ON THE RELATION BETWEEN THE CAUSAL BESSEL DERIVATIVE AND THE MARCEL RIESZ ELLIPTIC AND HYPERBOLIC KERNELS

Transcription

1 ACENA Vo ON THE RELATION BETWEEN THE CAUSAL BESSEL DERIVATIVE AND THE MARCEL RIESZ ELLIPTIC AND HYPERBOLIC KERNELS Rub A. CERUTTI RESUMEN: Cosrao os úcos Rsz coo casos artcuars úco causa utrahrbóco Bss s stua a rvaa causa Bss úcos Marc Rz ítco o hrbóco. Utzao a trasoraa ourr y to cuta roucto utcatvo as strbucos P± 0 λ y as orat aáogas P± 0 λ s vaua chas rvaas obtos rsutaos u coo casos artcuars cot a otros bos a Sako ara caso úco raa Rsz y a Rub ara caso úco ítco Bss. ABSTRACT:: By cosrg th Rsz kr as a artcuar cas o th causa Bss utrahyrboc kr th rato btw th causa Bss rvatv a th Marc Rsz kr ar rst. W vauat va ourr trasor ths ratos takg to accout th utcatv rouct o crta strbutos. Paabras cavs: rvaa Bss otca Bss otca Rsz Ky wors: Bss rvatv Bss otta Rsz otta INTRODUCTION Lt t t t... t b a ot o t. Lt P P t b a o grat uaratc or varabs th or R b th soa ra sac a t P P t t... tp tp... t I. Th graz uctos P± o λ wr trouc by Ga c [5] by th oowg t P± 0 λ P± t I λ 0 whr > 0 λ a cox ubr a t t t. Accorg Ga c [5].87 a ro th srs vot o th Bss uctos Tro c [0] th causa atcausa Bss kr by th xrsso: Dartato Matatca. acuta Ccas Exactas y Naturas y Agrsura. Uvrsa Nacoa Norst Ava. Lvrta Corrts Argta. -a: rcrutt@xa.u.u.ar

2 04 ACENA Vo. 005 P 0 P 0 P 0 C K G I 3 whr C Γ s a ostv ra ubr a z k. Th ourr trasor o a ucto s by to < t > t t [ ] whr t R t Th ourr trasor o G P± 0 K γ s th o Bss ucto o th thr s c [0] [ ] G 0 Q I 4 λ whr th strbuto Q ± 0 s a aaogu ar that I by th oowg t. λ Q ± 0 Q ± t λ I 5 0 wr s a ra ostv ubr c[5]. Ths strbutos ar tr strbutoa ucto o λ a ths s th rca rc wth th P± 0 λ aaogus strbutos that ar aaytc λ xct at λ k k 0... Tro c [9] th utrahyrboc Bss orator o orr o a ucto ϕ or Bss otta o orr by th covouto ± 0 ϕ B ϕ G P I 6 whr th covouto s takg th strbutoa ss. Ths Bss orator B ϕ s aaogu to th o by covouto wth th tc Bss kr trouc by Aroszaj-Sth c [] a Caró c []. Rub c [6] vrt ths orator by usg hyrsguar tgras wth wght rcs. Takg to accout ths rocur to vrt th causa Bss orator w hav aothr rvous ar c [3] th causa hyrsguar tgra wth wght rcs o a ucto as oows T x x k P 0 { t λ } t P t I 8 R

3 O th rato btw th causa Bss rvatv a Rubé A. CERUTTI 05 whr 0... ;... 0 ; 0 > > k k R usg th auxary wght ucto gv by 0 P P 0 P 4 K o Γ λ I 7 c [3] whch s a aaogu causa o th wght ucto u to Rub c [6]. W hav vauat th ourr trasor o T c [3] [ ] [ ] T Q M I 9 whr 4 M a Γ Γ Γ Γ A A c [3] Takg t or 0 I 9 w obsrv that th t th rght ha br xst a s ua to [ ] 0 Q. I 0 Th th t th t ha br xsts a by to s th orator ot by T. ro ths rsut w hav th graz causa Bss rvatv o orr D by D T c [3] I a ro I 0 w gt [ ] [ ] Q D 0 c [3]. I

4 06 ACENA Vo. 005 Lt A t II. THE CAUSAL BESSEL DERIVATIVE O THE ELLIPTIC BESSEL KERNEL b th tc Bss kr trouc by Aroszaj-Sth c [] Laurt Schwartz c [8] a A.P. Caró c [] by t t K A t Γ Its ourr trasor s c [] [ A t ] kr A t. ; II. II W obsrv that th causa Bss kr I 3 s a aaogu causa o th ϕ S th tc Bss otta o orr B s By covouto wth a ucto obta. Morovr s kow that th sac S s varat rat to th th B ϕ bogg to S. ro th abov cosratos w hav th oowg thors. Thor Lt Th D b as I a t A t [ ] 0 D A t Q B orator b th ucto by II. II 3 th ss o strbutos. Proo. It rsut ro I a II. Thor Lt D as I a t B ϕ b th tc Bss ottas o orr. Th [ D B ϕ] Q 0 [ ϕ]. II4 Proo. It rsut ro I or B ϕ; ϕ S. Coroary. I I w cosr 0 ro I 5 a II 4 w obta [ ] D B ϕ [ ϕ] [ ϕ]. II5 That s th sa rsut that th o u by Rub c [6] or tc Bss ottas.

5 O th rato btw th causa Bss rvatv a Rubé A. CERUTTI 07 Th causa Bss rvatv o tc Rsz kr Takg to accout th cotuty rsct to o th Q ± 0 λ strbuto a that wh 0 Q ± 0 λ rucs to Q ± 0 λ a th causa Bss kr I 3 s th causa Rsz kr c [7] w hav th oowg. Thor 3 Lt R P± o b th causa Rsz kr as oows. c [0]. Γ R P 0 P± 0 ± III Γ a t D as I. Th [ D R P± 0 ] Q 0 Q 0 Proo Sc th ourr trasor o R P± 0 [ R P± 0 ] Q 0. III s c[0] III 3 w hav III. Coroary. I w cosr 0 III w obta th causa Rsz rvatv o causa Rsz kr a [ D R P ± 0 ]. III 4 Proo: It rsut ro th cotuty rsct to o th Q 0 λ strbuto. Coroary Takg I 0 ro I 5 a II 7 w hav [ D R ] t t t. III 5 Coroary 3. I w cosr 0 a 0 III w hav [ ] D R t t t. III 6 Whch s th sa rsut that th o u by Sako or raa Rsz kr c [7].

6 08 ACENA Vo. 005 REERENCES [] ARONZAJN N. a K. SMITH. Thory o Bss ottas. Part I. A. Ist. ourr. 98. [] CALDERÓN A.P. Lbsg sacs o rtab uctos a strbutos. Syos. Pur Math [3] CERUTTI R.A. Th utrahyrboc Bss orator. A vrso thor. Mathatca a coutr og. Vo. Nº. Washgto Uvrsty. Mssour [4] CERUTTI R.A. Th utrahyrboc Bss orator: so basc rorts. Math. A Coutr Mog. Vo [5] GELAND I. a G. SHILOV. Graz uctos. Vo. I. Acac Prss [6] RUBIN B. Dscrto a vrso o Bss ottas by usg hyrsguar tgras wth wght rcs. Drsa y Uravya. Vo [7] SAMKO S. O sacs o Rsz ottas. Math USSR Izvstya vo 0 Nº [8] SCHWARTZ L. Thor s strbutos. Hrra [9] TRIONE S.E. Ivrso o utrahyrboc Bss orators. Rvsta a Uó Matátca. Argta. Vo [0] TRIONE S.E. Dstrbutoa roucts. Cuaros Matátca N 3. Isttuto Argto Matátca. CONICET Rcbo/Rcv/: 04-Abr-06 Actao/Acct/: 0-Ju-06