Internat. J. Math. & Math. Sci.
VOL. 18 NO. 4 (1995) 777-788
777
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE
ORDER IN THE HOLDER SPACES H (x)
BERTRAM ROSS
University of New Haven
300 Orange Avenue, West Haven, CT 06516, USA
and
STEFAN SAMKO
Rostov State University
105, Bol’shaya Sadovaya, Rostov-on-Don, 344711, Russia
Currently Fulbright Scholar at the University of New Haven,USA
(Received June 17, 1993 and in revised form September 20, 1993)
ABSTRACT.The
fractional
I+(x)0
integrals
of
considered. A theorem on mapping properties of
H
x(x)
is
proved,
this
being
a
variable
Ia+
(x)
generalization
order
e(x)
are
in Holder-type spaces
of
the
well
known
Hardy-Li ttlewood theorem.
KEYWORDS-
Fractional
integration,
variable
fractional
order,
mapping
properties, Holder continuous functions, Hardy-Littlewood theorem.
1991 AHS SUBJECT CLASSIFICATION CODES.
I.
26 A33, 26 A16
NTROOUCTION
In the paper [I] the authors introduced and investigated
the fractional
integrals
l(X)0
of variable order
’F[(x)]
e(x)>O
e(t)(x-t )(x)- dt
(1)
and considered the corresponding versions of
fractional differentiation as well.
In this paper we prove the theorem on the behaviour of the operator
l(X)
in the generalized Holder spaces
H )’(*)
the order of which also
778
B. ROSS AND S. SAMKO
depends on the point
This is a generalization of the Hardy
x
known in the case of constant orders
theorem, well
Littlewood
e(x)==const
and
and Littlewood [2]; see also Samko et al [3], p. 53-54).
x(x)=X=const (Hardy
and differentiation of a variable order is
Our interest in integration
motivated not only by the desire to generalize the classical notion, but
by
some far reaching goals as well. There is the well known theory of fractional
Sobolev type spaces
see its elements e.g. in
[3], sections 26-27. These
spaces consist of functions whose smoothness property can be expressed either
globally or locally in terms of the existence of fractional derivatives.
The
smoothness property of a function may, however, vary from point to point. The
construction of the corresponding Sobolev type spaces is an open question. The
notion introduced in (1) is the appropriate tool for this purpose.
In this
paper we deal only with the question of improving the smoothness property,
expressed in terms of the Holder type condition, by the operator (1), and the
theorem
proved
may
as
considered
be
a
starting
required definitions and some auxilliary lemmas,
further
In Section
investigations of functions with varying order of smoothness.
give all
for
point
we
while Section 2
contains the statement and the proof of the main result.
In what follows the letter
c
may
denote different positive constants.
2. PRELIMINARIES
Let
Q
.
[a,b]
H
the Holder space
)"
0 <
< a < b <
0.
The following is a generalization of
),
DEFINITION I. We say that
f(x)
E
H
)’(x)
(Q)
where
X(x)
is a positive
(not necessarily continuous) function, 0 < X(x) < I, if
If(x+h)-f(x)l
for all
x
x+h
E
cihl
x()
(2)
[a,b]
It is easily seen that (2) implies that
f(x+h)-f(x)lclhl
x(x+h)
So, it is not difficult to show that the definition of the class
H)’()(C))
by
(2) is equivalent to the definition by means of the following symmetrical
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER
779
nequal ty
If(x )-f(x )!
clx-x
2
2
ImaX{X(Xl)’X(x2 ))
HX()(O)
It is easily seen that
(3)
is a ring with respect to the usual
multiplication. It is a Banach space with respect to the norm
IlfllHx
f(x+h)-f(x)l
sup
sup
xeO
(x)
Ihl
h<
x(x)
h+ xEf
f(x )-f(x )1
2
sup
IX
E
2
Xl’
--X
g <= c
f
denotes the equivalence-
where
f(x)
DEFINITION 2. We say that
E
H x(x)’W(*)(O)
,
< p(x) <
are given functions, 0 < x(x) < 1,
clhl
x, x+h
under assumption that
We shal
E
)’(x)
where
),(x)
(x)
and
if
i__11/(x)
In
x+h
Ihl
<-2-
g
x,t,t+s(x-a)
E
s,t
d
const > 0
A
(5)
h
O
s
(6)
(x-a)
(x)
E
[0,1]
0 < d -^ s g
with a constant
satisfy the condition
Then the function
0
E
C(O)
E
A
<
l+ln
where
2
need the fol lowing auxi 11 iary assertions.
IX(x+h)-X(x)l
x
c >0, c >0.
O
LEMMA I. Let the function X(x)
for all
f<g;czf,
we give the following
Generalizing Definition
If(x+h)-f(x)l
(4)
Imax((xl ’(x2)
2
is bounded from zero and infinity-
(x)
max{e,b-a,I/(b-a)}
_<
d^ <
,
not depending on
(7)
s,t and
x
B. ROSS AND S. SAMKO
780
PROOF. Since
In g
{).[t+s(x-a)]
(x)
)‘(t)} In(x-a),
Iln g s, (x)l
In(b-a) > 0
c
A(l+c)(l+c-y) -1
> 0, so that f(y) < f(c)
If 1+c < 0
for y
_<
LEMNA 2. For any function (x)
+h)
Ay(l+c-y)-l
Ay
<
Therefore,
we have f (y)
A
f(y)
A(l+c)(l+c-y) -2
1, Iln(b-a)l
0 < (x)
such that
xCX()
cxcx)
y s c
A max{1,11n(b-a)l} ,whence (6) follows.
g(x)l
in all cases. Therefore,’ln
f(y)
A
c which gives f(y) < Alcl= A I]n(b-a)l. So,
A max
f(y)
where-
Then y < 0 and
c < 0
Let now
1+c z 0
if
A
<
-,
,
is
1, ln(b-a)
A max
f(y)
-
y>O we have
Alyl(l+c+lyl)
Aln(b-a). If y<0, then f(y)
x---E
A lyl(l+c-y)
f(y)
first. For
b-a z
Alln(x-a)l
b:a
1+1n
the maximum of the right-hand side
Really, let
ln(b-a). Suppose
in the case
Alln(x-a)l
b-a
1+1n s(xa
<
simple calculations show that
max{1,11n(b-a)l}
by (5) we have
h
cx(x)
the inequality
(8)
h>O, x>O
holds.
PROOF. By dividing a]] members in (8) by
inequality (8)
is equivalent
(l+c)(x)_ ((x)>
0
for a]]
h
a(x)
we see that the
to the inequality f(C)
C >
which is evident because
0
with
f(()
f (() < 0
and
f(O)
LEMNA 3. Let 0 <
<
and
[ tX[tx--(t+l) (z-]
O
(
dt
< X <
sin
sin (4:k)
Then
B(, 1+),)
(9)
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER
A(.,)
Let
PR(X)F.
substitution
S-1
t+l
781
left-hand side in (9)
denote the
After the
we have
)), (1-s)a- -1
(10)
ds
s
o
Hence
A(,,=)
(1-s)" l+(l_s)p
(1-s )X
[
1++0
0
with
/
(1-s)/-1
(1-s )X
ds +
ds
S
0
not determined as yet. Hence, by (10)
A(;k =)
A()+/,=-/) +
fo (l-s) X
(1-s)P-1
ds
s
The second integral here is evaluated by means of analytical continuation with
respect to
(l-s) x
[
0
(1-s)P-1
SI+X+(
B(X++I
ds
-X-a)
under the appropriate conditions on the parameters
A(),,)
A(),+/,-p) + r(-),-()
Simple calculations show that
A(X,)
:X+I
A(0,a)
1/a
B(X+I
M
[F(x++l
and
F(X+I
LF(+l_a)
So,
X
I]
So, choosing /=-), we have
+ F(-X-)
F( 1-
which gives (9) after easy calculations.
3. THE HAIN THEOREN
Considering the fractional
function
and
x(x)
0(x)
H )’(x)
integral
Ia(X)0
defined in (1), of the
we shall assume the following conditions on
to be satisfied-
(x)
782
B. ROSS AND S. SAMKO
i) 0 < (x)
H/(x)(o)
(x)
and
inf (x) > 0
XEO
0 < 6 < p(x) <
m
with
ii) 0 < X(x) <
X(x)
and
satisfies the condition (5)-
iii) (x) + (x) <
THEOREM. Let the conditions
HX(x)(O)
Then
I()0
be satisfied and let
iii)
i)
(x)
_
has the form
(a)
I()0
F[I /(((x)]
+ f(x)
(x-a
(11)
where
f(x)
if
E
max[X(x)+(x)] <
H(X)(o)
(12)
min{),(x)+({(x), p(x)}
(x)
and
xEO
f(x)
if
max[X(x)+(x)]
E
H Y(x)’(O)
(13)
In both cases
x()*()
x(’()+(’()
< c(x-a)
If(x)l < c(x-a)
(14)
REMARK. The assertion (13) can be exactified:
If(x+h)-f(x)l
for all
x e 0 such that
(x) + X(x) <
If(x+h)-f(x)l
for those
x
c
7(x)
1-(x)-(x) Ihl
constant not depending on x
and
clhl 7(x) In
which give the equality
and
h
I[o(x)]
E
HP(x)(Q)
(16)
Ihl <
IIII
c being a positive
(x) + x(x)
(see the proof of the theorem).
PROOF OF THE THEOREM. From the conditions
(x)
(15)
Really,
i)
and
17(x+h)-7(x)l
ii)
<
it follows that
l(x+h)
(x)l
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER
ax
c Ihl pc)
(lI"’(x)/F(x)l)
So,
sufficient to consider the
is
it
783
integral
(x-t )((x
g(x)
-1
0(t)dt
we have to prove that
(a) (x-a)( (x) +
(X)’
g(x)
(17)
fo(x)
where
f (x) c H
)’cx)+cx)
or
0
if
(18)
0
respectively
max[),(x)+(x)]=l
or
max[X(x)+(x)]<l
xcx)+cx)’1
f (x) c H
(with
the
exactification (15)-(16), if we will). The derivation itself of the equali.ty
(17) with the function
[ _(t)-(a)_
1-cx)
(x-t)
f (x)
o
is obvious. For the function
If 0 (x)l
g
s,a
o
we prove first the estimate (14). We have
I111 H x()[
C(x_a)e(x),X(a
where
f (x)
(x-t )e(x)- (t-a)
<
c (x-a)
dt
g
[1(l_S)e(x)_isX[a,s(x_a)}
o
(x)- is the function (6). By lemma
If o (x)l
dt
cx)+Xca)
[ (1-s )m-
(x) ds
we have
ds
o
c (x-a)
(x)+)’ca)
2
Hence, to obtain the estimate (14), it remains to observe that
(*)/()
(x_a)(x)*x(a) <c(x-a)(x)*X(X)c (x-a)(a)*x()c 2 (x_a)
which
follows
from
the
lemma
(we remark that
the
condition
(5)
for
784
B. ROSS AND S. SAMKO
X(x)+(x)
in
ii)
x(x)
is fulfilled because it is satisfied for
(x)
and for
by the assumption
i) ).Thus, (14) is proved.
by the assumption in
To prove the statements (18) we consider the difference
h < O, by denoting
positive. (If
h
taking
x
x+h
f (x+h)
0
f (x)
0
x +(-h) we reduce
x
to the case of positive increment). We represent this
the consideration
difference as
f(x+h)
f(x)
Po(t)_
e(t)
JJ
(t+h) (*+h)-
e(a)
o max
t-(x)
0
(t+h) (*)-m
dt
(19)
o
0
We estimate
Jeo(t)l
0 (x-t)tit
t-cCx)
(t+h)a-c)
-h
i
(t+h)-(x(x+h)
-h
Co(X-t)
-h
with
I
o (x-t)dt
I
J
0
first. We have
I(t+h)(x*h)-- (t+h)(x)-l
dt
-h
x+h-a
(x+h)-I
tu-t
Since
tlnt
(u-v)
t(x)-
between u and v
with
dt
we obtain
x+h-a
IJ
!-o
tF’-lln
cl(x+h)-(x)l
tl dt
0
(x+h)
and
e(x)
with
(
between
with
A
max{1,(b-a) -m}, we have
so that
z m > 0
Since
t(-ls
At m-I
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER
cAh/(x)
IJol
As regards the term
2(b-a)
tm-lln
c h/(*)
tl dt
(20)
o
in (19) it should be decomposed similarly to the
J
Littlewood theorem for the case
proof of the Hardy
x
I
785
const
(x)
X(x)
see Samko et al [3]. We have
const
J=J
+J
+J
2.
3
wi th
Oo(X)
J3
I
xia+h
(t+h) (x)- dt
J
[ go(X-t)-o(X)
2
-h
[(t+h) (x)-I
t (x)-l]
(t+h)-(x)
[o(X-t) 0o(X)]
dt
dt
o
In the case h
The estimate of J
x-a
o(X)
J
(x)
c(x)
(x_a+h)
for the term
(x-a)
(xcx)]
we use the inequal ty
IOo(X)l
and the inequality
IJ
(1+t)/-1
c
<
<
IIOllHX(X
t
with
(’x-a’X()+((x)
(x_a) x(x)
(21)
0</1 and t > 0
We have
C(x)
(x)
(22)
c(x_a)X(x)+C(x)
h
-a
c(x-a)
x()+c()-
h
c h
B. ROSS AND S. SAMKO
786
If
x-a
we use (21) again and have
h
IJl c
Hence, by (8)
IJ I<
The estimate of J 2
(x+h-a)(x)
(x)
-
c (x-a) X(X)h(X)
c
--m hX(X)+(x)
<
J
The estimate of
(x-a)(x)
is completely like in the
case
2
of constant order:
o
I
c
Id 2
Itl
-h
The estimate of J 3
IJ
C
h
13
X(x)
dt
C
(t+h)-(x)
h
X(x)+(x)
(23)
We have
<c
I
tX(x)l(t+h)(x)-I- t(x)’l
dt
o
tX(x)
X(X)/(X)
[t(x)-I (t+l)(x)-l]
(24)
dt
o
If
x-a < h
we have
j31
c
hX(X)*"x)
< c h
Let
x-a z h
Then
x(x)+(x)
I
tX(x)
[te(x)-I (t+l)(x)-l]
+
o
I
(tm-1+1)dt
o
c h
dt
FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER
787
[tX(x) [t=(x)-l (t+l)=(x)-l]
sc hX(X)*=(x)
IJ dr3
o
X(x) + (x) <
under the assumption that
IJ 3
<
Then by the 1emma 3 we have
sin[(x)]
c h x(X)/e(x)
sin[e(x)+X(x)]
c(1-t) "1, we have
because B(=(x),l+x(x)) const. Since 1/(sint)
c h x(X)+(X)
IJI
3
If
X(x)+=(x)
(25)
we split the integration in (24) fro 0 to
and fro
to
(x-a)/h and havehave f ro (24)
IJ 3
since
because
2
Itx)-I (t+l)(x)-ll
J31
x-a
s c h
+
c h
[1
c
t(x)-Z
+ in
_]
t -1 dt
h
for
<
t >
So,
c h in
(26)
h
Gathering the estimates (20), (22), (23), (24), (25) and (26) we obtain
the inequalities (15)
(16) which proves the theorem.
ACLEDGEMENT. This work was partially funded by a Fulbright
grant.
The authors are thankful
to the referee for a careful
reading of the manuscript and his suggestions.
788
B. ROSS AND S. SA_MKO
REFERENCES
ROSS,B. AND SAMKO,S.
fractional
order,
Integration
Integral
and differentiation to a variable
Transforms and Special Functions,
1(1993), N 3.
2. HARDY,G.H. AND LITTLEWOOD,J.E. Some properties of fractional integrals, I,
Math. Zeit., 27(1928), N 4,
3.
SAMKO,S. ,KILBAS
A.
AND
565-606.
MARICHEV,O.
Integrals
and
Derivatives
of
Fractional Order. Theory and Applications. Gordon & Breach Sci.
(Russian edition by "Nauka
Tekhnika",
Publ., 1993 (to appear).
Minsk, 1987).