Anatoly A. Kilbas. tn 1. t 1 a. dt 2. a t. log x

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1 J. Koren Mh. So ) No HADAMARD-TYPE FRACTIONAL CALCULUS Anoly A. Kilbs Absr. The er is devoed o he sudy of frionl inegrion nd differeniion on finie inervl [ b] of he rel xis in he frme of Hdmrd seing. The onsruions under onsiderion generlize he modified inegrion /x)µ f)d/ nd he modified differeniion δ + µ δ xd D d/dx) wih rel µ being ken n imes. Condiions re given for suh Hdmrd-ye frionl inegrion oeror o be bounded in he se X b) of Lebesgue mesurble funions f on R + 0 ) suh h b f) d < < ) ess su b [u f) ] < ) for R ) in riulr in he se L 0 ) ). The exisene lmos everywhere is esblished for he orresonding Hdmrd-ye frionl derivive for funion gx) suh h x µ gx) hve δ derivives u o order n on [ b] nd δ n [x µ gx)] is bsoluely oninuous on [ b]. Semigrou nd reirol roeries for he bove oerors re roved.. Inroduion The urose of his er is o develo frionl inegrion nd differeniion in he Hdmrd seing. For nurl n N { 2 } nd rel µ nd 0 suh n roh is bsed on he nh inegrl of he form.) Jµ+f)x) n x µ d n )! x d 2 2 ) µ log x n µ nf n ) d n ) n f) d n x > ) Reeived June Mhemis Subje Clssifiion: 26A33 47B38. Key words nd hrses: Hdmrd-ye frionl inegrion nd differeniion weighed ses of summble nd bsoluely oninuous funions.

2 92 Anoly A. Kilbs nd he orresonding derivive.2) Dµ+g)x) δ + µ)f) x) xf x) + µfx) δ x d dx Dµ+g n Dµ+D µ+ n g) n 2 3 ) x > ). The frionl versions of he inegrl.) nd he derivive.2) re given by.3) J+µf)x) α ) µ log x ) α d f) α > 0; x > ) Γα) x nd.4) ) D+µg)x) α x µ δ n x µ J0+µ n α g x) δ x d dx α > 0; n [α] + µ R) reseively [α] being inegrl r of α. When µ 0.3) nd.4) ke he forms.5) J+f)x) α log x ) α du fu) α > 0; x > ) Γα) u u nd.6) D+y)x) α δ n J+µ n α g) x) δ x d α > 0; n [α] + ). dx The inegrl.5) ws inrodued by Hdmrd [5] in he se 0 nd herefore J+f α nd D+f α re ofen referred o s Hdmrd frionl inegrl nd derivive of order α [6 Seion 8.3 nd Seion 23. noes o Seion 8.3]. Therefore we my ll he more generl onsruions in.3) nd.4) Hdmrd-ye frionl inegrl nd derivive of order α. I is well develoed n roh o frionl lulus by Riemnn nd Liouville bsed on he generlizion of usul inegrion f)d nd differeniion D d/dx see for exmle [6 Chers 2 nd 3]. Hdmrd frionl lulus roh is sudied less. Some fs for he Hdmrd lulus oerors.5) nd.6) were resened in [6 Seion 8.3]. The Mellin roh ws suggesed in [] o sudy he roeries of he oerors J0+µ α nd Dα 0+µ defined on he he hlf-xis R + 0 ). Some roeries of he oeror J0+µ α nd hree of is modifiions were invesiged in []-[3]. The im of his er is o sudy he roeries of he Hdmrd-ye frionl oerors.3) nd.4) on finie inervl [ b] of he rel line R ) for > 0. The er is orgnized s follows. Firs

3 Hdmrd-ye frionl lulus 93 in Seion 2 we give ondiions for he oeror J+µf α o be bounded in he se X b) R ) of hose omlex-vlued Lebesgue mesurble funions f on [ b] for whih f X < where b.7) f X f) d ) / < R) nd.8) f X ess su b [ f) ] R). In riulr when / ) he se X b) oinides wih he lssil L b)-se: L b) X / b) wih.9) b / f f) d) < ) f ess su b f). Nex in Seion 3 we rove h he Hdmrd-ye frionl derivive D+µg α exiss lmos everywhere for funion gx) ACδ;µ n [ b] suh h x µ gx) hve δ xd D d/dx) derivives u o order n on [ b] nd δ n [x µ gx)] is bsoluely oninuous on [ b]:.0) AC n δ;µ [ b] {h : [ b] C : δn [x µ hx)] AC[ b] µ R; δ x x dx }. Here AC[ b] is he se of bsoluely oninuous funions on [ b] whih oinide wih he se of rimiives of Lebesgue mesurble funions:.) hx) AC[ b] hx) + see for exmle [6.4)]. ψ)d ψ) L b) In onlusion in Seion 4 we esblish semigrou nd reirol roeries for he oerors J α +µ nd D α +µ. We noe h he orresonding resuls for he Hdmrd frionl lulus oerors.5) nd.6) re lso resened in Seions Hdmrd-ye frionl inegrion in he se X b) In his seion we show h he Hdmrd-ye frionl inegrion oeror J α +µ is defined on X b) for µ. To formule he resul

4 94 Anoly A. Kilbs we need he inomlee gmm-funion γν x) defined for ν > 0 nd x > 0 by [4 6.92)]: 2.) γν x) 0 ν e d. Theorem 2.. Le α > 0 0 < < b < nd le µ R nd R be suh h µ. Then he oeror J α +µ is bounded in X b) nd 2.2) J α +µf X K f X where 2.3) K for µ while 2.4) K Γα) µ ) α γ for µ >. log b ) α Γα + ) [ α µ ) log )] b Proof. Firs onsider he se <. Sine f) X b) hen / f) L b) nd we n ly he generlized Minkowsky inequliy see for exmle [6.33)]. In ordne wih.3) nd.7) we hve b ) / J+µf α X x Γα) b Γα) / x ) µ log x ) α f) d dx x x ) du x / u µ log u) α f dx u u ) / nd hene where b/ b/ b u µ log u) α x f x ) ) / dx d u x ) / b/u u µ log u) α f) d du M J α +µf X M f X b/ u µ log u) α du.

5 Hdmrd-ye frionl lulus 95 Dire lulion show h M oinides wih K given in 2.3) nd 2.4) when µ nd µ > reseively. Thus 2.2) is roved for <. Le now. By.3) nd.8) we hve x J+µf)x) α ) µ log x ) α f) d Γα) x nd hus 2.5) x J α +µf)x) Kx) f X where Kx) / When µ hen for ny x b 2.6) Kx) log x Γα + ) u µ ) log u) α du u. ) α Γα + ) log b ) α. If µ > hen mking he hnge of vrible µ )u y nd king 2.) ino oun we find Kx) [ x )] Γα) µ ) α γ α µ ) log. γν x) is inresing funion nd hus 2.7) Kx) Γα) µ ) α γ [ α µ ) log )] b for ny x b. I follows from 2.5)-2.7) h for ny x b 2.8) x J α +µf)x) K f X where K is given by 2.3) nd 2.4) when µ nd µ > reseively. Hene in ordne wih.8) from 2.8) we obin he resul in 2.2) for. This omlees he roof of heorem. Puing / in Theorem 2. nd king.9) ino oun we dedue he boundedness of he oeror J α +µ in he se L b). Corollry 2.2. Le α > 0 0 < < b < nd le µ R be suh h µ /. Then he oeror J α +µ is bounded in L b) nd 2.9) J α +µf K f

6 96 Anoly A. Kilbs where K is given by 2.3) for µ / while µ ) α [ γ α 2.0) K Γα) for µ > /. µ ) log )] b Seing µ 0 in Theorem 2. we obin he orresonding semen for he Hdmrd frionl oeror J α + in.5). Theorem 2.3. Le α > 0 0 < < b < nd le 0. Then he oeror J α + is bounded in X b) nd 2.) J α +f X K 2 f X where 2.2) K 2 for 0 while 2.3) K 2 Γα) ) α γ for < 0. log b ) α Γα + ) [ α log )] b Remrk 2.4. I follows from [ Theorem 4)] h if µ > hen he Hdmrd-ye frionl oeror J α 0+µ is bounded in X R + ) nd J α 0+µf X K 3 f X. Suh resul formlly follow from 2.2) nd 2.4) if we u 0 b nd ke ino oun he relion 2.4) γν ) Γν). Remrk 2.5. I follows from Corollry 2.2 h he oeror J α +µ is bounded in L b) for µ /. Similr resul n no be obined from Theorem 2.3 for he Hdmrd frionl oeror J α +. This f leds o onjeure h he oeror J α + is robbly bounded from L b) ino noher se. Remrk 2.6. The resuls in Theorem 2. nd Theorem 2.3 for Hdmrd-ye nd Hdmrd frionl inegrion oerors re nlogues of hose for he lsssil Riemnn-Liouville frionl inegrls see [6 Theorem 2.6]). We only noe h he weighed se X b) is suible for he former while he se L b) for he ler.

7 Hdmrd-ye frionl lulus Hdmrd-ye frionl differeniion in he se ACδ;µ n [ b] In his seion we give suffiien ondiions for he exisene of he Hdmrd-ye frionl derivive D+µg α in.4). Sine he resul will be se in erms of he se ACδ;µ n [ b] defined in.0) we firs hrerize his se. Theorem 3.. The se ACδ;µ n [ b] onsiss of hose nd only hose funions gx) whih re reresened in he form [ ] 3.) gx) x µ x log x ) n n ϕ)d + k log x ) k n )! where ϕ) L b) nd k k 0 n ) re rbirry onsns. Proof. Firs rove neessiy. Le gx) ACδ;µ n [ b] where δ xd D d/dx). Then by.0) δ n [x µ gx)] AC[ b] nd hene by.) 3.2) δ n [x µ gx)] ϕ)d + n where ϕ) L b) nd n is n rbirry onsn. Rewrie 3.2) in he form d dx δn 2 [x µ gx)] ϕ)d + n x x. Chnging x o nd o u nd inegrion boh sides of his relion we hve δ n 2 [x µ gx)] k0 log x ϕ)d + n 2 + n log x where n 2 nd n re rbirry onsns. Reeing his roedure m m n ) imes we obin δ n m [x µ gx)] log x ) m ϕ) m )! d 3.3) m + k0 n +k m k! log x ) k where n m n re rbirry onsns. Now 3.3) wih m n yields 3.) nd neessiy is roved.

8 98 Anoly A. Kilbs Le now gx) is reresened by 3.) or x µ gx) log x ) n ϕ) km n n )! d + k0 k log x ) k. Tking δ-derivive m m n ) imes we hve δ m [x µ gx)] log x ) n m ϕ) n m )! d n k! k + log x ) k m. k m)! From here for m n we obin 3.2) wih n )! n ) nd hene in ordne wih.0) nd.) gx) ACδ;µ n [ b]. This omlees he roof of heorem. Noe h i follows from our roof h ϕ) nd k re given by 3.4) ϕ) g n ) k g k) k 0 n ) k! where 3.5) g k x) δ k [x µ gx)] k 0 n ) g 0 x) x µ gx). Hene 3.) n be rewrien in he form 3.6) gx) x µ [ log x ) n g n ) n n )! d + k0 g k ) k! ] log x ) k Now we redy o rove he resul giving suffiien ondiions for he exisene of he Hdmrd-ye frionl derivive.4). Theorem 3.2. Le α > 0 n [α] + µ R nd gx) ACδ;µ n [ b]. Then he Hdmrd-ye frionl derivive D+µg α exiss lmos everywhere on [ b] nd my be reresened in he form 3.7) D α +µg)x) x µ[ Γn α) n + k0 log x ) n α g n )d g k ) Γk α + ) where g k ) k 0 n ) re given by 3.5). log x ) k α ]

9 Hdmrd-ye frionl lulus 99 Proof. Sine gx) ACδ;µ n [ b] we hve reresenion 3.6). Subsiuing his relion ino.4) we hve D+µg)x) α x µ δ n x log x ) n α Γn α) [ 3.8) log ) n g n u) u n )! du n g k ) + log ) k α ]d k!. k0 Inerhnging he order of inegrion nd lying he Dirihle formul see for exmle [6.33)]) we hve log x ) n α d log ) n g u n u)du g n u)du u log x ) n α log u ) n d. The inner inegrl is evlued by he hnge of vrible y log/u)/ logx/u) nd using he formuls [4.5) nd.55)] for he be funion: u nd hene log x ) n α log u ) n d log x ) n α d Γn α)γn) Γ2n α) Γn α)γn) Γ2n α) log x ) n α u log ) n g u n u)du log x u) 2n α g n u)du. Subsiuing his relion ino 3.8) nd king δ n -differeniion we obin 3.7). Thus heorem is roved. Corollry 3.3. If 0 < α < µ R nd gx) ACδ;µ [ b] hen D+µg α exiss lmos everywhere on [ b] nd 3.9) D α +µg)x) x µ Γ α) [ log x ) α [ µ g)] d + lim + [µ g)] log x ) α ].

10 200 Anoly A. Kilbs When µ 0 from Theorem 3.2 we dedue suffiien ondiions for he exisene of he Hdmrd frionl derivive.6). Theorem 3.4. Le α > 0 n [α] + nd gx) ACδ;0 n [ b]. Then he Hdmrd frionl derivive D+g α exiss lmos everywhere on [ b] nd my be reresened in he form 3.0) D α +g)x) Γn α) n + k0 δ k g)) Γk α + ) log x ) n α δ n g))d log x ) k α. Corollry 3.5. If 0 < α < nd gx) ACδ;0 [ b] hen Dα +g exiss lmos everywhere on [ b] nd D+g)x) α x log x ) α g ) d Γ α) 3.) + g) Γ α) log x ) α. Remrk 3.6. The resuls in Theorem 3.2 nd Theorem 3.4 for Hdmrd-ye nd Hdmrd frionl differeniion oerors re nlogues of hose for he lsssil Riemnn-Liouville frionl derivives see [6 Theorem 2.2]). We only noe h he weighed se ACδ;µ n [ b] is suible for he former while he se ACn [ b] for he ler. 4. Semigrou nd reirol roeries of Hdmrd-ye frionl lulus oerors Firs we rove he semigrou roery for he Hdmrd-ye frionl inegrion oeror J α +µ in.3). Theorem 4.. Le α > 0 β > 0 0 < < b < nd le µ R nd R be suh h µ. Then for f X b) he semigrou roery holds 4.) J α +µj β +µ f J α+β +µ f.

11 Hdmrd-ye frionl lulus 20 Proof. Firs we rove 4.) for suffiienly good funions f. Alying Fubini s heorem we find 4.2) J+µJ α β +µ f)x) u µ log Γα) x) x ) α du u u u ) µ log u ) β d f) Γβ) u x µ µ f)d log x ) α log u ) β du Γα)Γβ) u u. The inner inegrl is evlued by he hnge of vrible τ log u/)/ logx/): log x ) α log u ) β du u u log x ) α+β Γα)Γβ) Γα + β). Subsiuing his relion ino 4.2) nd king.3) ino oun we hve J+µJ α β +µ f)x) x u µ log Γα + β) x) x ) α+β du u u J α+β +µ f)x) nd hus 4.) is roved for suffiienly good funions f. If µ hen by Theorem 2. he oerors J+µ α J β α+β +µ nd J+µ re bounded in X b) hene he relion 4.) is rue for f X b). This omlees he roof of heorem. Corollry 4.2. Le α > 0 β > 0 0 < < b < nd le µ R be suh h µ /. Then for f L b) he semigrou roery 4.) holds. When m 0 from Theorem 4. nd Theorem 2.3 we obin he semigrou roery for he Hdmrd frionl inegrion oerors.5). Theorem 4.3. Le α > 0 β > 0 0 < < b < nd 0. Then for f X b) he semigrou roery holds 4.3) J α +J β + f J α+β + f.

12 202 Anoly A. Kilbs Nex we onsider he omosiion beween he oerors of Hdmrdye frionl differeniion.4) nd frionl inegrion.3). Theorem 4.4. Le α > β > 0 0 < < b < nd le µ R nd R be suh h µ. Then for f X b) here holds 4.4) D β +µ J α +µf J α β +µ f. In riulr if β m N hen 4.5) D m +µj α +µf J α m +µ f. Proof. Le m < β m m N). If β m hen by.4) 4.6) D+µy)x) m x µ x d ) m x µ yx) dx nd hene D+µJ m +µf)x) α x µ x d ) m x d dx dx Γα) u µ log x u ) α fu) du u. Alying he formul of differeniion under he inegrl sign nd using he relion for he gmm-funion [4.2)] nd.3) we obin D+µJ m +µf)x) α x µ x d ) m x x u µ dx Γα) x x µ x d ) m x dx Γα ) x µ x d ) m x µ J+µ α dx f)x). log x ) α du fu) u u u µ x log x u ) α 2 fu) du u Reeing his roedure k k m) imes we hve D+µJ m +µf)x) α x µ x d ) m k x µ J+µ α k dx f)x) nd 4.5) follows for k m. If m < β < m hen 4.4) follows from 4.) nd 4.5): D β +µ J +µf α D+µJ m m β +µ J +µf α D+µJ m m+α β +µ f J α β +µ f. Thus heorem is roved.

13 Hdmrd-ye frionl lulus 203 Corollry 4.5. Le α > β > 0 0 < < b < nd le µ R be suh h µ /. Then for f L b) he relion 4.4) holds. In riulr 4.5) is vlid for β m N. Theorem 4.6. Le α > β > 0 0 < < b < nd 0. Then for f X b) he relion holds 4.7) D β + J α +f J α β + f. In riulr if β m N hen 4.8) D m +J α +f J α m + f. Theorem 4.4 is lso rue for α β whih mens h he Hdmrdye frionl differeniion.6) nd inegrion.5) re reirol oerions if he former is lied firs. The resul below is roved similrly o he roof of Theorem 4.4. Theorem 4.7. Le α > 0 0 < < b < nd le µ R nd R be suh h µ. Then for f X b) here holds 4.9) D α +µj α +µf f. In riulr if µ / hen 4.0) is vlid for f L b). Theorem 4.8. Le α > 0 0 < < b < nd le 0. Then for f X b) here holds 4.0) D α +J α +f f. Remrk 4.9. I follows from [2 Theorem )] h if α > 0 β > 0 nd µ > hen for f X R + ) he semigrou roery 4.) J α 0+µJ β 0+µ f J α+β 0+µ f holds. Suh resul follows from 4.) if we u 0 nd b nd ke ino oun h he oerors J0+µ α J β α+β 0+µ nd J0+µ re bounded in X R + ) when µ >. Remrk 4.0. The resuls resened in Theorems 4.7 nd 4.8 show h he Hdmrd-ye nd Hdmrd frionl differeniion.4) nd.6) nd inegrion.3) nd.5) re reirol oerions if he formers re lied firs. I is he roblem when he lers n

14 204 Anoly A. Kilbs be lied firs. Suh roblem is solved for he Riemnn-Liouville frionl lulus oerors see [6 Theorem 2.4]). Aknowledgemen. The resen invesigion ws rly suored by he Belrusin Fundmenl Reserh Fund. Referenes [] P. L. Buzer A. A. Kilbs nd J. Trujillo Frionl lulus in he Mellin seing nd Hdmrd-ye frionl inegrls J. Mh. Anl. Al. o er) [2] Comosiions of Hdmrd-ye frionl inegrion oerors nd he semigrou roery J. Mh. Anl. Al. o er) [3] Mellin rnsform nd inegrion by rs for Hdmrd-ye frionl inegrls J. Mh. Anl. Al. o er) [4] A. Erdelyi W. Mgnus F. Oberheinger nd F.G. Triomi Higher Trnsendenl Funions Vol. MGrw-Hill Boo. Coo. New York 953; Rerined Krieger Melbourne Florid 98. [5] J. Hdmrd Essi sur l eude des fonions donnees r leur develomen de Tylor J. Mh. Pures e Al. Ser ) [6] S. G. Smko A. A. Kilbs nd O. I. Mrihev Frionl Inegrls nd Derivives. Theory nd Aliions Gordon nd Breh Yverdon e libi 993. Dermen of Mhemis nd Mehnis Belrusin Se Universiy Minsk Belrus E-mil: kilbs@mmf.bsu.unibel.by