FRACTIONAL INTEGRATION OPERATOR OF VARIABLE ORDER IN THE HOLDER SPACES H

The fractional integrals I+(x)0 of variable order e(x) are considered. A theorem on mapping properties of Ia+ in Holder-type spaces H is proved, this being a generalization of the well known Hardy-Li ttlewood theorem. KEYWORDSFractional integration, variable fractional order, mapping properties, Holder continuous functions, Hardy-Littlewood theorem. 1991 AHS SUBJECT CLASSIFICATION CODES. 26 A33, 26 A16 I. NTROOUCTION In the paper [I] the authors introduced and investigated the fractional integrals l(X)0 )(x)’F[(x)] e(t)(x-t dt (1) of variable order e(x)>O 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 x This is a generalization of the Hardy Littlewood theorem, well known in the case of constant orders e(x)==const and x(x)=X=const (Hardy and Littlewood [2]; see also Samko et al [3], p. 53-54). Our interest in integration and differentiation of a variable order is 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 be considered as a starting point for further investigations of functions with varying order of smoothness. In Section we give all required definitions and some auxilliary lemmas, 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] . < a < b < 0. The following is a generalization of the Holder space H )" 0 < ), 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 cihl x() (2) for all x x+h E [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 ImaX{X(Xl)’X(x2 )) 2 2 (3) It is easily seen that HX()(O) is a ring with respect to the usual multiplication. It is a Banach space with respect to the norm f(x+h)-f(x)l sup sup IlfllHx (x) xeO h< Ihl x(x) h+ xEf f(x )-f(x )1 2 sup E IX --X Imax((xl ’(x2) Xl’ 2 2 (4) where denotes the equivalencef g <= c f<g;czf, c >0, c >0. 2 Generalizing Definition we give the following DEFINITION 2. We say that f(x) E Hx(x)’W(*)(O) where ),(x) are given functions, 0 < x(x) < 1, , < p(x) < if and (x) i__11/(x) If(x+h)-f(x)l clhl )’(x) In Ihl <-2under assumption that x, x+h E O We shal need the fol lowing auxi 11 iary assertions. LEMMA I. Let the function X(x) E C(O) satisfy the condition for all IX(x+h)-X(x)l < A A const > 0 l+ln h x x+h E 0 Then the function (5)

depends on the point x This is a generalization of the Hardy Littlewood theorem, well known in the case of constant orders e(x)==const and x(x)=X=const (Hardy and Littlewood [2]; see also Samko et al [3], p. 53-54).
Our interest in integration and differentiation of a variable order is 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 be considered as a starting point for further investigations of functions with varying order of smoothness.In Section we give all required definitions and some auxilliary lemmas, while Section 2 contains the statement and the proof of the main result.
In what follows the letter c may denote different positive constants.

PRELIMINARIES Let
The following is a generalization of the Holder space H )" 0 < ), DEFINITION I. We say that f(x) E H )'(x) (Q) where X(x) is a positive (not necessarily continuous) function, 0 < X(x 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 nequal ty (3) It is easily seen that HX()(O) is a ring with respect to the usual multiplication.It is a Banach space with respect to the norm where denotes the equivalence-f g <= c f<g;czf, c >0, c >0.
LEMMA I. Let the function X(x) E C(O) satisfy the condition for all where x,t,t+s(x-a) E O s E [0,1] is bounded from zero and infinity- with a constant d max{e,b-a,I/(b-a)} not depending on s,t and x PROOF.Since In g (x) {).[t+s(x-a)] )'(t)} In(x-a), by (5) we have Ay(l+c-y)-l Ay < Aln(b-a).If y<0, then f(y) Alyl(l+c+lyl) -< A Therefore, f(y) A max 1, ln(b-a) in the case c In(b-a) > 0 Let now c < 0 Then y < 0 and f(y) A > 0, so that f(y) < f(c) for y _< c which gives f(y) < Alcl= A I]n(b-a)l.So, f(y) A max 1, Iln(b-a)l in all cases.Therefore,'ln g(x)l A max{1,11n(b-a)l} ,whence (6) follows.

THE HAIN THEOREN
Considering the fractional integral Ia(X)0 defined in (1), of the function 0(x) H )'(x) we shall assume the following conditions on (x) and x(x) to be satisfied- where REMARK.The assertion (13) can be exactified: for all x e 0 such that (x) + X(x) < and If(x+h)-f(x)l clhl 7(x) In Ihl < (16) IIII for those x which give the equality (x) + x(x) c being a positive constant not depending on x and h (see the proof of the theorem).
PROOF OF THE THEOREM.From the conditions i) and ii) it follows that E HP(x)(Q) Really, 17(x+h)-7(x)l < l(x+h) (x)l (x) we have to prove that g( (X)' (x-a)( where respectively (with the exactification (15)-( 16), if we will).The derivation itself of the equali.ty is obvious.For the function f (x) we prove first the estimate (14).We have o )e(x)- where g (x)is the function (6).By lemma we have s,a 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 (x_a) (*)/() 2 which follows from the lemma (we remark that the condition (5) for X(x)+(x) is fulfilled because it is satisfied for x(x) by the assumption in ii) and for (x) by the assumption in i) ).Thus, ( 14) is proved.
To prove the statements (18) we consider the difference f (x+h) f (x) 0 0 taking h positive.(If h < O, by denoting x+h x x x +(-h) we reduce the consideration to the case of positive increment).We represent this difference as with Po(t)_ e(t) e(a) We estimate J first.We have 0 JJ o max Jeo(t)l I I(t+h)(x*h)--(t+h)(x)-l dt As regards the term J in (19) it should be decomposed similarly to the proof of the Hardy Littlewood theorem for the case (x) const X(x) x const see Samko et al [3].We have J=J +J +J 2.
The estimate of J In the case h x-a for the term o(X) The estimate of J We have 3 IJ 13 <c I tX(x)l(t+h)(x)-I-t(x)'l dt o C h X(X)/(X) o tX(x) [t(x)-I (t+l)(x)-l] dt (24) If x-a < h we have j31 c hX(X)*"x) I tX(x) [te(x)-I + (t+l)(x)-l] dt o < c h x(x)+(x) I (tm-1+1)dt c h o Let x-a z h Then simple calculations show that the maximum of the right-hand side is A max{1,11n(b-a)l} Really, let f(y) A lyl(l+c-y) -, where-, y s c ln(b-a).Suppose b-a z first.For y>O we have f(y) 1+),)   [ tX[tx--(t+l)(z-] dt sin (4:k) lln tl dt with ( between (x+h) and e(x) so that z m > 0 Since t(-ls At m-I with A max{1,(b-a) -m}, we have 2(b-a) IJol cAh/(x) I o tm-lln tl dt c h/(*) c(x_a)X(x)+C(x) h c(x-a) x()+c()-h c h -a If x-a h we use (21) again and have IJl c (x) (x+h-a)(x) (x-a)(x)