3.10} with .X = 1 is fulfilled, then
(4.8)
When the condition in {3.10) with = .X= 1 is not satisfied, E1,17 (C.,,r) is a subset of the right
hand side of {4-8).
Theorem 6. Let 1 < .X< 2, a; = 1 - A, O< u< 1 and 1 < r < oo.
(a) lf the condition in {3.10) is satisfied, or if 1 < r ::; 2, then the transform E>. ." 1s
one-to-one on Cv,r.
(b) Let
w = (2 - .X)~ - μ+u - 1, (4.9)
and let ~, ( E C be ch osen as
1
(2- .X)Re(O 2': ¡(r) + 2(1- .X)(v - 1) - Re(u) + 2, Re(~) > v - 1 (4.10)
Re(()< 1 - v. (4.11)
If the condition in (3.10) is satisfied, then
E>.,a (Cv,r) = ( Mi¡2+w/(2a;)H-2a;,2a;( +w-1 L>.-2,1/2+{-w/(2a;J) (C3¡2-v+Re(w)/(2a;),r) · ( 4.12)
When the condition in {3.10} is not satisfied , then E>. ,a (Cv,r) is a subset of the right hand side
of {4-12).
5. lnversion of the Mittag-Leffier transform
On the basis of the results of Section 2, we deduce the inversion of the Mittag-Leffier
transform ( 1.1) from the inversion of the H-transform (1.6) . As it was indicated in Introduction,
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the inversion of the H-transform (1.6) in the spaces Lv,r was investigated in (19]. Here we
present sorne results from this paper. Let
{
max [- Re(bm+1 )-1 + l . . . _ R e(b9 )- l + l] P-+1 ' ' pq
ªº =
-oo (q = m);
(q > m),
{
min r~ + 1 . . . Re(ap) + l] an+i ' ' ap
/30 =
oo (p = n).
(p > n) ,
The inversion of the H-transform can have the respective form (2.8) or (2.9):
or
¡ q-m+l p-n x 00 [ 0 Hp+l ,q+i xt
f(x) = xl-(w+l)/h__<!_x(w+l)/h
dx
(1 - a; - a;, ai)n+l,p, (1 - a; - a;, a;)1,n, (-w, h)
(5.1)
(5.2)
l (H/)(t)dl
(5.3)
l {H/)(t)dl
(5.4)
These formulas are true provided that a• = O under sorne additional conditions which are
different in the cases ~ = O, ~ > O and ~ < O. We give only the result for ~ > O.
Theorem H ([19, Theorem 4.1]). Let 1 < r < oo, -oo < a < 1 - 11 < /3, ao < 11 <
min{/30, [Re(μ+ 1/2)/ ~] + 1}, a• =O, ~ > O and ~(1 - 11) +Re(μ) ::; 1/2 + ¡(r), where ¡(r)
is given by (2.16). Let w E C and h >O.
lf f E C,,,r, then the relation (5.3) holds far Re(w) > 11h - 1, while the formula (5.4) is
validforRe(w) < 11h- l.
According to (3.8), (3.9), (5.1) and (5.2)
a= O, /3 = 1, a* = 2 - A, μ = ~ - O", a0 = 1 - Re;O"), /30 = oo. (5.5)
Since a• = O for A = 2, then from Theorem H we deduce the following result which yields the
inversion of the Mittag-Leffier transform (1.11) in the space Lv,r·
Theorem 7. Let 1 < r < oo, 11 E R and O" E C be such that
max [o, 1 - Re~O")] < 11 < min [1, ~ - Ret)] , 211 + Re(O") :::: 2+ ¡(r). (5.6)
lf f E Lv,r and Re(w) > 11h - 1, then the inversion formula
[°" [ (-w, h) , (0,1)
x Jo H~·~ xt
o ' (0"-2,2),(0,1), (-w-1,h) l (E,,,f)(t)dl (5 .7)
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is valid, while far Re(w) < vh - 1 there holds the relation
J(x) = -xl-(w+l)Jh_!!_x(w+l )/h
dx
x ¡00 H32,·3º [xt o
(0,1),(-w ,h) l (E2 ,a f)(t)dt.
(-w - 1, h)(u - 2, 2), (O, 1)
(5.8)
Rernark 2. Taking v = l/r in Theorem 1 and Theorems 2-6, we can obtain the results
characterizing the theory of the Mittag-Leffier transform E,\ ,a in the spaces L2 (R+) and Lr (R+)
(r 2'. 1), while from Theorem 7 we obtain the inversion of the integral transform E2 ,0 / in
Lr(R+)·
Acknowledgernent
The present investigation was initiated during the first author's stay at University of La
Laguna as Visiting Professor and it was partly supported by Belarusian Fundamental Research
Fund.
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