EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 6-2 Yıl:
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1 EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl: ON SEMIGOUP GENEAED Y OUIE- ESSEL ANSOM AND IESZ POENIAL ASSOCIAED WIH SEMIGOUP OUIE- ESSEL DÖNÜŞÜMÜ AAINDAN ÜEİLEN SEMİGUU VE - SEMİGUP AAINDAN DOĞULAN İESZ POANSİYELİ ÜZEİNE Şa KELEŞ * Sim arakçı Akiz Üivrsisi akülsi Mamaik ölümü Aala ürki. Akiz Üivrsisi akülsi Mamaik ölümü Aala ürki. ASAC Gliş arihi: 6 Nisa 3 Kabul arihi: Kasım 3 - smigroup associa wih sigular Laplac- ssl irial opraor k k is irouc. prsaios o h isz poials via h -smigroup ar obai. Mahmaics Subjc Classiicaio: 6A33 44A35 K Wors a Phrass: isz Poials Graliz Shi ourir- ssl rasorm Laplac- ssl Dirial Opraor. ÖZE u çalışmaa öclikl k k Laplac- ssl Dirasil opraorü araıa üril - smigrup aımlamışır. aımlamış ola - smigrup vasıasıla isz poasilii i bir gösrimi l ilmişir. Aahar Klimlr: Gllşmiş kama opraörü isz Poasili ourir- ssl öüşümü Laplac- ssl irasil opraörü. *Sorumlu Yazar: saalikol@gmail.com
2 76. INODUCION Klş v arakçı I h ourir- Harmoic Aalsis som o h impora ools as wll as sigular igral opraors ar h isz ssl Parabolic Poials c. h isz Poaials rprsig h gaiv racioal powrs o h Laplac- ssl Diraial opraor ar i i rms o ourir- ssl muliplir b S ar appli o h hor o ucios parial irial quaios a ohr aras o mahmaics as wll as harmoic aalsis. Samko a Kilbas 987; Si 97; ubi 996. h sigular igral opraors a poials associa wih h Laplac- ssl irial opraor Lvia 95; Muckhoup a Si 965; Kipriaov 967. k k ar kow o b impora irial opraors i aalsis a is applicaios hav b rsarch aras or ma mahmaicias such as I. Kipriaov a M. Kluchasv 97 L.Lakhov 984 A. D. Gajiv a I. A. Aliv V. S. Guliv 998 a ohrs. I. A. Aliv a S. Szr hav sui a w characrizaio o h isz Poaial gra b ouir rasorm. h hav irouc h so- call smigroup gra b h raial krl W i..... a usig his - smigroup W h EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl:
3 77 O Smıgroup Gra ourır- ssl rasorm a ısz Poıal Assocıa wıh Smıgroup hav obai a w igral rprsaio o h isz poials. No ha; or a. W coicis wih h wll kow Poisso a Gauss- Wirsrass igrals rspcivl. h purpos o his aricl is o i - smigroup gra b ourir- ssl rasorm a obai a w rprsaio o h isz poials wih ai o his - smigroup.. AUXILIAY DEINIIONS NOAIONS AND ESULS a o b S h L... : spac o ucios which ar h rsricios o ar v i h las EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl: o h s ucios o Schwarz ha variabl. h spac C o coiious ucios is i similiar. h closur o h spac S i h smiorm p p p L L whr is a i paramr p a.... is o b p p L o h sigular irial opraor o Laplac- ssl as allows: k k 3 Do b h graliz shi opraor GSO acig accorig o h law ' ' ; cos si
4 78 whr ' Klş v arakçı a '. Lvia 95; Kipriaov a Kluchasv 97; Kluchasv 97. W rmark ha is closl coc wih ssl irial opraor Lvia 95 h covoluio - covoluio gra b GSO is i o h spac S b whr.... I is as o prov h ollowig Youg s iquali r p q whr p q r a b usig lmma o p q r Muchhoup a E. Si 965 abou covoluio or h Marcikiwicz irpolaio horm. 4 h ourir- ssl rasormaio Kipriaov 967 is i o h spac o s S b ucios i. z z j z 3 a is ivrs is i b z c z whr. z z... z c. 4 EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl:
5 79 O Smıgroup Gra ourır- ssl rasorm a ısz Poıal Assocıa wıh Smıgroup h ucio j p p i 3 is coc wih h ssl ucio o h irs ki J p as ollows Lvia 95 j p J p p p p h acig o h ourir- ssl rasormaio o - covoluio is as ollows: S 5 L P... b a polomial i h variabls wih cosa coicis a v accorig i h las variabl. h P... P i... i P... P i... i I paricular or h Laplac- ssl irial opraor S 6 is prov arakci a Aliv ON HE - SEMIGOUPS GENEAED Y OUIE- ESSEL ANSOM h - smigroup associa wih is a igral opraor o covoluio p gra b h graliz shi. h krl. o his opraor is i as h ivrs ourir- ssl rasormaio o h ucio EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl:
6 8 Klş v arakçı EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl: whr. Lmma L. h 8 Proo rom 3 4 a 7 w g c z z z j c z i z. Chagig h variabl as i i m z w g. h - smigroup gra b h krl is i b 9 whr....
7 8 O Smıgroup Gra ourır- ssl rasorm a ısz Poıal Assocıa wıh Smıgroup 4. ON HE IESZ POENIALS GENEAED Y HE LAPLACE- ESSEL DIEENIAL OPEAO AND HEI EPESENAION VIA HE - SEMIGOUP Accorig o h ormula 6 w hav or all S hus w i h gaiv racioal powrs o h Laplac- ssl opraor b W o h las opraor b 988; I I as i Gajiv a Aliv S Ma kow rsuls or h classic isz poials such as h Har- Lilwoo- Sobolv horm c ar also vali or h isz poial I. S or ampl Gajiv a Aliv 988; Guliv 998. horm Gajiv a Aliv 988 or all S a whr I EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl:
8 8 A ormalizig coici Klş v arakçı I is chos i such a wa ha As mio abov our mai purpos is o rmi a w ormula rprsig h isz Poials associa wih - smigroups. 4. MAIN HEOEM horm 3 ; p a I b h isz poaial associa wih h Laplac- ssl irial opraor. h L p I No: A similar ormula rprsig a rlaio bw h calssical - smigroup a h isz Poial associa wih h Laplac irial opraor Aliv. k k was prov b S. Szr a I. A. Proo. chagig h orr o igraio w hav = EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl:
9 83 O Smıgroup Gra ourır- ssl rasorm a ısz Poıal Assocıa wıh Smıgroup EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl: hror w hav whr
10 84 Klş v arakçı EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl: I accorac wih w mus show ha W will show h las quali o b sraigh calculaio bu iircl usig h ourir- ssl rasorm. Sic hols or all L p p i hols i paricular or Schwarz s ucios. hus assumig S w g =. z I
11 85 O Smıgroup Gra ourır- ssl rasorm a ısz Poıal Assocıa wıh Smıgroup EEENCES Lvia. M. 95 ssl ucio pasios i sris a ourir Igrals Uspkhi Ma. Nauk 6 No -43 I ussia. Si E. M. 96 h characrizaio o ucios arisig as poials ull. Amr. Mah. Soc. 67 No -4. Muckhoup. Si E. 965 Classical pasios a hir rlaio o cojuga harmoic ucios ras. Amr. Ma. Soc Kipriaov I. A. 967 h ourir ssl rasormaio a mbig horms or wigh class ru Ma. Is. AN SSS I ussia Wh. L. 969 O hprsigular Igrals a Lbsgu Spacs o iriabl ucios II. ras. Amr. Mah. Soc. 39 No r Lizorki P. I. 97 O scripio o L spac i h rm o irc sigular igrals Mah. Sborik 8 No Si E. M. 97 Sigular igrals a iriabili Propris o ucios Prico Uiv. Prss Prico N. J. Kipriaov I. A. Kluchasv M. I. 97 O Sigular Igrals gra b h graliz shi opraor II. Sibirsk. Ma. Zh I ussia; Eglish rasla i: Sibria Mah. J. Kluchasv M. I.97 O Sigular igrals gra b h graliz shi opraor I. Sibirsk. Ma. Zh. 8-8 I ussia ishr M. J. 97 Sigular Igrals a racioal powrs o opraors ras. Amr. Mah. Soc. 6 No Samko S. G. 976 O isz poials spacs Izv. AN SSS Sr. Ma. 4 No Samko S. G. 977 h spacs p r Su. Mah. PL 6 No EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6- Yıl: p L a h hprsigular igrals Lakhov L. N. 984 O classs o sphrical ucios a sigular psoirial opraors Dokl. Aka Nauk I ussia Samko S. G. Kilbas A. A. Marichv O.I. 987 Igrals a rivaivs o racioal orr a som o hir applicaios Nauk. k. Misk. Aliv I. A. 987 isz rasorm gra b a graliz shi opraor Izvsia Aka Nauk Azarbaija p. Sr. iz. ch. Mah. Nauk 7-3 I ussia
12 86 Klş v arakçı Gajiv A. D. Aliv I. A. 988 h isz a ssl p poails gra b a graliz shi opraor a hir ivrs Proc. IV All- Uio Wir Co. hor o ucios a Approimaio Saraov. Gajiv A. D. Aliv I. A. 988 O a class o poial p opraor gra b a graliz shi opraor pors Smiar o h I. N. Vkua Is. Appl. Mah. 3 No bilisi. Aliv I. A. Gajiv A. D. 994 Wigh simas o muliimsioal sigular igrals gra b h graliz shi opraor ussia Aca. Sci. Sb. Mah. 77 No ubi. S. 996 racioal Igrals a Poials Ess- U. K. Aiso Wsl- Logma. Guliv V. S. 998 Sobolv s horm or isz - Poials Dokl. us. Aka. Nauk 358 No Mourou M. A. rimch K. 998 Calro s rproucig ormula associa wih h ssl opraor J. Mah. Aal. a Appl arakci S. Aliv I. A. 998 O ivrsio o - llipic poials b h mho o alakrisha- ubi racioal Calculus a Applis Aalsis Vol No Szr S. Aliv I. A. A w characrizaio o h isz poial spacs wih h ai o a composi wavl rasorm Joural o Mahmaical Aalsis a Applicaios **** EÜED - ilimlri Esiüsü Drgisi Cil-Saı: 6-Yıl:
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