Диссертация (1103157), страница 10
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Ôîðìóëû (1.26) è (1.27) äîêàçûâàþòñÿ ïîëíîñòüþ àíàëîãè÷íî ôîðìóëàì (1.22) è (1.23).Îöåíêè (1.24) è (1.25). Èñïîëüçóÿ íåðàâåíñòâî Éåíñåíà, ïîëó÷èì ñëåäóþ-48ùóþ îöåíêó:|(Rqh µ)(p)|r≤kµkr−1cZh(qp (x))r x(r−1)c |µ|(dx)(1.73)Rn+rRq|h| (x(r−1)c |µ|) (p).= kµkr−1cÑëåäîâàòåëüíî, ñïðàâåäëèâû ñëåäóþùèå ñîîòíîøåíèÿ:(1.73)kRqh µkrr,c ≤ kµkr−1cZrprc−I |(Rq|h| x(r−1)c |µ|)(p)| dpRn+(1.61)==nkµkrc (2π) 2 (M e−q )(rc)khkrr,α.Γ(αr)Îòñþäà ñëåäóåò îöåíêà (1.24).
Çàìåòèì òàêæå, ÷òî ïðè h ≥ 0, µ ≥ 0, r = 1íåðàâåíñòâî â ôîðìóëå (1.73) ñòàíîâèòñÿ ðàâåíñòâîì.Îöåíêà (1.25) ñëåäóåò èç îöåíêè (1.24), åñëè ó÷åñòü, ÷òî ïðè h(t) = max{0, p0 −t} ñïðàâåäëèâà ôîðìóëàkhkrr,αZ=01.5p0(α+1)r(p0 − t)r trα−1 dt = p0Γ(αr)Γ(r + 1).Γ((α + 1)r + 1)Äîêàçàòåëüñòâî òåîðåì 1.3 è 1.4Äëÿ äîêàçàòåëüñòâà òåîðåìû 1.3 íàì ïîòðåáóåòñÿ äîêàçàòü îäíî âñïîìîãàòåëüíîå óòâåðæäåíèå.Ïóñòü ôóíêöèè q1 , q2 óäîâëåòâîðÿþò (1.8), (1.10) è (M e−q2 )(z) 6= 0 ï.â. ïðèRe z = c, ãäå c ∈ Rn+ . Ïóñòü ôóíêöèè h1 , h2 ∈ L2α (R1+ ) è (M h2 )(s) 6= 0 ï.â. ïðèRe s = c1 + · · · + cn . Ïîëîæèì ïî îïðåäåëåíèþσqh11,q,h22 (z)(M e−q1 )(z) · (M h1 )(z1 + · · · + zn ),=(M e−q2 )(z) · (M h2 )(z1 + · · · + zn )Re z = c.Ëåììà 1.8. Ïóñòü q1 , q2 óäîâëåòâîðÿþò (1.8), (1.10). Ïóñòü h1 , h2 ∈ L2α (R1+ )ïðè α = c1 + · · · + cn , ãäå c ∈ Rn+ , è ïóñòü kµkc < ∞.
Ïðåäïîëîæèì, ÷òî(M e−q2 )(z) 6= 0 ï.â. ïðè Re z = c, à (M h2 )(s) 6= 0 ï.â. ïðè Re s = α. Òîãäàñïðàâåäëèâû ñëåäóþùèå óòâåðæäåíèÿ:49(A) Èìååò ìåñòî ôîðìóëàRqh11 µ = Mc−1 σqh11,q,h22 M Rqh22 µ .(1.74)(B) Ïóñòü σqh11,q,h22 ∈ L2 (c + iRn ) èëè σqh11,q,h22 ∈ L∞ (c + iRn ).
Ïðåäïîëîæèì, ÷òî(M e−q1 )(z) 6= 0 ï.â. ïðè Re z = c, (M h1 )(s) 6= 0 ï.â. ïðè Re s = α. Ïóñòüòàêæå kf k2,c < ∞ è Rqh11 µ = Mc−1 σqh11,q,h22 M f . Òîãäà f = Rqh22 µ.Äîêàçàòåëüñòâî. Óòâåðæäåíèå (A). Ìû âîñïîëüçóåìñÿ òåîðåìîé 1.2. Èç ðàâåíñòâà (1.26), çàïèñàííîãî äëÿ Rqh11 µ è äëÿ Rqh22 µ, ñëåäóåò ôîðìóëà(M Rqh11 µ)(z) = σqh11,q,h22 (z)(M Rqh22 µ)(z) äëÿ ï.â. z ∈ c + iRn .(1.75)Èç òåîðåìû 1.2 òàêæå ñëåäóåò, ÷òî Rqh11 µ ∈ L2c (Rn+ ). Ñëåäîâàòåëüíî, ïðèìåíÿÿ êðàâåíñòâó (1.75) îáðàòíîå ïðåîáðàçîâàíèå Ìåëëèíà Mc−1 è ïîëüçóÿñü ôîðìóëîé(1.59), ïîëó÷èì ôîðìóëó (1.74).Óòâåðæäåíèå (B).
Ïóñòü f ∈ L2c (Rn+ ) è Rqh11 µ = Mc−1 σqh11,q,h22 M Rqh22 f . Èçôîðìóëû (1.74) ñëåäóåò ðàâåíñòâîMc−1 σqh11,q,h22 M f = Mc−1 σqh11,q,h22 M Rqh22 µ .(1.76)Èç òåîðåìû 1.2 è ôîðìóëû (1.51) ñëåäóåò, ÷òî M f , M Rqh22 µ ∈ L2 (c + iRn ). Ó÷èòûâàÿ, ÷òî σqh11,q,h22 ∈ L2 (c+iRn ) èëè σqh11,q,h22 ∈ L∞ (c+iRn ), ïîëó÷àåì, ÷òî σqh11,q,h22 M f ,σqh11,q,h22 M Rqh22 µ ∈ L2 (c + iRn ). Îòñþäà, èñïîëüçóÿ ôîðìóëû (1.52) è (1.76), ïîëó÷àåì, ÷òî M f = M Rqh22 µ.
Èç èíúåêòèâíîñòè M íà L2c (Rn+ ) ñëåäóåò, ÷òî f = Rqh22 µ.Óòâåðæäåíèå (B) äîêàçàíî.Ïåðåéä¼ì òåïåðü ê äîêàçàòåëüñòâó òåîðåìû 1.3.Äîêàçàòåëüñòâî òåîðåìû 1.3. Íåîáõîäèìîñòü. Ïóñòü f = Rqh µ, ãäå h, q , µóäîâëåòâîðÿþò óñëîâèÿì òåîðåìû 1.3. Ïîëüçóÿñü òåîðåìîé 1.2, ìû ïîëó÷àåì,÷òî kf k2,c < ∞.Çàìåòèì, ÷òî ñïðàâåäëèâî ðàâåíñòâî ρhq = σqhLL,q,h , ãäå hL (t) = e−t è qL (x) =x1 + · · · + xn . Çàìåòèì òàêæå, ÷òî RqhLL ïðåîáðàçîâàíèå Ëàïëàñà:(RqhLL µ)(p) =ZRn+e−px µ(dx).(1.77)50Èñïîëüçóÿ ëåììó 1.8 (A), ìû ïîëó÷àåì, ÷òî Tqh f ≡ Mc−1 ρhq M f ) = RqhLL µ. Èç òåîðåìû õàðàêòåðèçàöèè äëÿ ïðåîáðàçîâàíèÿ Ëàïëàñà [14, Theorem 4.2.1] ñëåäóåò,÷òî Tqh f âïîëíå ìîíîòîííà. Íàêîíåö, èç òåîðåìû 1.2, ïðèìåí¼ííîé ê ôóíêöèèTqh f = RqhLL µ, ïîëó÷àåì, ÷òî kTqh f k1,c < ∞.Äîñòàòî÷íîñòü. Òàê êàê kf k2,c < ∞ è ρhq ∈ L2 (c + iRn ) ∪ L∞ (c + iRn ), òîôóíêöèÿ Tqh f êîððåêòíî îïðåäåëåíà êàê ýëåìåíò L2c (Rn+ ).
Ïî óñëîâèþ ôóíêöèÿTqh f ≡ Mc−1 ρhq M f ) âïîëíå ìîíîòîííà. Ïî òåîðåìå õàðàêòåðèçàöèè äëÿ ïðåîáðàçîâàíèÿ Ëàïëàñà [14, Theorem 4.2.1] íàéä¼òñÿ íåîòðèöàòåëüíàÿ áîðåëåâñêàÿìåðà µ íà Rn+ òàêàÿ, ÷òî Tqh f = RqhLL µ, ãäå îïåðàòîð RqhLL îïðåäåë¼í â ôîðìóëå(1.77).Ó÷èòûâàÿ, ÷òî kTqh f k1,c < ∞ è ÷òî h ≥ 0, ìû ïîëó÷àåì èç òåîðåìû 1.2, ÷òîkµkc < ∞. Òåîðåìà 1.3 äîêàçàíà.Òåïåðü ïåðåéä¼ì ê äîêàçàòåëüñòâó òåîðåìû 1.4.Äîêàçàòåëüñòâî òåîðåìû 1.4. Íåîáõîäèìîñòü.
Ïóñòü Π = Πq µ, q = qα . Èçêîíå÷íîñòè çíà÷åíèé ôóíêöèè Π ñëåäóåò, ÷òî µ(Lq (p0 , p)) < ∞ ïðè p0 > 0,p ∈ Rn+ , ãäå ìíîæåñòâî Lq (p0 , p) îïðåäåëåíî â ôîðìóëå (1.38)Âûïóêëîñòü ôóíêöèè Π ñëåäóåò èç ôîðìóëû (1.6) ñ ó÷¼òîì âîãíóòîñòè ôóíêöèè qp . Ïîëîæèòåëüíàÿ îäíîðîäíîñòü ôóíêöèè Π òàêæå ñëåäóåò èç ôîðìóëû(1.6) ñ ó÷¼òîì òîãî, ÷òî qλp (x) = λqp (x) ïðè λ > 0, p ∈ Rn+ , x ∈ Rn+ .Ñâîéñòâî Π(+0, p) = 0 ñëåäóåò èç ñëåäóþùåé öåïî÷êè íåðàâåíñòâ:Z0 ≤ Π(p0 , p) =max(0, p0 − qp (x)) µ(dx) ≤ p0 · µ Lq (p0 , p) .Rn+Ïîëüçóÿñü ôîðìóëîé (1.39), ó÷èòûâàÿ íåïðåðûâíîñòü ìåðû µ è ñâîéñòâî µ({0}) =∂Π0, ìû ïîëó÷àåì, ÷òî ∂p(+0, p) = 0.0∂Π(p, p0 ) ÿâëÿåòñÿ íåóáûâàÒàê êàê ôóíêöèÿ Π âûïóêëà, òî å¼ ïðîèçâîäíàÿ ∂p0nþùåé ôóíêöèåé p0 ïðè êàæäîì p ∈ R+ .
Èç ôîðìóëû (1.39) òàêæå ñëåäóåò, ÷òî∂Π∂p0 (p, p0 ) íåïðåðûâíà ñëåâà ïî ïåðåìåííîé p0 íà ìíîæåñòâå [0, +∞) ïðè ôèêñèðîâàííîì p ∈ Rn+ . Ñëåäîâàòåëüíî, äëÿ êàæäîé íåîòðèöàòåëüíîé ôóíêöèèRu ∈ D(R1+ ) îïðåäåë¼í èíòåãðàë ËåáåãàÑòèëòüåñà [0,∞) u(t)d ∂Π∂t (t, p) è ñïðàâåä-51ëèâà öåïî÷êà ðàâåíñòâ:Zu(t)d ∂Π∂t (t, p) = −Z=== −∂Π(t, p)u0 (t) dt∂t0[0,∞)(1.39)Z∞∞u0 (t)Z Z∞Zµ(dx) dt = −u0 (t) dt µ(dx)0qp (x)≤tRn+ qp (x)Z=u(qp (x)) µ(dx).Rn+Èç ýòîé ôîðìóëû, âûáèðàÿ ïîñëåäîâàòåëüíîñòü íåîòðèöàòåëüíûõ u ∈ D 0 (Rn+ ),ìîíîòîííî ñõîäÿùèõñÿ ê exp(−tα ), ïîëó÷èì ðàâåíñòâîZ1α∂Πexp(−tα )dt ∂p(t, p ) =0[0,∞)Zexp(−p1 xα1 − · · · − pn xαn ) µ(dx), p ∈ Rn+ .(1.78)Rn+Îòñþäà ñëåäóåò, ÷òî Fα (p) âïîëíå ìîíîòîííà.Äîñòàòî÷íîñòü.
Ïðåäïîëîæèì, ÷òî ôóíêöèÿ Π óäîâëåòâîðÿåò ñâîéñòâàì(1)(4) èç óñëîâèÿ òåîðåìû 1.4. Èñïîëüçóÿ òåîðåìó Áåðíøòåéíà î âïîëíå ìîíîòîííûõ ôóíêöèÿõ â ñëó÷àå ôóíêöèè Fα , ïîëó÷àåì, ÷òî íàéä¼òñÿ òàêàÿ íåîòðèöàòåëüíàÿ áîðåëåâñêàÿ ìåðà µ íà Rn+ , óäîâëåòâîðÿþùàÿ (1.34), ÷òî ñïðàâåäëèâî∂Πðàâåíñòâî (1.78). Èç ýòîãî ðàâåíñòâà, â ñèëó óñëîâèÿ ∂p(+0, p) = 0, p ∈ Rn+ , ñëå0äóåò, ÷òî µ({0}) = 0.e = Πq µ.
Ìû ïîêàæåì, ÷òî Π = Πe.Îïðåäåëèì ΠeÇàìåòèì, ÷òî èç óæå äîêàçàííîé íåîáõîäèìîñòè ñëåäóåò, ÷òî ôóíêöèÿ Πòàêæå óäîâëåòâîðÿåò ðàâåíñòâó (1.78). Ñëåäîâàòåëüíî, ñïðàâåäëèâî ðàâåíñòâîZ[0,∞)1α∂Πexp(−tα )dt ∂p(t, p ) =0Z1∂Πexp(−tα )dt ∂p(t, p α ),0ep ∈ Rn+ .[0,∞)e è äåëàÿ â ïîñëåäíåìÓ÷èòûâàÿ ïîëîæèòåëüíóþ îäíîðîäíîñòü ôóíêöèé Π è Π11ðàâåíñòâå çàìåíó ïåðåìåííûõ t = λ α s α , ãäå λ > 0 ôèêñèðîâàííîå ÷èñëî,52ïîëó÷èì ñëåäóþùåå ñîîòíîøåíèå:Z[0,∞)1α1α∂Π(s , p ) =exp(−λs)ds ∂p0Z11∂Πexp(−λs)ds ∂p(s α , p α ),0eλ > 0, p ∈ Rn+ .[0,∞)Èç åäèíñòâåííîñòè àíàëèòè÷åñêîãî ïðîäîëæåíèÿ ñëåäóåò, ÷òî ýòî ðàâåíñòâîe∂Π∂Πòàêæå âåðíî ïðè λ ∈ C, Re λ > 0.
Ñëåäîâàòåëüíî, ôóíêöèè ∂pè∂p0 ñîâïàäàþò.0e ñèëó óñëîâèé Π(+0, p) = 0 è Π(+0,p) = 0 äëÿ âñåõ p ∈ Rn+ , ìû ïîëó÷àåì, ÷òîe . Òåîðåìà 1.4 äîêàçàíà.Π=Π5322.1Îáðàùåíèå îáîáù¼ííîãî ïðåîáðàçîâàíèÿÐàäîíàÎñíîâíûå ðåçóëüòàòû ýòîì ïàðàãðàôå ìû ïðèâîäèì ðåøåíèå çàäà÷ îáðàùåíèÿ 1.2 è 1.4 äëÿ îïåðàòîðîâ Rq , Rqh è Πq . Ìû ðàññìàòðèâàåì çàäà÷ó îáðàùåíèÿ ïðè îäíîì èç äâóõïðåäïîëîæåíèé íà ôóíêöèþ q . Ìû ñ÷èòàåì, ÷òî ëèáî ôóíêöèÿ q óäîâëåòâîðÿåò(1.8), (1.10), ëèáî q = qα , ãäå qα îïðåäåëåíî â ôîðìóëå (1.9).Íàïîìíèì, ÷òî â ãëàâå 1 íàìè áûëè ïîëó÷åíû òåîðåìû 1.1 è 1.2, êîòîðûåñâîäÿò çàäà÷ó îáðàùåíèÿ äëÿ îïåðàòîðîâ Rq , Rqh è Πq ê çàäà÷å îáðàùåíèÿ ïðåîáðàçîâàíèÿ Ìåëëèíà (1.16).
Ýòè òåîðåìû ÿâëÿþòñÿ àíàëîãàìè êëàññè÷åñêîéïðîåêöèîííîé òåîðåìû, ñâîäÿùåé çàäà÷ó îáðàùåíèÿ ïðåîáðàçîâàíèÿ Ðàäîíà êçàäà÷å îáðàùåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå. Èç ýòèõ òåîðåì âûòåêàþò ñëåäóþùèåôîðìóëû îáðàùåíèÿ.Îïðåäåëèì ñëåäóþùåå ïðîñòðàíñòâî:CcN,σ (Rn+ ) = f ∈ C N (Rn+ ) | kf kN,σ,c < ∞ ,(2.1)ãäå N ∈ N ∪ 0, σ > 0, c ∈ Rn èkf kN,σ,c |α|∂u(y),= max sup (1 + |y|n ) ∂y α |α|≤N y∈Rnσnu(y) = ecy f (ey ).(2.2)Çàìåòèì, ÷òî CcN,σ (Rn+ ) ⊂ L1c (Rn+ ) ïðè N ≥ 0, σ > n, c ∈ Rn .Ïóñòü BR = {x ∈ Rn | |x| ≤ R , à Ωn−1 = 2π n/2 Γ(n/2)−1 îáîçíà÷àåò ïëîùàäüåäèíè÷íîé ñôåðû â Rn .Òåîðåìà 2.1.
Ïóñòü q óäîâëåòâîðÿåò (1.8) è (1.10), c ∈ Rn+ , α = c1 + · · · + cn .Ïóñòü h ∈ L1α (R1+ ), (M h)(s) 6= 0 ï.â. ïðè Re s = α, (M e−q )(z) 6= 0 ï.â. ïðèN,σRe z = c. Ïðåäïîëîæèì, ÷òî f ∈ CI−c(Rn+ ), ãäå N ≥ n + 1 è σ > n ≥ 2. Òîãäàôóíêöèÿ f ìîæåò áûòü íàéäåíà ïî ôóíêöèÿì Rq f , Rqh f , Πq f ïîñðåäñòâîìñëåäóþùèõ ôîðìóë:jjf (x) = fappr(x, R) + ferr(x, R),x ∈ Rn+ , R > 0, j = 1, 2, 3,(2.3)54−n1Zxz−Ifappr (x, R) = (2πi)(M Rq f )(z)Γ(s) dz,(M e−q )(z)c+iBR−n− 21 −n2fappr (x, R) = (2π)Ziz−Ix(M Rqh f )(z) Γ(s)dz,(M e−q )(z) (M h)(s)(2.4)c+iBR3fappr(x, R) = (2πi)−nZxz−I(M Πq f )(z)Γ(s + 2) dz,(M e−q )(z)c+iBRn−1 2 Nj|ferr(x, R)| ≤xc−I(Ω ) nkfkN,σ,I−c N −n ,(2π)n (σ − n)(N − n)Rj = 1, 2, 3,(2.5)ãäå s = z1 + · · · + zn .Ìû äîêàçûâàåì òåîðåìó 2.1 â 2.2.Òåïåðü ìû ïåðåéä¼ì ê çàäà÷å õàðàêòåðèçàöèè òàêèõ ôóíêöèé h è q , ïðèêîòîðûõ îïåðàòîðû Rq , Rqh è Πq ÿâëÿþòñÿ îáðàòèìûìè.
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