Диссертация (1103157), страница 13
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Ïóñòü ìíîæåñòâî íóëåé ôóíêöèè M e−q íèãäå íå ïëîòíî â ïëîñêîñòèHcn . Ìû ïîêàæåì îò ïðîòèâíîãî, ÷òî Rq èíúåêòèâåí â L1I−c (Rn+ ). Ïðåäïîëîæèì,îò ïðîòèâíîãî, ÷òî ñóùåñòâóåò ôóíêöèÿ f ∈ L1I−c (Rn+ ) òàêàÿ, ÷òî f 6≡ 0, Rq f ≡0.Ñïðàâåäëèâû ñëåäóþùèå óòâåðæäåíèÿ:nnHI−c\ ZI−c (M f ) îòêðûòî â HI−cè íåïóñòî,Hcn \ Zc (M e−q ) îòêðûòî è âñþäó ïëîòíî â Hcn .Îòñþäà ñëåäóåò, ÷òî ñóùåñòâóåò îòíîñèòåëüíî îòêðûòîå ìíîæåñòâî U ⊂ Hcn ,U 6= ∅, òàêîå ÷òî(M f )(I − z) (M e−q )(z) 6= 0,z ∈ U.Ó÷èòûâàÿ ýòó ôîðìóëó è ïîëüçóÿñü ðàâåíñòâîì (1.21), ìû ïîëó÷àåì, ÷òî (M Rq f )(z) 6=0 ïðè z ∈ U .
Ýòî ïðîòèâîðå÷èò ïðåäïîëîæåíèþ, ÷òî Rq f ≡ 0.Ñëó÷àé r = 2. ( =⇒ ). Ïðåäïîëîæèì, îò ïðîòèâíîãî, ÷òî ñóùåñòâóåò îãðàíè÷åííîå ìíîæåñòâî U ⊂ Hcn ïîëîæèòåëüíîé ìåðû Ëåáåãà (â Hcn ) òàêîå, ÷òîn(M e−q )(z) = 0 ïðè z ∈ U . Îïðåäåëèì ôóíêöèþ χ íà HI−côîðìóëîé1, z ∈ U,χ(I − z) =0, z ∈6 U.Çàòåì îïðåäåëèì ôóíêöèþ χb íà Rn+ ôîðìóëîé (2.25). Èç ôîðìóë (1.57), (2.25)è èç âëîæåíèÿ FL2 (Rn ) ⊂ L2 (Rn ) ñëåäóåò, ÷òîχb ∈ L2I−c (Rn+ ) è χb 6≡ 0.Èç ôîðìóë (1.55), (2.25) òàêæå ñëåäóåò, ÷òî ñïðàâåäëèâî ðàâåíñòâî (2.27).Ècïîëüçóÿ ôîðìóëó (1.21) ïðè f = χb è ó÷èòûâàÿ, ÷òî (M χb)(I−z)·(M e−q )(z) =0 ïðè z ∈ Hcn , ìû ïîëó÷àåì, ÷òî (M Rq χb)(z) = 0 ïðè z ∈ Hcn .
Îòñþäà ñëåäóåò,÷òî Rq χb ≡ 0, ÷òî ïðîòèâîðå÷èò èíúåêòèâíîñòè Rq â ïðîñòðàíñòâå L2I−c (Rn+ ).(⇐=). Ïðåäïîëîæèì, ÷òî M e−q íå îáíóëÿåòñÿ â ïëîñêîñòè Hcn ïî÷òè âñþäó.66Ïóñòü ôóíêöèÿ f ∈ L2I−c (Rn+ ) òàêîâà, ÷òî Rq f ≡ 0.Çàìåòèì, ÷òî xI−2c f (x) ∈ L2c (Rn+ ). Ïîëüçóÿñü ëåììîé 2.3, ìû íàõîäèì òàêèåak ∈ R, pk ∈ Rn+ , ÷òîxI−2c f (x) =X∞k=1ak exp −qpk (x) â L2c (Rn+ ).Èç ýòîé ôîðìóëû è èç ôîðìóëû êîïëîùàäè (1.36) âûòåêàåò ñëåäóþùåå ðàâåíñòâî, äîêàçûâàþùåå, ÷òî f ≡ 0:kf k22,I−cZI−2c 2x=f (x) dx =Rn+∞X∞Zak0k=1t−1 e−t (Rq f )( ptk ) dt = 0.Ñëó÷àé r = ∞. ( =⇒ ). Ïðåäïîëîæèì, îò ïðîòèâíîãî, ÷òî ñóùåñòâóåò z 0 ∈0Hcn òàêîé, ÷òî (M e−q )(z 0 ) = 0. Ïîëîæèì χb(x) = xz −I .nnÇàìåòèì, ÷òî χb ∈ L∞I−c (R+ ) è ÷òî äëÿ êàæäîãî p ∈ R+(Rq χb)(p) = p−z 0(1.63)(Rq χb)(I) == p−z 0(M e−q )(z 0 )= 0,Γ(z10 + · · · + zn0 )ãäå z 0 = (z10 , . .
. , zn0 ). Ïîñëåäíåå ðàâåíñòâî ïðîòèâîðå÷èò èíúåêòèâíîñòè Rq íànL∞I−c (R+ ).n2(I−c) −|x|(⇐=). Ïóñòü f ∈ L∞e f (x) ∈I−c (R+ ) è ïóñòü Rq f ≡ 0. Çàìåòèì, ÷òî xL1c (Rn+ ). Ïîëüçóÿñü ëåììîé 2.4, ìû íàõîäèì òàêèå ak ∈ R, pk ∈ Rn+ , ÷òîx2(I−c) −|x|ef (x) =∞Xexp −qkp (x) â L1c (Rn+ ).k=1Îòñþäà è èç ôîðìóëû (1.36) âûòåêàåò ñîîòíîøåíèåZx2(I−c) −|x| 2Rn+ef (x) dx =∞Xk=1Zak0∞t−1 e−t (Rq f )( ptk ) dt = 0,âëåêóùåå f ≡ 0.Äîêàçàòåëüñòâî òåîðåìû 2.3. Ñëó÷àé r = 1.
( =⇒ ). Ïðåäïîëîæèì, îò ïðîòèâíîãî, ÷òî Rqh èíúåêòèâåí â L1I−c (Rn+ ), íî ëèáî Zc (M e−q ) èìååò íåïóñòóþâíóòðåííîñòü â Hcn , ëèáî Zα (M h) èìååò íåïóñòóþ âíóòðåííîñòü â Hα1 . Òîãäàñóùåñòâóåò íåïóñòîå îòíîñèòåëüíî îòêðûòîå ìíîæåñòâî U ⊂ Zc ((M e−q )(M h)0 ),ãäå (M h)0 (z) = (M h)(z1 + · · · + zn ), z = (z1 , .
. . , zn ).67n) òàêóþ ÷òî χ 6≡ 0, χ(I −z) =Âûáåðåì ïðîèçâîëüíóþ ôóíêöèþ χ ∈ C ∞ (HI−c0 ïðè z 6∈ U è îïðåäåëèì χb ∈ L1I−c (Rn+ ) ôîðìóëîé (2.25). Ïîâòîðÿÿ äîêàçàòåëüñòâî ñëó÷àÿ r = 1 â òåîðåìå 2.2 (ïðè ýòîì ïîëüçóÿñü ôîðìóëîé (1.22) âìåñòîb ≡ 0, ÷òî ïðîòèâîðå÷èò èíúåêòèâíîñòè Rqh â(1.21)), ìîæíî ïîêàçàòü, ÷òî Rqh χïðîñòðàíñòâå L1I−c (Rn+ ).(⇐=). ×òîáû äîêàçàòü îáðàòíîå óòâåðæäåíèå, äîñòàòî÷íî ïîêàçàòü, ÷òî åñëèRq èíúåêòèâåí â L1I−c (Rn+ ) è Zc (M h) íèãäå íå ïëîòíî â Hα1 , òî Rqh èíúåêòèâåíâ L1I−c (Rn+ ).Ïðåäïîëîæèì, ÷òî îïåðàòîð Rq èíúåêòèâåí â L1I−c (Rn+ ) è ÷òî Zc (M h) íèãäåíå ïëîòíî â Hα1 .
Ïðåäïîëîæèì, ÷òî ôóíêöèÿ f ∈ L1I−c (Rn+ ) òàêîâà, ÷òî Rqh f ≡ 0.Ïîëüçóÿñü ôîðìóëàìè (1.21) è (1.22), ìû ïîëó÷àåì ñîîòíîøåíèå(M Rqh f )(z) = (M Rq f )(z) (M h)(s),ãäå z = (z1 , . . . , zn ) ∈ Hcn , s = z1 + · · · + zn . Îòñþäà è èç ôîðìóëû (1.55) ñëåäóåò,÷òî M Rq f ≡ 0 â Hc1 êàê íåïðåðûâíàÿ ôóíêöèÿ, îáíóëÿþùàÿñÿ íà îòêðûòîìâñþäó ïëîòíîì ìíîæåñòâå. Ñëåäîâàòåëüíî, Rq f ≡ 0 è f ≡ 0.Ñëó÷àé r = 2. Èç ëåììû 2.3 ñëåäóåò, ÷òî îïåðàòîð Rqh èíúåêòèâåí â L2I−c (Rn+ )òîãäà è òîëüêî òîãäà, êîãäà(M (h ◦ q))(z) 6= 0,(h ◦ q)(x) = h(q(x)),(2.28)äëÿ ïî÷òè âñåõ z ∈ Hcn . Ïîëüçóÿñü ôîðìóëîé (1.61) ñ p = I , ìû ïîëó÷àåì, ÷òî(2.28) âûïîëíåíî äëÿ ïî÷òè âñåõ z ∈ Hcn òîãäà è òîëüêî òîãäà, êîãäà M e−q íåðàâíî íóëþ ïî÷òè âñþäó â Hcn è M h íå ðàâíî íóëþ ïî÷òè âñþäó â Hα1 .nÑëó÷àé r = 3.
Èç ëåììû 2.4 ñëåäóåò, ÷òî Rqh èíúåêòèâåí â L∞I−c (R+ ) òîãäàè òîëüêî òîãäà, êîãäà íåðàâåíñòâî (2.28) âûïîëíåíî äëÿ âñåõ z ∈ Hcn . Ñ äðóãîéñòîðîíû, ó÷èòûâàÿ òîæäåñòâî (1.61), ìîæíî âèäåòü, ÷òî (2.28) âûïîëíåíî äëÿâñåõ z ∈ Hcn òîãäà è òîëüêî òîãäà, êîãäà M e−q íå îáíóëÿåòñÿ â Hcn è M h íåîáíóëÿåòñÿ â Hα1 .2.5Äîêàçàòåëüñòâî òåîðåì 2.4, 2.5 è ïðåäëîæåíèé 2.1, 2.2Ìû íà÷í¼ì ýòîò ïàðàãðàô ñ äîêàçàòåëüñòâà òåîðåìû 2.4.Äîêàçàòåëüñòâî òåîðåìû 2.4. Óòâåðæäåíèå (1). Ïóñòü µ áîðåëåâñêàÿ ìåðà(ñî çíàêîì), èíòåãðèðóåìàÿ ñ âåñîì exp(−A|x|α ) ïðè íåêîòîðîì A > 0 è ïóñòü68(Πq µ)(p0 , p) = 0, p0 > 0, p ∈ Rn+ , ãäå q = qα , α ∈ (0, 1].
Íå îãðàíè÷èâàÿ îáùíîñòè,ìû ñ÷èòàåì, ÷òî a1 = · · · = an = 1, òàê êàê îáùèé ñëó÷àé ñâîäèòñÿ ê ýòîìóçàìåíîé ïåðåìåííûõ.Ïîëüçóÿñü ôîðìóëîé (1.39), ìû ïîëó÷àåì, ÷òî µ(Lq (p0 , p)) = 0 ïðè p0 > 0,p ∈ Rn+ , ãäå ìíîæåñòâî Lq (p0 , p) îïðåäåëåíî â ôîðìóëå (1.38). Ïóñòü µ = µ1 − µ2 ðàçëîæåíèå Æîðäàíà ìåðû µ, òàê ÷òî µ1 è µ2 íåîòðèöàòåëüíûå áîðåëåâñêèåìåðû è ñïðàâåäëèâî ðàâåíñòâîµ1 (Lq1 (p0 , p)) = µ2 (Lq2 (p0 , p)),p0 > 0, p ∈ Rn+ .Âûáåðåì h(t) = exp(−C(A)tα ), ãäå C(A) > 0 íåêîòîðàÿ äîñòàòî÷íî áîëüRøàÿ êîíñòàíòà. Òîãäà Rn h(qp (x))µj (dx) < ∞, j = 1, 2, äëÿ âñåõ p ∈ Rn+ .+Ïîëüçóÿñü ëåììîé 1.4, ìû ïîëó÷àåì, ÷òîZexp −C(A)((p1 x1 )α + · · · + (pn xn )α ) µ(dx) = 0,p ∈ Rn+ .Rn+Ñäåëàåì çàìåíó ïåðåìåííûõ yj = xαj è çàìåíó ïàðàìåòðîâ vj = C(A)pαj .
Ïðèýòîì ìåðà µ ïåðåéä¼ò â ìåðó µ0 , äëÿ êîòîðîé ñïðàâåäëèâî ðàâåíñòâîZexp(−v1 y1 − · · · − vn yn µ0 (dy),v ∈ Rn+ .Rn+Îòñþäà ñëåäóåò, ÷òî µ0 = 0 è µ = 0.Óòâåðæäåíèå (2). Ïóñòü f ∈ L1 (Rn+ ) ∩ C(Rn+ ) è ïóñòü Πq f = 0. Èç ôîðìóë(1.43) è (1.44) ñëåäóåò, ÷òî Rq f = 0.Çàìåòèì, ÷òî òîæäåñòâî (Rq f )(p) = 0, p ∈ Rn+ , ýêâèâàëåíòíî òîæäåñòâóZf (u) u1 du2 ∧ · · · ∧ dun = 0,t > 0.(2.29)au1 1 ···uann =taÑäåëàåì çàìåíó ïåðåìåííûõ xj = uj j , ãäå aj îïðåäåëåíû â (1.9), è ââåä¼ì êîîðäèíàòû (y1 , . . . , yn−1 , h) è ôóíêöèþ fh , êàê â ôîðìóëå (2.6). Ìû ïîëó÷èì, ÷òîRòîæäåñòâî (2.29) ýêâèâàëåíòíî òîìó, ÷òî Rn−1 fh (y)dy = 0 ïðè âñåõ h ∈ R.Óòâåðæäåíèå (3).
Ïóñòü f ∈ L1I−c (Rn+ ) ∩ C(Rn+ ) è ïóñòü Πq f = 0, ãäå q = qα ,α ∈ (−∞, 0), c ∈ Rn+ (ïîêîìïîíåíòíî). Ìû òàêæå ñ÷èòàåì, íå îãðàíè÷èâàÿîáùíîñòè, ÷òî a1 = · · · = an = 1. Ïîëüçóÿñü ôîðìóëîé (1.43), ïîëó÷àåì, ÷òî69Rq f = 0.11Ïîëîæèì g(x) = f (x1α , . . . , xnα ) è çàìåòèì, ÷òî(2.30)g ∈ L1(I−c)/α (Rn+ ).Çàìåòèì, ÷òî ñïðàâåäëèâà ôîðìóëàZ(Rq f )(p) =f (x)qp−1 (1)x1 dx2 ∧ · · · ∧ dxn.pα1 xα1Äåëàÿ çàìåíó ïåðåìåííûõ yj = (pj xj )α , ïðèâåä¼ì ýòó ôîðìóëó ê ñëåäóþùåìóâèäó:Z11y1αynαn−1(Rq f )(p) = (−1) f,...,p1pny1 y2 · · · y n α1 −1 dy2 ∧ · · · ∧ dyn.αn−1 p1 · · · pn(2.31)y1 +···+yn =10≤yj ≤1Ó÷èòûâàÿ ôîðìóëû (2.30) è (2.31) è âû÷èñëÿÿ ïðåîáðàçîâàíèå Ìåëëèíà îò òîæäåñòâà−1−1−1−1p1 α · · · pn α (Rq f )(p1 α , . . .
, pn α ) = 0ïî ïåðåìåííûì p1 , . . . , pn , ïîëó÷èì òîæäåñòâîΓ( α1 − z1 ) · · · Γ( α1 − zn )(M g)(z)= 0,Γ( αn − z1 − · · · − zn )Re z =I −c.αÎòñþäà ñëåäóåò, ÷òî M g(z) = 0 ïðè Re z = (I −c)/α. Ó÷èòûâàÿ ôîðìóëó (2.30),ïîëó÷èì, ÷òî g = 0 è f = 0.Óòâåðæäåíèå (4). Ïóñòü f ∈ L1 (Rn+ ) ∩ C 1 (Rn+ ) è ïóñòü Πq f = 0, ãäå q =q−∞ . Äîñòàòî÷íî ðàññìîòðåòü ñëó÷àé, êîãäà a1 = · · · = an = 1, òàê êàê îáùèéñëó÷àé ñâîäèòñÿ ê íåìó çàìåíîé ïåðåìåííûõ p.
Èç ôîðìóëû (1.43) ñëåäóåò, ÷òî(Rq f )(p) = 0, p ∈ Rn+ , ãäå∞(Rq f )(p) =n ZXj=1 1Z∞ x1xj−1 1 xj+1xn (dx)∧j,...,, ,,...,,··· fp1pj−1 pj pj+1pn p1 · · · p n(2.32)1è (dx)∧j = dx1 · · · dxj−1 dxj+1 · · · dxn . Çàìåòèì, ÷òî â äàííîì ñëó÷àå ôóíêöèÿq íå ÿâëÿåòñÿ ãëàäêîé è ìû íå ìîæåì èñïîëüçîâàòü ôîðìóëó (1.12), ÷òîáû70îïðåäåëèòü Rq f .−1Äèôôåðåíöèðóÿ òîæäåñòâî (Rq f )(p−11 , . . . , pn ) ≡ 0 ïî ïåðåìåííûì p1 , . .
. ,pn , ìû ïîëó÷àåì ñëåäóþùåå òîæäåñòâî:p1∂f∂f(p) + · · · + pn(p) = −nf (p),∂x1∂xnp ∈ Rn+ .Èñïîëüçóÿ ìåòîä õàðàêòåðèñòèê, ìû íàõîäèì, ÷òî â ñôåðè÷åñêèõ êîîðäèíàòàõ(ϑ, r) ñïðàâåäëèâà ôîðìóëà f (ϑ, r) = C(ϑ)r−n . Ó÷èòûâàÿ, ÷òî f ∈ L1 (Rn+ ), ìûïîëó÷àåì, ÷òî C = 0 è f = 0. Òåîðåìà 2.4 äîêàçàíà.Ñëåäóþùåé íàøåé öåëüþ ÿâëÿåòñÿ äîêàçàòåëüñòâî ïðåäëîæåíèÿ 2.1. Íàìïîòðåáóåòñÿ îäíà âñïîìîãàòåëüíàÿ ëåììà, êîòîðàÿ óñòàíàâëèâàåò, ÷òî ñâîéñòâîèíúåêòèâíîñòè äëÿ îïåðàòîðà ïðèáûëè Πq íàñëåäóåòñÿ ïî îòíîøåíèþ ê ÷àñòè÷íîé êîìïîçèöèè (1.11).1Ëåììà 2.5. Ïóñòü ôóíêöèè q : Rk+1→ R1+ , k ≥ 1, φ ∈ Rm++ → R+ , m ≥2, óäîâëåòâîðÿþò (1.8), (1.10).
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