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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).