Диссертация (1103157), страница 12
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Ïóñòü f ∈ CcN,σ (Rn+ ), ãäå c ∈ Rn+ , N ≥ n + 1 è σ > n ≥ 2. Òîãäàñïðàâåäëèâû ñëåäóþùèå ôîðìóëû:Znf (x) = (2π)− 2x−(c+iξ) (M f )(c + iξ) dξ + ferr (x),BRx ∈ Rn+ ,(2.11)59ferr (x) ≤(Ωn−1 )2 nNx−ckf kN,σ,c N −n ,(2π)n (σ − n)(N − n)Rx ∈ Rn+ , R > 0.(2.12)Äëÿ äîêàçàòåëüñòâà ëåììû 2.1 íàì ïîòðåáóåòñÿ îäíî âñïîìîãàòåëüíîå óòâåðæäåíèå. Íàïîìíèì, ÷òî ïðåîáðàçîâàíèå Ôóðüå F îïðåäåëÿåòñÿ ôîðìóëîé (1.49).Îïðåäåëèì ñïåöèàëüíûé êëàññ ôóíêöèé:C N,σ (Rn ) = u ∈ C N (Rn ) | kukN,σ < ∞ , N ∈ N ∪ 0, σ > 0, |α|σ ∂u(y).kukN,σ = max sup (1 + |y|n ) n αn∂y |α|≤N y∈R(2.13)Ëåììà 2.2.
Ïóñòü u ∈ C N,σ (Rn ), ãäå σ > n ≥ 2. Òîãäà äëÿ ïðåîáðàçîâàíèÿÔóðüå Fu ñïðàâåäëèâà îöåíêà− n2 Ωn−1 NnkukN,σ |ξ|−N ,σ−n|Fu(ξ)| ≤ (2π)ξ ∈ Rn \ 0,(2.14)ãäå Ωn−1 îáîçíà÷àåò ïëîùàäü åäèíè÷íîé ñôåðû â Rn .Äîêàçàòåëüñòâî. Çàìåòèì, ÷òî èç óñëîâèÿ u ∈ C N,σ (Rn ) ñëåäóåò, ÷òî âñå ÷àñòíûå ïðîèçâîäíûå ôóíêöèè u äî ïîðÿäêà N âëþ÷èòåëüíî ïðèíàäëåæàò L1 (Rn )è ñïðàâåäëèâà ñëåäóþùàÿ ôîðìóëà:− n2Ze(2π)−iξx ∂|α|u(x)dx = (iξ1 )α1 · · · (iξn )αn (Fu)(ξ),α∂xRnãäå ξ ∈ Rn , α ∈ Zn+ .
Áåðÿ ìîäóëü îò ëåâîé è ïðàâîé ÷àñòåé, äîìíîæàÿ íà ïîëèíîìèàëüíûå êîýôôèöèåíòû è ñóììèðóÿ ïî âñåì α ∈ Zn+ , ïîëó÷èì ðàâåíñòâîZ|α|X N n∂u(x)−−iξx(2π) 2 e = (|ξ1 | + · · · + |ξn |)N |Fu(ξ)|,dxαα∂x|α|=N(2.15)Rnãäå ξ ∈ Rn . Òåïåðü ó÷ò¼ì, ÷òî èç óñëîâèÿ u ∈ C N,σ (Rn ) âûòåêàåò ñëåäóþùàÿîöåíêà ïðè âñåõ α ∈ Zn+ , |α| = N :ZZ|α|n−1∂u(x) ≤ kukN,σ (1 + |x|n )− nσ dx = kukN,σ Ω e−iξxdx.∂xασ−nRnRnÓ÷èòûâàÿ ýòó îöåíêó è îöåíêó |ξ1 | + · · · + |ξn | ≥ |ξ|, à òàêæå ïðèíèìàÿ âî60âíèìàíèå, ÷òî ñóììà âñåõ ïîëèíîìèàëüíûõ êîýôôèöèåíòîâ|α| = N , ðàâíà nN , ïîëó÷èì èç (2.15) îöåíêó (2.14).Nαïî α ∈ Zn+ ,Äîêàçàòåëüñòâî ëåììû 2.1. Ïóñòü f ∈ CcN,σ (Rn+ ), ãäå N , σ è c óäîâëåòâîðÿþò óñëîâèÿì ëåììû 2.1.
Îïðåäåëèì u(y) = (Ec f )(y), ãäå îïåðàòîð Ec çàäà¼òñÿ ôîðìóëîé (1.53). Çàìåòèì, ÷òî f ∈ CcN,σ (Rn+ ) òîãäà è òîëüêî òîãäà, êîãäàu ∈ C N,σ (Rn ). Îòñþäà ñëåäóåò, â ÷àñòíîñòè, ÷òî u ∈ L1 (Rn ) ∩ L2 (Rn ), òàê ÷òîñïðàâåäëèâà ôîðìóëà îáðàùåíèÿ ïðåîáðàçîâàíèÿ ÔóðüåZ− n2u(y) = (2π)e−iξy (F −1 u)(ξ) dξ, y ∈ Rn .RnÏðèìåíèì ê ýòîìó ðàâåíñòâó îïåðàòîð Ec−1 è âîñïîëüçóåìñÿ ðàâåíñòâîì (F −1 u)(ξ) =(M f )(c + iξ) (ñì. ôîðìóëó (1.55)). Ìû ïîëó÷èì ôîðìóëó− n2Zf (x) = (2π)x−(c+iξ) (M f )(c + iξ) dξ,x ∈ Rn+ .(2.16)RnÒåïåðü çàìåòèì, ÷òî ñïðàâåäëèâà ñëåäóþùàÿ öåïî÷êà íåðàâåíñòâ: ZZ (1.55) −c−(c+iξ)x(M f )(c + iξ) dξ ≤ xRn \BR|(Fu)(ξ)| dξRn \BRn−1 Nn− n2 −c Ω(2.14)≤ (2π)xσ−nZkukN,σ|ξ|−N dξ(2.17)Rn \BR(Ωn−1 )2 nNkf kN,σ,c Rn−N .x(σ − n)(N − n)− n2 −c= (2π)Èç ôîðìóë (2.16) è (2.17) ñëåäóþò ôîðìóëû (2.11) è (2.12).
Ëåììà 2.1 äîêàçàíà.2.3Àíàëîãè òàóáåðîâûõ òåîðåì ÂèíåðàÄîêàçàòåëüñòâà òåîðåì 2.2 è 2.3 îñíîâàíû íà ìíîãîìåðíûõ òåîðåìàõ Âèíåðà îáàïïðîêñèìàöèè ôóíêöèé, äëÿ ôîðìóëèðîâêè êîòîðûõ íàì ïîòðåáóåòñÿ ââåñòèîäíî îáîçíà÷åíèå. Äëÿ çàäàííîé ôóíêöèè f íà Rn îáîçíà÷èì ÷åðåç Sf ëèíåéíóþ61îáîëî÷êó å¼ àääèòèâíûõ ñäâèãîâ:Sf = span fa | fa (x) = f (x − a), a ∈ Rn .(2.18)Íàïîìíèì òàêæå, ÷òî ïðåîáðàçîâàíèå Ôóðüå F îïðåäåëÿåòñÿ ôîðìóëîé (1.49).Ìíîãîìåðíûå òåîðåìû Âèíåðà îá àïïðîêñèìàöèè ôóíêöèé ôîðìóëèðóþòñÿ ñëåäóþùèì îáðàçîì.Òåîðåìà 2.6. Ïóñòü f ∈ L2 (Rn ). Òîãäà Sf âñþäó ïëîòíî â L2 (Rn ) òîãäà èòîëüêî òîãäà, êîãäà Ff 6= 0 ï.â.Òåîðåìà 2.7.
Ïóñòü f ∈ L1 (Rn ). Òîãäà Sf âñþäó ïëîòíî â L1 (Rn ) òîãäà èòîëüêî òîãäà, êîãäà (Ff )(ξ) 6= 0 äëÿ ëþáîãî ξ ∈ Rn .Òåîðåìû 2.6 è 2.7 â ñëó÷àå n = 1 âïåðâûå áûëè ïðåäñòàâëåíû â êíèãå [79].Äîêàçàòåëüñòâî òåîðåìû 2.6, ïðèâîäèìîå â [79], ëåãêî îáîáùàåòñÿ íà ñëó÷àén ≥ 2. Àíàëîãè÷íî, äîêàçàòåëüñòâî íåîáõîäèìîñòè â òåîðåìå 2.7 ëåãêî ïåðåíîñèòñÿ íà ìíîãîìåðíûé ñëó÷àé. Ñ äðóãîé ñòîðîíû, àâòîðó íåèçâåñòíà ññûëêà íàðàáîòó, ãäå áûëà áû äîêàçàíà äîñòàòî÷íîñòü â òåîðåìå 2.7 â ñëó÷àå n ≥ 2.
Ïîýòîé ïðè÷èíå, äîêàçàòåëüñòâî áóäåò ïðèâåäåíî íèæå.Äîêàçàòåëüñòâî òåîðåìû 2.7 (äîñòàòî÷íîñòü). Ïóñòü (Ff )(ξ) 6= 0 äëÿ ëþáîãî ξ ∈ Rn . Ìû õîòèì ïîêàçàòü, ÷òî ëþáàÿ ôóíêöèÿ h ∈ L1 (Rn ) ìîæåò áûòüàïïðîêñèìèðîâàíà â L1 (Rn ) ôóíêöèÿìè èç Sf . Çàìåòèì, ÷òî ôóíêöèÿ h ìîæåòáûòü àïïðîêñèìèðîâàíà â L1 (Rn ) ôóíêöèÿìè, ÷üå ïðåîáðàçîâàíèå Ôóðüå èìååòêîìïàêòíûé íîñèòåëü.
Ïîýòîìó, íå îãðàíè÷èâàÿ îáùíîñòè, ìû áóäåì ñ÷èòàòü,÷òî ôóíêöèÿ Fh èìååò êîìïàêòíûé íîñèòåëü.Èç òåîðåìû ÕàíàÁàíàõà ñëåäóåò, ÷òî äîñòàòî÷íî ïîêàçàòü, ÷òî äëÿ âñÿêîéôóíêöèè K ∈ L∞ (Rn ) èç ðàâåíñòâà f ∗ K = 0 ñëåäóåò ðàâåíñòâî h ∗ K = 0,ãäå ∗ îáîçíà÷àåò ñâ¼ðòêó.  ÷àñòíîñòè, äîñòàòî÷íî ïîêàçàòü, ÷òî ñóùåñòâóåòôóíêöèÿ g ∈ L1 (Rn ) òàêàÿ, ÷òî f ∗ g = h.Ìû áóäåì èñïîëüçîâàòü òåîðèþ êîììóòàòèâíûõ áàíàõîâûõ àëãåáð, íåîáõîäèìûå îïðåäåëåíèÿ ñì., íàïðèìåð, â [47]. Îáîçíà÷èì ÷åðåç L1 (Rn , C) ïðîñòðàíñòâî êîìïëåêñíîçíà÷íûõ èíòåãðèðóåìûõ ôóíêöèé.Ïóñòü Ω ⊂ Rn îòêðûòîå îãðàíè÷åííîå ìíîæåñòâî ñ çàìûêàíèåì Ω è ñîäåðæàùåå supp Fh. Çàìåòèì, ÷òî L1 (Rn , C) ÿâëÿåòñÿ êîììóòàòèâíîé áàíàõîâîé(ñâ¼ðòî÷íîé) àëãåáðîé. ÎïðåäåëèìI = g ∈ L1 (Rn , C) | (Fg)(ξ) = 0, ξ ∈ Ω .62Çàìåòèì, ÷òî I çàìêíóòûé èäåàë â L1 (Rn , C), è ïîëîæèì A = L1 (Rn , C)/I .
Aÿâëÿåòñÿ êîììóòàòèâíîé áàíàõîâîé àëãåáðîé ñ åäèíèöåé e+I , ãäå e ∈ L1 (Rn , C),Fe ≡ 1 íà Ω, à e + I îáîçíà÷àåò êëàññ ñìåæíîñòè e â A.Ìîæíî ïîêàçàòü, ÷òî âñÿêèé íåíóëåâîé ìóëüòèïëèêàòèâíûé ëèíåéíûé ôóíêöèîíàë íà A èìååò âèä ϕξ : a + I 7→ a(ξ), ãäå ξ ∈ Ω, a ∈ L1 (Rn , C). Èñïîëüçóÿòåîðåìó ÃåëüôàíäàÌàçóðà ([47, Theorem 1.2.9]) è ó÷èòûâàÿ, ÷òî ϕξ (f + I) 6= 0äëÿ ëþáîãî ξ ∈ Ω, ìû ïîëó÷àåì, ÷òî ýëåìåíò f + I îáðàòèì â A. Ýòî îçíà÷àåò,÷òî ñóùåñòâóåò ôóíêöèÿ g0 ∈ L1 (Rn , C) òàêàÿ, ÷òî Ff · Fg0 ≡ 1 íà Ω.
Ïîëîæèìg = h ∗ Re g0 . Òîãäà f ∗ g = h. Òåîðåìà äîêàçàíà.Ìû ïåðåíåñ¼ì òåîðåìû 2.6 è 2.7 íà ñëó÷àé àíàëèçà â íåîòðèöàòåëüíîì îðòàíòå Rn+ , ãäå ðîëü ïðåîáðàçîâàíèÿ Ôóðüå F èãðàåò ïðåîáðàçîâàíèå ÌåëëèíàM èç ôîðìóëû (1.16). Äëÿ çàäàííîé ôóíêöèè k íà Rn+ îáîçíà÷èì ëèíåéíóþîáîëî÷êó å¼ ìóëüòèïëèêàòèâíûõ ñäâèãîâ ÷åðåç Tk :Tk = span kp | kp (x) = k(p1 x1 , .
. . , pn xn ), p ∈ Rn+ .(2.19)Ñëåäóþùèå äâå òåîðåìû ÿâëÿþòñÿ ìíîãîìåðíûìè àíàëîãàìè òåîðåìû [50, ChapterII, Theorem 8.1] äëÿ ñëó÷àÿ àíàëèçà â ïîëîæèòåëüíîì îðòàíòå Rn+ .Ëåììà 2.3. Ïóñòü k ∈ L2c (Rn+ ), ãäå c ∈ Rn+ . Ñëåäóþùèå óòâåðæäåíèÿ ýêâèâàëåíòíû:(1) Tk âñþäó ïëîòíî â L2c (Rn+ ).(2) (M k)(z) 6= 0 ï.â. ïðè Re z = c.(3) ÓðàâíåíèåZkp (x)f (x) dx = 0,Rn+p ∈ Rn+ ,èìååò òîëüêî òðèâèàëüíîå ðåøåíèå f = 0 â L2I−c (Rn+ ).Äîêàçàòåëüñòâî. (1 ⇐⇒ 2). Ïóñòü îïåðàòîð Ec îïðåäåë¼í ôîðìóëîé (1.53).Ñïðàâåäëèâà ñëåäóþùàÿ ôîðìóëà:(Ec kp )(y) = p−c (Ec k)(y + ln p),(2.20)ãäå kp îïðåäåëåíî â ôîðìóëå (2.19), p = (p1 , . . .
, pn ) ∈ Rn+ , ln p = (ln p1 , . . . , ln pn ).63Èç ôîðìóëû (2.20), â ÷àñòíîñòè, ñëåäóåò ñîîòíîøåíèå(2.21)Ec Tk = SEc k ,ãäå ìíîæåñòâà SEc k , Tk îïðåäåëåíû â ôîðìóëàõ (2.18) è (2.19) ñîîòâåòñòâåííî.Ó÷èòûâàÿ, ÷òî Ec èçîìåòðèÿ L2c (Rn+ ) íà L2 (Rn+ ) (ñì. ôîðìóëó (1.57)), ìûïîëó÷àåì, ÷òî Tk âñþäó ïëîòíî â L2c (Rn+ ) òîãäà è òîëüêî òîãäà, êîãäà SEc k âñþäóïëîòíî â L2 (Rn ).
Íàêîíåö, ïîëüçóÿñü òåîðåìîé 2.6, ìû ïîëó÷àåì, ÷òî SEc k âñþäóïëîòíî â L2 (Rn ) òîãäà è òîëüêî òîãäà, êîãäà(1.55)(F −1 Ec k)(ξ) == (M k)(c − iξ) 6= 0 ï.â. ïðè x ∈ Rn .(1 ⇐⇒ 3). Ýòî ñëåäñòâèå èç òåîðåìû ÕàíàÁàíàõà.Ëåììà 2.4. Ïóñòü k ∈ L1c (Rn+ ), ãäå c ∈ Rn+ . Ñëåäóþùèå óòâåðæäåíèÿ ýêâèâàëåíòíû:(1) Tk âñþäó ïëîòíî â L1c (Rn+ ).(2) (M k)(z) 6= 0 ïðè Re z = c.(3) ÓðàâíåíèåZkp (x)f (x) dx = 0,Rn+p ∈ Rn+ ,(2.22)nèìååò òîëüêî òðèâèàëüíîå ðåøåíèå f = 0 â L∞I−c (R+ ).Äîêàçàòåëüñòâî. (2 ⇐⇒ 3). Îïðåäåëèì îïåðàòîð Ec ôîðìóëîé (1.53).
Ñïðàâåäëèâà ñëåäóþùàÿ ôîðìóëà:ZZu(x)v(x) dx =(Ec u)(y) (EI−c v)(y) dy,Rn+Rn(2.23)päëÿ âñåõ u ∈ Lrc (Rn+ ), v ∈ LI−c (Rn+ ), r, p ∈ [1, ∞], 1r + p1 = 1.Ó÷èòûâàÿ ôîðìóëû (1.57), (2.20), (2.23), ìû ïîëó÷àåì, ÷òî óðàâíåíèå (2.22)nèìååò òîëüêî òðèâèàëüíîå ðåøåíèå â L∞I−c (R+ ) òîãäà è òîëüêî òîãäà, êîãäà óðàâíåíèåZ(Ec k)(y − a) Φ(y) dy = 0, a ∈ Rn ,(2.24)Rnèìååò òîëüêî òðèâèàëüíîå ðåøåíèå Φ ≡ 0 â L∞ (Rn ). Ïî òåîðåìå ÕàíàÁàíàõàýòî ýêâèâàëåíòíî òîìó, ÷òî ìíîæåñòâî SEc k âñþäó ïëîòíî â L1 (Rn ).
Ïî òåîðåìå642.7 ýòî ýêâèâàëåíòíî òîìó, ÷òî(1.55)(F −1 Ec k)(ξ) == (M k)(c − iξ) 6= 0 ïðè x ∈ Rn .(1 ⇐⇒ 3). Ýòî ñëåäóåò èç òåîðåìû ÕàíàÁàíàõà.2.4Äîêàçàòåëüñòâî òåîðåì 2.2 è 2.3Äîêàçàòåëüñòâî òåîðåìû 2.2. Çàìåòèì, ÷òî óòâåðæäåíèå òåîðåìû 2.2, êàñàþùååñÿ îïåðàòîðà Πq , áóäåò ñëåäîâàòü èç òåîðåìû 2.3 â ñëó÷àå îïåðàòîðà Rqh ñh(t) = max{0, p0 − t}, åñëè ó÷åñòü âûðàæåíèå (1.68) äëÿ M h.Ïîýòîìó íàì äîñòàòî÷íî äîêàçàòü òåîðåìó 2.2 äëÿ ñëó÷àÿ îïåðàòîðà Rq . Äëÿêðàòêîñòè îáîçíà÷èìHcn = z ∈ Cn : Re z = c ,c ∈ Rn .Äëÿ çàäàííîé ôóíêöèè ϕ íà Hcn ïîëîæèìZc (ϕ) = z ∈ Hcn : ϕ(z) = 0 .Ñëó÷àé r = 1.
( =⇒ ). Ïðåäïîëîæèì, îò ïðîòèâíîãî, ÷òî íàéä¼òñÿ òàêîåîòíîñèòåëüíî îòêðûòîå ìíîæåñòâî U ⊂ Hcn , U 6= ∅, ÷òî (M e−q )(z) = 0 äëÿâñåõ z ∈ U .nÏóñòü χ ∈ C ∞ (HI−c) îòëè÷íàÿ îò íóëÿ ôóíêöèÿ òàêàÿ, ÷òî χ(I − z) = 0ïðè z 6∈ U . Îïðåäåëèì ôóíêöèþ χb íà Rn+ ñëåäóþùåé ôîðìóëîé:−1χb = E(I−c)FT(I−c) χ,(2.25)ãäå îïåðàòîðû E(I−c) è T(I−c) îïðåäåëÿþòñÿ â ôîðìóëàõ (1.53), (1.54), à F ïðåîáðàçîâàíèå Ôóðüå, îïðåäåë¼ííîå â ôîðìóëå (1.49).Èñïîëüçóÿ, ÷òî FCc∞ (Rn ) ⊂ L1 (Rn ) è ó÷èòûâàÿ ôîðìóëó (1.57), ìû ïîëó÷àåì, ÷òîχb ∈ L1I−c (Rn+ ) è χb 6≡ 0.(2.26)Èç ôîðìóë (1.55) è (2.25) ñëåäóåò, ÷òî(M χb)(I − c) = χ(I − z),z ∈ Hcn .(2.27)65Èñïîëüçóÿ ôîðìóëó (1.21) ñ f = χb è ó÷èòûâàÿ, ÷òî χ(I − z) = 0 ïðè z 6∈ U ,à (M e−q )(z) = 0 ïðè z ∈ U , ìû ïîëó÷àåì, ÷òî (M Rq χb)(z) = 0 ïðè z ∈ Hcn .Ñëåäîâàòåëüíî, Rq χb ≡ 0, ÷òî ïðîòèâîðå÷èò èíúåêòèâíîñòè Rq â ïðîñòðàíñòâåL1I−c (Rn+ ).(⇐=).
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