Диссертация (1103157), страница 8
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. . , 1).Òåîðåìû 1.1 è 1.2 äîêàçûâàþòñÿ â 1.4.Ïåðåéä¼ì òåïåðü ê çàäà÷å õàðàêòåðèçàöèè. Èç òåîðåìû 1.2 ñëåäóåò, ÷òî îïåðàòîð Rqh , ãäå q óäîâëåòâîðÿåò (1.8), (1.10), à h ∈ L2α (R1+ ), íåïðåðûâíî îòîáðàæàåò ïðîñòðàíñòâî íåîòðèöàòåëüíûõ áîðåëåâñêèõ ìåð ñ êîíå÷íîé íîðìîé k · kc ,ãäå c ∈ Rn+ , α = c1 + · · · + cn , â ïðîñòðàíñòâî L2c (Rn+ ). Ýòî îòîáðàæåíèå íåÿâëÿåòñÿ ñþðúåêòèâíûì, è ìû ñîáèðàåìñÿ îïèñàòü åãî îáðàç.Íàì ïîòðåáóåòñÿ ââåñòè íåñêîëüêî îïðåäåëåíèé. Ïóñòü q óäîâëåòâîðÿåò (1.8),(1.10) è (M e−q )(z) 6= 0 ï.â. ïðè Re z = c, c ∈ Rn+ .
Ïóñòü h ∈ L2α (R1+ ), ãäåα = c1 + · · · + cn , è (M h)(s) 6= 0 ï.â. ïðè Re s = α. Ïðè ôèêñèðîâàííîì p0 > 0ïîëîæèìρhq (z) =Γ(s)Γ(z1 ) · · · Γ(zn ),(M e−q )(z) · (M h)(s)ρq (z) =Γ(s + 2)Γ(z1 ) · · · Γ(zn ),(M e−q )(z)ps+10(1.28)ãäå z ∈ c + iRn , s = z1 + · · · + zn . Îïðåäåëèì îïåðàòîðû Tqh è Tq ôîðìóëàìèTqh f = Mc−1 ρhq M f,Tq f = Mc−1 ρq M f.(1.29)37Íàïîìíèì, ÷òî ôóíêöèÿ g : Rn+ → R íàçûâàåòñÿ âïîëíå ìîíîòîííîé, åñëè g ∈C ∞ (Rn+ ) è ñïðàâåäëèâû ñëåäóþùèå íåðàâåíñòâà:(−1)|α| ∂|α|g(p)≥ 0,∂pαα ∈ Zn+ , p ∈ Rn+ .Ñïðàâåäëèâà ñëåäóþùàÿ òåîðåìà.Òåîðåìà 1.3.
Ïóñòü c ∈ Rn+ , α = c1 + · · · + cn . Ïðåäïîëîæèì, ÷òîq óäîâëåòâîðÿåò (1.8), (1.10), (M e−q )(z) 6= 0 ï.â. ïðè Re z = c,h ∈ L2α (R1+ ),h ≥ 0,(M h)(s) 6= 0 ï.â. ïðè Re s = α,ρhq ∈ L2 (c + iRn ) ∪ L∞ (c + iRn )(1.30)(1.31)(1.32)Òîãäà ôóíêöèÿ f : Rn+ → R ïðåäñòàâèìà â âèäå f = Rqh µ, ãäå µ íåîòðèöàòåëüíàÿ áîðåëåâñêàÿ ìåðà, kµkc < ∞, åñëè è òîëüêî åñëèkf k2,c < ∞,kTqh f k1,c < ∞,Tqh f âïîëíå ìîíîòîííà,(1.33)Êðîìå òîãî, ôóíêöèÿ h(t) = max{0, p0 − t} óäîâëåòâîðÿåò óñëîâèÿì (1.31) èñïðàâåäëèâû ðàâåíñòâà Rqh = Πq (p0 , ·), ρhq = ρq , Tqh = Tq .Òåîðåìà 1.3 ðåøàåò çàäà÷è 1.1 è 1.3 ïðè ïðåäïîëîæåíèè, ÷òî ôóíêöèÿ qóäîâëåòâîðÿåò óñëîâèÿì (1.8) è (1.10).
Òåîðåìà 1.3 äîêàçûâàåòñÿ â 1.5.Òåïåðü îáðàòèìñÿ ê ñëó÷àþ, êîãäà q = qα , ãäå qα îïðåäåëåíî â ôîðìóëå(1.9). Íàïîìíèì, ÷òî â îáîáù¼ííîé ìîäåëè ÕàóòåêêåðàÈîõàíñåíà ýòî ñîîòâåòñòâóåò ïðåäïîëîæåíèþ î ïîñòîÿííîé ýëàñòè÷íîñè çàìåùåíèÿ ðåñóðñîâ íà ìèêðîóðîâíå. Ìû ïðèâåä¼ì òåîðåìó õàðàêòåðèçàöèè äëÿ ñëó÷àÿ îïåðàòîðà Πq , òàêêàê èìåííî ýòîò ñëó÷àé èìååò íàèáîëüøåå ïðàêòè÷åñêîå çíà÷åíèå. Ìû áóäåìðàññìàòðèâàòü íåîòðèöàòåëüíûå áîðåëåâñêèå ìåðû µ, äëÿ êîòîðûõZRn+e−px µ(dx) < ∞ ïðè âñåõ p ∈ Rn+ .(1.34)Òåîðåìà 1.4. Ïóñòü q = qα , α ∈ (−∞, 1] \ 0, ãäå ôóíêöèÿ qα îïðåäåëåíà âôîðìóëå (1.9).
Ôóíêöèÿ Π(p0 , p) : R1+ × Rn+ → R1+ ïðåäñòàâèìà â âèäå Π = Πq µ,ãäå µ íåîòðèöàòåëüíàÿ áîðåëåâñêàÿ ìåðà íà Rn+ , óäîâëåòâîðÿþùàÿ (1.34) èòàêàÿ, ÷òî µ({0}) = 0, òîãäà è òîëüêî òîãäà, êîãäà(1) Π(p0 , p) âûïóêëà.38(2) Π(λp0 , λp) = λΠ(p0 , p) ïðè λ > 0, p0 > 0, p ∈ Rn+ .(3) Π(+0, p) =∂Π∂p0 (+0, p)= 0, p ∈ Rn+ .(4) ÔóíêöèÿZFα (p) =[0,∞)1∂Π(t, p α ),exp(−tα )d ∂p0111p α = (p1α , . . . , pnα ),(1.35)âïîëíå ìîíîòîííà.Òåîðåìà 1.4 ðåøàåò çàäà÷ó õàðàêòåðèçàöèè 1.1 äëÿ îïåðàòîðà Πq ïðè ïðåäïîëîæåíèè, ÷òî q = qα . Ìû äîêàçûâàåì òåîðåìó 1.4 â 1.5.Ñäåëàåì íåñêîëüêî çàìå÷àíèé îòíîñèòåëüíî íàøåãî ïîäõîäà ê çàäà÷å õàðàêòåðèçàöèè îïåðàòîðîâ Rqh (â ÷àñòíîñòè, îïåðàòîðà Πq ).
Íàèáîëåå ðàñïðîñòðàí¼ííûì â ëèòåðàòóðå ïîäõîäîì ê õàðàêòåðèçàöèè îáîáù¼ííîãî ïðåîáðàçîâàíèÿÐàäîíà ÿâëÿåòñÿ ïîäõîä òåîðèè óðàâíåíèé â ÷àñòíûõ ïðîèçâîäíûõ.  ýòîì ïîäõîäå îáðàç îáîáù¼ííîãî ïðåîáðàçîâàíèÿ Ðàäîíà îòîæäåñòâëÿåòñÿ ñ ÿäðîì íåêîòîðîãî îïåðàòîðà â ÷àñòíûõ ïðîèçâîäíûõ. Ýòîò ñïîñîá âîñõîäèò ê ðàáîòå [32](â ñëó÷àå ïðåîáðàçîâàíèÿ Ðàäîíà ïî ïðÿìûì â R4 ) è ïðèìåíÿåòñÿ, íàïðèìåð, âñòàòüÿõ [16] (â ñëó÷àå ïðåîáðàçîâàíèÿ Ðàäîíà íà îäíîðîäíûõ ïðîñòðàíñòâàõ) è[48, 34] (â ñëó÷àå ïðåîáðàçîâàíèÿ Ðàäîíà íî k -ïëîñêîñòÿì â ïðîåêòèâíîì ïðîñòðàíñòâå). Ýòîò ïîäõîä òàêæå èñïîëüçóåòñÿ äëÿ õàðàêòåðèçàöèè èíòåãðàëüíûõîïåðàòîðîâ òèïà Ðàäîíà (îïåðàòîðîâ âèäà Rqh èç ôîðìóëû (1.13)).
Íàïðèìåð, âñòàòüå [12] îáðàç (îäíîìåðíîãî) ïðåîáðàçîâàíèÿ Ëàïëàñà íåîòðèöàòåëüíûõ ìåðîòîæäåñòâëÿåòñÿ ñ ìíîæåñòâîì ðåøåíèé áåñêîíå÷íîé ñèñòåìû äèôôåðåíöèàëüíûõ íåðàâåíñòâ.  ðàáîòàõ [14] è [41] ïðèâîäÿòñÿ ìíîãîìåðíûå àíàëîãè ýòîãîðåçóëüòàòà. Íàêîíåö, â ñòàòüå [41] çàäà÷à õàðàêòåðèçàöèè ôóíêöèè ïðèáûëè âìîäåëè ÕàóòåêêåðàÈîõàíñåíà ñâîäèòñÿ ê óæå ðåø¼ííîé çàäà÷å õàðàêòåðèçàöèè(ìíîãîìåðíîãî) ïðåîáðàçîâàíèÿ Ëàïëàñà.
Íàø ïîäõîä ê õàðàêòåðèçàöèè îïåðàòîðîâ Rqh àíàëîãè÷åí. Ìû ñâîäèì îïåðàòîðû Rqh ê ïðåîáðàçîâàíèþ Ëàïëàñà èïðèìåíÿåì òåîðåìó õàðàêòåðèçàöèè äëÿ ïðåîáðàçîâàíèÿ Ëàïëàñà.Îòìåòèì, ÷òî äðóãîé øèðîêî ðàñïðîñòðàí¼ííûé ñïîñîá îïèñàíèÿ îáðàçàîáîáù¼ííîãî ïðåîáðàçîâàíèÿ Ðàäîíà îñíîâàí íà òàê íàçûâàåìûõ ìîìåíòíûõóñëîâèÿõ (òàêæå èçâåñòíûõ êàê óñëîâèÿ Êàâàëüåðè). Ýòîò ïîäõîä âîñõîäèò êðàáîòàì [91] è [38]. Ìû íå áóäåì îñòàíàâëèâàòüñÿ íà ýòîì ïîäõîäå.391.3Ôîðìóëà êîïëîùàäè è ñëåäñòâèÿ èç íå¼Â íàñòîÿùåì ðàçäåëå ìû äîêàæåì íåñêîëüêî âñïîìîãàòåëüíûõ óòâåðæäåíèé,êîòîðûå áóäóò èñïîëüçîâàíû ïðè ðåøåíèè çàäà÷ îáðàùåíèÿ è õàðàêòåðèçàöèèäëÿ îïåðàòîðîâ Rq è Rqh , âêëþ÷àÿ ñëó÷àé îïåðàòîðà Πq .Êëþ÷åâûì èíñòðóìåíòîì ÿâëÿåòñÿ ôîðìóëà êîïëîùàäè, êîòîðàÿ ÿâëÿåòñÿîáîáùåíèåì òåîðåìû Ôóáèíè â Rn íà ñëó÷àé êðèâîëèíåéíûõ êîîðäèíàò. Èñòîðèÿ ýòîé ôîðìóëû âîñõîäèò ê ðàáîòå [95], ãäå îíà áûëà ïîëó÷åíà â äâóìåðíîìñëó÷àå.
 ðàáîòå [28] ýòà òåîðåìà áûëà ïðèâåäåíà â îáùåì ñëó÷àå äëÿ ëèïøèöíåïðåðûâíûõ êîîðäèíàò. Íàì ïîòðåáóåòñÿ ýòà ôîðìóëà â ñëåäóþùåé ôîðìå.Ëåììà 1.1 (ôîðìóëà êîïëîùàäè). Ïóñòü q óäîâëåòâîðÿåò (1.8). Ïóñòü f ∈L1 (Rn+ ) ∩ C(Rn+ ). Òîãäà äëÿ âñåõ p ∈ Rn+ ñïðàâåäëèâî ðàâåíñòâîZ∞Zf (x) dx =Rn+t−1 (Rq f )( pt ) dt.(1.36)0Äîêàçàòåëüñòâî.
Çàôèêñèðóåì p ∈ Rn+ . Ïîëüçóÿñü ïîëîæèòåëüíîé îäíîðîäíîñòüþ ôóíêöèè qp , ïîëó÷èì â ñèëó òîæäåñòâà Ýéëåðà, ÷òîqp (x) = x∇qp (x),x ∈ Rn+ .Îòñþäà, ó÷èòûâàÿ ïîëîæèòåëüíîñòü qp , ïîëó÷àåì, ÷òî ∇qp (x) 6= 0 ïðè x ∈Rn+ . Áîëåå òîãî, èç óñëîâèÿ (1.8) ñëåäóåò, ÷òî ∇qp (λx) = ∇qp (x) ïðè λ > 0 èx ∈ Rn+ . Îòñþäà ñëåäóåò, ÷òî äëÿ ëþáîãî êîìïàêòà K ⊂ Rn+ íàéäóòñÿ òàêèåêîíñòàíòû C1 (K) > 0 è C2 (K) > 0, ÷òî äëÿ âñåõ x ∈ K ñïðàâåäëèâû îöåíêèC1 (K)|x| ≤ qp (x) ≤ C2 (K)|x|.
 ÷àñòíîñòè, îòîáðàæåíèå qp |K : K → R ÿâëÿåòñÿëèïøèöåâûì.Ïðèìåíÿÿ ôîðìóëó êîïëîùàäè [28, Theorem 3.2.12], ïîëó÷àåì, ÷òî äëÿ âñÿêîé íåîòðèöàòåëüíîé íåïðåðûâíîé ôóíêöèè fK : Rn+ → R, supp fK ⊂ K , ñïðàâåäëèâà ôîðìóëàZ∞ ZZf (x) dx =Rn+f (x)dSxdt,|∇qp (x)|(1.37)0 qp−1 (t)ãäå dSx ïîâåðõíîñòíàÿ ìåðà íà qp−1 (t) (òî åñòü, (n − 1)-ìåðíàÿ ìåðà Õàóñäîðôà). Ïîëüçóÿñü òåîðåìîé î ìîíîòîííîé ñõîäèìîñòè, ïîëó÷àåì, ÷òî ôîðìó-40ëà (1.37) ñïðàâåäëèâà äëÿ ïðîèçâîëüíîé íåîòðèöàòåëüíîé f ∈ C(Rn+ ). Íàêîíåö, ïðåäñòàâëÿÿ ïðîèçâîëüíóþ f ∈ L1 (Rn+ ) ∩ C(Rn+ ) â âèäå f = f+ − f− , ãäåf+ = max(f, 0), f− = max(−f, 0) è ïîëüçóÿñü àääèòèâíîñòüþ èíòåãðàëà Ëåáåãà,ïîëó÷èì, ÷òî ôîðìóëà (1.37) ñïðàâåäëèâà äëÿ âñåõ f ∈ L1 (Rn+ ) ∩ C(Rn+ ).
Íàêî−1(1) è ∇qp (x) = t∇qp/t (x)íåö, èç ôîðìóëû (1.37), ñ ó÷¼òîì ðàâåíñòâ qp−1 (t) = qp/tè îïðåäåëåíèÿ (1.12), ñëåäóåò ôîðìóëà (1.36). ñëåäóþùåé ëåììå ìû ïîëó÷èì ñâÿçü ìåæäó ôóíêöèåé ïðèáûëè Πq µ èìåðîé µ ìíîæåñòâLq (p0 , p) = x ∈ Rn+ | qp (x) ≤ p0 .(1.38)Ëåììà 1.2. Ïóñòü q óäîâëåòâîðÿåò (1.8). Ïóñòü µ áîðåëåâñêàÿ ìåðà ñîçíàêîì òàêàÿ, ÷òî |µ|(Lq (p0 , p)) < ∞ ïðè p0 > 0, p ∈ Rn+ . Òîãäà ñïðàâåäëèâàñëåäóþùàÿ ôîðìóëà:µ(Lq (p0 , p)) =∂∂p0 (Πq µ)(p0+ 0, p),p0 > 0, p ∈ Rn+ .(1.39)Êðîìå òîãî, åñëè µ qp−1 (p0 ) = 0, òî â ôîðìóëå (1.39) âìåñòî ïðàâîé ïðîèçâîäíîé ìîæíî ïèñàòü îáûêíîâåííóþ ïðîèçâîäíóþ.Äîêàçàòåëüñòâî.
Ïóñòü ∆ > 0. Ñïðàâåäëèâà ñëåäóþùàÿ öåïî÷êà ðàâåíñòâ:Z=(Πq µ)(p0 + ∆, p) − (Πq µ)(p0 , p)Z(p0 + ∆ − qp (x))µ(dx) −(p0 − qp (x))µ(dx)Lq (p0 +∆,p)Lq (p0 ,p)(1.40)Z= ∆ · µ(Lq (p0 + ∆, p)) +(p0 − qp (x)) µ(dx).Lq (p0 +∆,p)\Lq (p0 ,p)Çàìåòèì, ÷òî ñïðàâåäëèâà ôîðìóëàLq (p0 + ∆, p) \ Lq (p0 , p) = x ∈ Rn+ | 0 < qp (x) − p0 ≤ ∆ ,Ñëåäîâàòåëüíî, ñïðàâåäëèâà ñëåäóþùàÿ îöåíêà:Z(p0 − qp (x)) µ(dx)Lq (p0 +∆,p)\Lq (p0 ,p)Z≤|p0 − qp (x)| |µ|(dx) ≤ ∆ · |µ| Lq (p0 + ∆, p) \ Lq (p0 , p) .Lq (p0 +∆,p)\Lq (p0 ,p)(1.41)41Èç ôîðìóë (1.40) è (1.41), ñ ó÷¼òîì íåïðåðûâíîñòè ìåðû µ, ñëåäóåò ðàâåíñòâî(Πq µ)(p0 + ∆, p) − (Πq µ)(p0 , p) = ∆ · µ(Lq (p0 , p)) + o(∆),∆ → +0.(1.42)Îòñþäà ñëåäóåò ôîðìóëà (1.39). Åñëè æå µ qp−1 (p0 ) = 0, òî ôîðìóëà (1.42)ñïðàâåäëèâà è ïðè ∆ → −0.Ñëåäóþùàÿ ëåììà ïîêàçûâàåò, ÷òî èçó÷åíèå îïåðàòîðà ïðèáûëè Πq èç ôîðìóëû (1.6) ñâîäèòñÿ ê èçó÷åíèþ îáîáù¼ííîãî ïðåîáðàçîâàíèÿ Ðàäîíà Rq èçôîðìóëû (1.12), è îáðàòíî.
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