Диссертация (Некоторые методы проекционного типа численного решения одного класса слабо сингулярных интегральных уравнений), страница 3

Îí áàçèðóåòñÿ íà ïðîñòîéðåãóëÿðèçàöèè, ïðåäëîæåííîé Êàíòîðîâè÷åì [64], [65].Ïóñòü ϕ ðåøåíèå óðàâíåíèÿ (1.3.1). Íàïîìíèì, ÷òî y = Λϕ ÿâëÿåòñÿðåøåíèåì óðàâíåíèÿy = $0 Λy + Λf,(1.3.6)îòëè÷àþùåãîñÿ îò èñõîäíîãî óðàâíåíèÿ òîëüêî òåì, ÷òî åãî ïðàâàÿ ÷àñòü fçàìåíåíà íà Λf . ßñíî, ÷òîϕ = $0 y + f.Ïóñòü Λf ∈ B . Íàéäåì ïðèáëèæåííîå ðåøåíèå y h óðàâíåíèÿ (1.3.6)ìåòîäîì Ãàëåðêèíày h = $0 P h Λy h + P h Λf,à çàòåì âû÷èñëèì ïðèáëèæåíèå ϕeh ê ðåøåíèþ ϕ ïî ôîðìóëåϕeh = $0 y h + f.(1.3.7)21Òàê ýòîò ìåòîä îñíîâàí íà àïïðîêñèìàöèè ôóíêöèè y = Λϕ, òî îí îáëàäàåòïðåèìóùåñòâîì ïåðåä ìåòîäîì Ãàëåðêèíà òîãäà, êîãäà ôóíêöèÿ y ÿâëÿåòñÿáîëåå ãëàäêîé, ÷åì ϕ.Çàìåòèì, ÷òîϕeh = $0 P h Λ($0 y h + f ) + f = $0 P h Λϕeh + fÏîýòîìó ìåòîä Êàíòîðîâè÷à ìîæåò áûòü çàïèñàí â âèäå(1.3.8)ϕeh = $0 P h Λϕeh + f.×åòâåðòûé ìåòîä ýòî èòåðèðîâàííûé ìåòîä Êàíòîðîâè÷à, â êîòîðîìïðèáëèæåííîå ðåøåíèå ϕbh îïðåäåëÿåòñÿ ñëåäóþùèì îáðàçîìϕbh = $0 y h + f,y h = $0 Λy h + Λf,ãäå y h ðåøåíèå óðàâíåíèÿ (1.3.7).Êàê íåòðóäíî âèäåòü, y h = P h y .

Ïîýòîìó y h ìîæíî ðàññìàòðèâàòü êàêïðèáëèæåííîå ðåøåíèå óðàâíåíèÿ (1.3.6), íàéäåííîå èòåðèðîâàííûì ìåòîäîìÃàëåðêèíày h = $0 ΛP h y h + Λf.Çàìå÷àíèå 1.3.1. Îáðàòèìâíèìàíèå íà òî, ÷òî äëÿ ÷èñëåííîéðåàëèçàöèè ïåðâûõ äâóõ ìåòîäîâ äîñòàòî÷íî èìåòü ìåòîä ðåøåíèÿêîíå÷íîìåðíîé çàäà÷è (1.3.2), à äëÿ ðåàëèçàöèè îñòàëüíûõ ìåòîäîâ âîñïîëüçîâàòüñÿ ýòèì ìåòîäîì äëÿ ðåøåíèÿ çàäà÷è (1.3.6), çàìåíèâïðàâóþ ÷àñòü f íà Λf .Ïóñòü ϕ ðåøåíèå çàäà÷è (1.3.1), à ϕh , ϕh , ϕeh è ϕbh ïðèáëèæåííûåðåøåíèÿ,íàéäåííûåìåòîäîìÃàëåðêèíà,èòåðèðîâàííûììåòîäîì22Ãàëåðêèíà, ìåòîäîì Êàíòîðîâè÷à è èòåðèðîâàííûì ìåòîäîì Êàíòîðîâè÷àñîîòâåòñòâåííî.Áóäåìèñïîëüçîâàòüñëåäóþùèåîáîçíà÷åíèÿäëÿïîãðåøíîñòåé ðàññìàòðèâàåìûõ ìåòîäîâ:εh = ϕh − ϕ,εh = ϕh − ϕ,εeh = ϕeh − ϕ,εbh = ϕbh − ϕ. äàëüíåéøåì íàì áóäåò ïîëåçíî ñëåäóþùåå óòâåðæäåíèå.Ëåììà 1.3.1.

Ïóñòü f ∈ B . Òîãäàk εh kB ≤ γ0 k(I − P h )ϕkB ,(1.3.9)k εh kB ≤ γ1 kΛ(I − P h )ϕkB , .(1.3.10)Ïóñòü Λf ∈ B . Òîãäàk εeh kB ≤ γ1 k(I − P h )ΛϕkB ,(1.3.11)k εbh kB ≤ γ2 kΛ(I − P h )ΛϕkB .(1.3.12)Äîêàçàòåëüñòâî. Ïðèìåíÿÿ ê óðàâíåíèþ (1.3.1) îïåðàòîð P h , èìååìP h ϕ = $0 P h Λϕ + P h f.Âû÷èòàÿ ýòî ðàâåíñòâî èç (1.3.2), ïðèõîäèì ê ðàâåíñòâó(I − $0 P h Λ)εh = −(I − P h )ϕ,èç êîòîðîãî ñ ó÷åòîì (1.3.3) ñëåäóåò îöåíêà (1.3.9).Âû÷èòàÿ (1.3.1) èç (1.3.4), ïðèõîäèì ê ðàâåíñòâó(I − $0 ΛP h )εh = −$0 Λ(I − P h )ϕ,èç êîòîðîãî ñ ó÷åòîì (1.3.5) ñëåäóåò îöåíêà (1.3.10).Çàìåòèì, ÷òîεeh = $0 (y h − y),εbh = $0 (y − y),23ãäå y h − y è y h − y ïîãðåøíîñòè ìåòîäà Ãàëåðêèíà è èòåðèðîâàííîãî ìåòîäàÃàëåðêèíà ðåøåíèÿ óðàâíåíèÿ (1.3.6). Ïîýòîìó îöåíêè (1.3.11) è (1.3.12)ïîëó÷àþòñÿ èç îöåíîê (1.3.9) è (1.3.10) çàìåíîé ϕ íà y = Λϕ.1.4Ìåòîäû, îñíîâàííûå íà èñïîëüçîâàíèè îïåðàòîðàïðîåêòèðîâàíèÿ íà ïðîñòðàíñòâî êóñî÷íî ïîñòîÿííûõôóíêöèé è íåêîòîðûå ñâîéñòâà îïåðàòîðàπhÄàäèì êðàòêîå îïèñàíèå èçâåñòíûõ ïðîåêöèîííûõ ìåòîäîâ ðåøåíèÿóðàâíåíèÿ 1.2.1, èñïîëüçóþùåãî ïðîåêòèðîâàíèÿ íà ïðîñòðàíñòâî êóñî÷íîïîñòîÿííûõ ôóíêöèé, ñëåäóÿ, â îñíîâíîì [59].Ââåäåì íà îòðåçêå J = [0, τ∗ ] ïðîèçâîëüíóþ íåðàâíîìåðíóþ ñåòêó J h ñóçëàìè0 = τ0 < τ1 < .

. . < τn−1 < τn = τ∗ .Ïîëîæèì hi = τi − τi−1 ,1 ≤ i ≤ n,hmax = max hi .1≤i≤nhÂâåäåì ïðîñòðàíñòâî S1/2(J), ýëåìåíòàìè êîòîðîãî ÿâëÿþòñÿ êóñî÷íîïîñòîÿííûå ôóíêöèè âèäà f (τ ) =nPj=1fj−1/2 χhj−1/2 (τ ), ãäå χhj−1/2 õàðàêòåðèñ-òè÷åñêàÿ ôóíêöèÿ èíòåðâàëà (τj−1 , τj ).hÂâåäåì îïåðàòîð ïðîåêòèðîâàíèÿ π h : Lp (J) → S1/2(J) ôîðìóëîé(π h ϕ)(τ ) = π h ϕi−1/2 = h−1iZτiϕ(τ 0 ) dτ 0 ,τ ∈ (τi−1 , τi ),1 ≤ i ≤ n.τi−1Çàìåòèì, ÷òî π h ÿâëÿåòñÿ îïåðàòîðîì îðòîãîíàëüíîãî ïðîåêòèðîâàíèÿ èçhL2 (J) íà S1/2(J). ðàáîòå [59] ïîäðîáíî èññëåäîâàíû ñâîéñòâà ñõîäèìîñòè ÷åòûðåõïðîåêöèîííûõ ìåòîäîâ ðåøåíèÿ óðàâíåíèÿ (1.3.1), ÿâëÿþùèìèñÿ ÷àñòíûìè24ñëó÷àÿìè ââåäåííûõ â ïðåäûäóùåì ïàðàãðàôå ìåòîäîâ, ñîîòâåòñòâóþùèõhâûáîðó B = Lp (J), P h = π h è S h = S1/2(J).Ïåðâûé èç íèõ ýòî êëàññè÷åñêèé ìåòîä Ãàëåðêèíà, â êîòîðîì ïðèáëèæåíhíîå ðåøåíèå ϕh ∈ S1/2(J) ÿâëÿåòñÿ ðåøåíèåì êîíå÷íîìåðíîé çàäà÷èϕh = $0 π h Λϕh + π h f.(1.4.1)Âòîðîé ìåòîä ýòî èòåðèðîâàííûé ìåòîä Ãàëåðêèíàϕh = $0 Λπ h ϕh + f.(1.4.2)Òðåòèé ìåòîä ìåòîä Êàíòîðîâè÷àϕeh = $0 π h Λϕeh + f.×åòâåðòûé ìåòîä åñòü èòåðèðîâàííûé ìåòîä Êàíòîðîâè÷àϕbh = $0 y h + f,y h = $0 Λπ h y h + Λf. [59] îöåíêè ïîãðåøíîñòåé ýòèõ ìåòîäîâ ïîëó÷åíû ïðè ñëåäóþùèõäîïîëíèòåëüíûõ ïðåäïîëîæåíèÿõ î ñâîéñòâàõ ÿäðà E :E ∈ Wr1 (δ, +∞) äëÿ âñåõ δ > 0 è âñåõ r ∈ [1, ∞),(1.4.3)DE(τ ) = o(τ −1 E(τ )) ïðè τ → 0+ .(1.4.4)Îáðàòèì âíèìàíèå íà òî, ÷òî èç (1.4.3) ñëåäóåò, ÷òî E ∈ C(R+ ).

Ïîñêîëüêóïðîèçâîäíàÿ DE îïðåäåëåíà ëèøü ïî÷òè âñþäó íà R+ , óñëîâèå (1.4.4)ïîíèìàåòñÿ ñëåäóþùèì îáðàçîìα(δ) = ess sup |τ DE(τ )/E(τ )| → 0 ïðè δ → 0+ .τ ∈(0,δ)Ïîñêîëüêó E íå âîçðàñòàåò è E ∈ Lr (R+ ) äëÿ âñåõ r ∈ [1, ∞), òî, êàêñëåäóåò èç [59], E(τ ) = o(τ −ε ) ïðè τ → 0+ äëÿ âñåõ ε > 0.25Èç ïðåäïîëîæåíèÿ (1.4.3) ñëåäóåò, ÷òî äëÿ âñåõ r ≥ 1δ∫ E r (τ ) dτ ∼ δE r (δ) ïðè δ → 0+ .(1.4.5)0Äåéñòâèòåëüíî, èíòåãðèðóÿ ïî ÷àñòÿì, èìååì:δδδ000∫ E r (τ ) dτ = δE r (δ) − r ∫ τ E r−1 (τ )DE(τ ) dτ ≤ δE r (δ) + rα(δ) ∫ E r (τ ) dτ.ÑëåäîâàòåëüíîδδE r (δ) ≤ ∫ E r (τ ) dτ ≤01δE r (δ)1 + rα(δ)è ïîýòîìó ñïðàâåäëèâî ñâîéñòâî (1.4.5). äàííîé ðàáîòå ìû îòêàæåìñÿ îò ïðåäïîëîæåíèé (1.4.3), (1.4.4).Áóäåì ïðåäïîëàãàòü ëèøü âûïîëíåíèå ñëåäóþùèõ ñâîéñòâ ÿäðà E :A1 ) E ÿâëÿåòñÿ ïîëîæèòåëüíîé óáûâàþùåé ôóíêöèåé, çàäàííîé íà R+ èòàêîé, ÷òî E(0+ ) = +∞;A2 ) E ∈ Lr (R+ ) äëÿ âñåõ r ∈ [1, +∞); êðîìå òîãî, kEkL1 (R+ ) ≤ 1/2.A3 ) Äëÿ âñåõ r ≥ 1 âûïîëíåíî ñâîéñòâî (1.4.5).Çàìå÷àíèå 1.4.1.

Îáðàòèì âíèìàíèå íà òî, ÷òî ÿäðî èíòåãðàëüíîãî1E1 óäîâëåòâîðÿåò ïðåäïîëîæåíèÿì2A1 ) A3 ). Ýòî ñëåäóåò èç èçâåñòíûõ ñâîéñòâ ôóíêöèè E1 :óðàâíåíèÿ ïåðåíîñà èçëó÷åíèÿ E =E1 (τ ) ∼ e−τ /τE1 (τ ) ∼ ln(1/τ ) ïðè τ → 0+ ;ïðèτ → +∞,DE1 (τ ) = −e−τ /τ ∼ −1/τ ïðè τ → 0+ .Ââåäåì â ðàññìîòðåíèå èíòåãðàëüíûé ìîäóëü íåïðåðûâíîñòè ÿäðà Eωr (E, η) = sup kE(| · +δ|) − E(| · |)kLr (R) ,0<δ≤η1 ≤ r < ∞.26Ëåììà 1.4.1. Ïðè âûïîëíåíèè ïðåäïîëîæåíèé A1 ), A2 ) ñïðàâåäëèâàîöåíêà(1.4.6)ωr (E, η) ≤ 41/r kEkLr (0,η/2) .Êàê ñëåäñòâèå, ïðè âûïîëíåíèè ïðåäïîëîæåíèé A1 ) A3 ) ñïðàâåäëèâàîöåíêàωr (E, η) ≤ 21/r η 1/r E(η/2)(1 + o(1)) ïðè η → 0+ .(1.4.7)Äîêàçàòåëüñòâî.

Çàìåòèì, ÷òîkE(| · +δ|) − E(| · |)krLr (R) = 2=2R∞|E(|τ + δ/2|) − E(|τ − δ/2|)|r dτ =0δ/2RR∞0δ/2[E(δ/2 − τ ) − E(τ + δ/2)]r dτ + 2[E(τ − δ/2) − E(τ + δ/2)]r dτ.Âîñïîëüçóåìñÿ ýëåìåíòàðíûì íåðàâåíñòâîì(1.4.8)(a − b)r ≤ ar − br ,ñïðàâåäëèâûì äëÿ âñåõ 0 ≤ b ≤ a, ïîëó÷èìkE(| · +δ|) − E(| ·|)krLr (R)+2R∞≤2δ/2RE (δ/2 − τ ) dτ − 2E r (τ ) dτ − 2=2rE (τ ) dτ − 20δ/2RE r (τ + δ/2) dτ +00δ/2δ/2RrR∞E r (τ + δ) dτ =δ/2RδrE (τ ) dτ + 2δ/2Rδ0rE (τ ) dτ = 4δ/2RE r (τ ) dτ.0Îòñþäàωr (E, η) = sup kE(| · +δ|) − E(| · |)kLr (R) ≤ 41/r kEkLr (0,η/2) .0<δ≤ηÅñëè âûïîëíåíî ñâîéñòâî A3 ), òî kEkLr (0,η/2) ∼ (η/2)1/r E(η/2) ïðè η → 0.Ïîýòîìó ñïðàâåäëèâà îöåíêà (1.4.7).Ëåììà äîêàçàíà.27Ïîëîæèìω r (E; J h ) = sup ωr (E(| · −t|); J h ),0<t<τ∗ãäå"ωr (E(| · −t|); J h ) =nXτi#1/rτih−1∫ ∫ |E(|τ 0 − t|) − E(|τ − t|)|r dτ 0 dτiτi−1 τi−1i=1Ëåììà 1.4.2.

Ïðè âûïîëíåíèè ïðåäïîëîæåíèé A1 ), A2 ) ñïðàâåäëèâàîöåíêà(1.4.9)ω r (E; J h ) ≤ 81/r kEkLr (0,hmax /2) .Êàê ñëåäñòâèå, ïðè âûïîëíåíèè ïðåäïîëîæåíèé A1 ) A3 ) ñïðàâåäëèâàîöåíêàω r (E; J h ) ≤ 41/r h1/rmax E(hmax /2)(1 + o(1)) ïðè hmax → 0.(1.4.10)Äîêàçàòåëüñòâî. Ôèêñèðóåì t ∈ (0, τ∗ ) è ïîëîæèì ti = τi − t, 0 ≤ i ≤ n.Çàìåòèì, ÷òî ti − ti−1 = hi , 1 ≤ i ≤ n èh rωr (E(| · −t|); J ) =nXIih (t),i=1ãäåIih (t)=h−1ititi∫ ∫ |E(|τ 0 |) − E(|τ |)|r dτ 0 dτti−1 ti−1Îáîçíà÷èì ÷åðåç ` è j ìèíèìàëüíîå è ìàêñèìàëüíîå èç òåõ çíà÷åíèé i, äëÿêîòîðûõ îòðåçîê [ti−1 , ti ] ïåðåñåêàåòñÿ ñ èíòåðâàëîì (−hmax /2, hmax /2). Òîãäàh rωr (E(| · −t|); J ) =`−1Xi=1Iih (t)+jXi=`Iih (t)+nXIih (t),i=j+1Ïóñòü j + 1 ≤ i ≤ n. Ïîëüçóÿñü ìîíîòîííîñòüþ ÿäðà E è íåðàâåíñòâîì(1.4.8), èìååìIih (t) ≤ hi [E(ti−1 ) − E(ti )]r ≤ hmax [E r (ti−1 ) − E r (ti )].28ÎòñþäànXXIih (t) ≤ hmaxi=j+1[E r (ti−1 ) − E r (ti )] ≤ hmax E r (tj ).(1.4.11)i=j+1Àíàëîãè÷íî`−1X(1.4.12)Iih (t) ≤ hmax E r (|t`−1 |).i=1Ïóñòü òåïåðü i òàêîâî, ÷òî îòðåçîê [ti−1 , ti ] ïåðåñåêàåòñÿ ñ èíòåðâàëîì(−hmax /2, hmax /2).

Åñëè [ti−1 , ti ] ⊂ (−hmax /2, hmax /2), òîtititiIih (t) ≤ h−1∫ ∫ [E r (|τ |) + E r (|τ 0 |)] dτ dτ 0 = 2 ∫ E r (|τ |) dτ.iti−1 ti−1ti−1Ïîñêîëüêó äëÿ i = j ñïðàâåäëèâû íåðàâåíñòâà −hmax /2 < tj−1 < hmax /2 < tj ,òî(Ijh (t) = h−1jhmax /2tj−1"≤tj−1#tj∫ [E(|τ |) − E(τ 0 )]r dτ 0 dτ +tj−1h−1j∫ [E(|τ |) − E(|τ 0 |)]r dτ 0 dτ +∫+2 ∫(hmax /2 hmax /2hmax /2tj∫)tj∫ [E(τ ) − E(τ 0 )]r dτ 0 dτhmax /2 hmax /2∫ [E r (|τ |) + E r (|τ 0 |)] dτ 0 dτ +∫tj−1hmax /2"+2 ∫tj−1tj−1#tjtj∫ [E r (|τ |) − E r (tj )]r dτ 0 dτ + ∫hmax /2)tj∫ [E r (hmax /2)−E r (tj )]dτ 0 dτhmax /2= h−12(hmax /2 − tj−1 ) ∫jhmax /2E r (|τ |)dτ +2(tj − hmax /2) ∫ [E r (|τ |) − E r (tj )]dτ +tj−1tj−1)+ (tj − hmax /2)2 [E r (hmax /2) − E r (tj )]hmax /2≤E r (|τ |) dτ + hmax [E r (hmax /2) − E r (tj )].tj−1Àíàëîãè÷íîI`h (t)=hmax /2 hmax /2(≤2 ∫≤hmax /2 hmax /2≤2τ`∫−hmax /2E r (|τ |) dτ + hmax [E r (hmax /2) − E r (|t` |)].29Òàêèì îáðàçîì,jXIih (t)hmax /2≤2∫E r (|τ |) dτ +−hmax /2i=`+ hmax [E r (hmax /2) − E r (|t` |)] + hmax [E r (hmax /2) − E r (tj )].

(1.4.13)Ñêëàäûâàÿ îöåíêè (1.4.11) (1.4.13), ïðèõîäèì ê íåðàâåíñòâó"hmax /2ω r (E; J h ) = sup ωr (E(| · −t|); J h ) ≤ 4 ∫0<t<τ∗#1/rE r (τ ) dτ + 2hmax E r (hmax /2).0Ó÷èòûâàÿ, ÷òîhmax /22hmax E r (hmax /2) ≤ 4 ∫E r (τ ) dτ,0çàâåðøàåì âûâîä îöåíêè (1.4.9).Ïðè âûïîëíåíèè óñëîâèÿ A3 ) èìååìkEkLr (0,hmax /2) ∼ (hmax /2)1/r E(hmax /2) ïðè hmax → 0.(1.4.14)Ïîýòîìó âåðíà îöåíêà (1.4.10).Ëåììà 1.4.3. Äëÿ âñåõ r ∈ [1, ∞) ñïðàâåäëèâû îöåíêèk(I − π h )EkLr (J) ≤ 41/r kEkLr (0,hmax /2) ,(1.4.15)k(I − π h )E ∗ kLr (J) ≤ 41/r kEkLr (0,hmax /2) .(1.4.16)êàê ñëåäñòâèå, ïðè âûïîëíåíèè ñâîéñòâà (1.4.14) ñïðàâåäëèâû îöåíêèk(I − π h )EkLr (J) ≤ 21/r h1/rmax E(hmax /2)(1+o(1)) ïðè hmax → 0,(1.4.17)k(I − π h )E ∗ kLr (J) ≤ 21/r h1/rmax E(hmax /2)(1+o(1)) ïðè hmax → 0.

(1.4.18)Äîêàçàòåëüñòâî. Çàìåòèì, ÷òîk(I − πh)EkrLr (J)h r≤ ωr (E; J ) =nXi=1Iih ,30ãäåτiτiIih = h−1∫ ∫ |E(|τ 0 |) − E(|τ |)|r dτ 0 dτ.iτi−1 τi−1Îáîçíà÷èì ÷åðåç j ìàêñèìàëüíîå èç òåõ çíà÷åíèé i, äëÿ êîòîðûõîòðåçîê [ti−1 , ti ] ïåðåñåêàåòñÿ ñ èíòåðâàëîì (0, hmax /2). Òîãäà àíàëîãè÷íîäîêàçàòåëüñòâó ëåììû 1.4.2k(I − π≤2hmaxR /2rh)EkrLr (J)≤jPIihi=1r+nPi=j+1E (τ ) dτ + hmax E (hmax /2) ≤ 40Iih ≤hmaxR /2E r (τ ) dτ.0Îöåíêà (1.4.15) äîêàçàíà. Îöåíêà (1.4.16) äîêàçûâàåòñÿ ñîâåðøåííîàíàëîãè÷íî è ôàêòè÷åñêè ÿâëÿåòñÿ ñëåäñòâèåì îöåíêè (1.4.15).Ëåììà 1.4.4.

Ïóñòü âûïîëíåíû ïðåäïîëîæåíèÿ A1 ), A2 ) è ïóñòü 1 ≤p ≤ q ≤ ∞, 1/s = 1 − 1/p + 1/q > 0. Òîãäàk(I − π h )ΛkLp (J)→Lq (J) ≤ 22/s+1/q kEkLs (0,hmax /2) ,(1.4.19)kΛ(I − π h )kLp (J)→Lq (J) ≤ 23/s−1/q kEkLs (0,hmax /2) ,(1.4.20)kΛ(I − π h )ΛkLp (J)→Lq (J) ≤ 22+3/s kEk2Lr (0,hmax /2) ,(1.4.21)ãäå r = 2(1 + 1/s)−1 ∈ [1, 2]. Îöåíêà (1.4.21) âåðíà è â ñëó÷àå 1/s = 0, òîåñòü ïðè p = 1 è q = ∞.Êàê ñëåäñòâèå, ïðè âûïîëíåíèè ïðåäïîëîæåíèé A1 ) A3 ) ñïðàâåäëèâûîöåíêèk(I − π h )ΛkLp (J)→Lq (J) ≤ 21/s+1/q h1/smax E(hmax /2)(1+o(1)) ïðè hmax → 0, (1.4.22)kΛ(I − π h )kLp (J)→Lq (J) ≤ 22/s−1/q h1/smax E(hmax /2)(1+o(1)) ïðè hmax → 0, (1.4.23)2kΛ(I − π h )ΛkLp (J)→Lq (J) ≤ 21+2/s h1+1/smax E (hmax /2)(1+o(1)) ïðè hmax → 0.