• No results found

∫ ∫ ∫ ∫ L é èêÖÑëíÄÇàåéëíà çÖäéíéêõï äéçìëéÇÇ à ùäëíêÄèéãüñàà éèÖêÄíéêéÇ çÄ äéçìëÄï

N/A
N/A
Protected

Academic year: 2021

Share "∫ ∫ ∫ ∫ L é èêÖÑëíÄÇàåéëíà çÖäéíéêõï äéçìëéÇÇ à ùäëíêÄèéãüñàà éèÖêÄíéêéÇ çÄ äéçìëÄï"

Copied!
4
0
0

Loading.... (view fulltext now)

Full text

(1)

ÑéäãÄÑõ ÄäÄÑÖåàà çÄìä, 2006, ÚÓÏ 406, ‹ 4, Ò. 1–4 1 ïÓÓ¯Ó ËÁ‚ÂÒÚ̇ Óθ ÚÓ˜Ì˚ı ÓˆÂÌÓÍ Í·ÒÒË-˜ÂÒÍËı ÓÔÂ‡ÚÓÓ‚ ‚ „‡ÏÓÌ˘ÂÒÍÓÏ ‡Ì‡ÎËÁÂ Ë ÒÏÂÊÌ˚ı ӷ·ÒÚflı. Ç ÔÓÒΉÌ ‚ÂÏfl ËÒıÓ‰fl ËÁ ÌÓ‚˚ı Á‡‰‡˜ ‡Ì‡ÎËÁ‡ ‚ÂҸχ ÔÓÔÛÎflÌ˚ÏË ÒÚ‡ÎË ÓˆÂÌÍË ÓÔÂ‡ÚÓÓ‚ Ì ̇ ‚ÒÂÏ ÔÓÒÚ‡ÌÒÚ‚Â, ‡ ̇ ÌÂÍÓÚÓ˚ı ÍÓÌÛÒ‡ı ‚ ˝ÚËı ÔÓÒÚ‡ÌÒÚ‚‡ı (ÒÏ., ̇-ÔËÏÂ, [1–4]). äÓÏ ÚÓ„Ó, ‚ ÚÂÓËË ËÌÚ„‡Î¸-Ì˚ı ÓÔÂ‡ÚÓÓ‚ Ò ÔÓÎÓÊËÚÂθÌ˚ÏË fl‰‡ÏË ıÓÓ-¯Ó ËÁ‚ÂÒÚ̇ ÚÂÓÂχ ˝ÍÒÚ‡ÔÓÎflˆËË òÛ‡ (ÒÏ., ̇ÔËÏÂ, [5]), ÍÓÚÓ‡fl „Ó‚ÓËÚ, ˜ÚÓ ËÌÚ„‡Î¸-Ì˚È ÓÔÂ‡ÚÓ Kx(t) = (t, s)x(s)ds Ò k(t, s) ≥ 0 Ó„‡-Ì˘ÂÌ ‚ ÔÓÒÚ‡ÌÒÚ‚Â Lp ÚÓ„‰‡ Ë ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ˘ÂÒÚ‚ÛÂÚ ÔÓÎÓÊËÚÂθ̇fl, ÍÓ̘̇fl Ô.‚. ÙÛÌ͈Ëfl u(t), ˜ÚÓ ÓÔÂ‡ÚÓ Ó„‡Ì˘ÂÌ ‚ Ô‡‡ı K: → Ë K: → , „‰Â v = u1/p – 1. éÚÏÂÚËÏ, ˜ÚÓ ‚ Ò‚flÁË Ò ‡Á΢Ì˚ÏË Á‡‰‡˜‡ÏË ‡Ì‡ÎËÁ‡ ËÌÚÂ-ÂÒ Í ÚÂÓÂÏ‡Ï ˝ÍÒÚ‡ÔÓÎflˆËË Á̇˜ËÚÂθÌÓ ‚ÓÁ-ÓÒ [6–8]. èÓ˝ÚÓÏÛ Ë ‚ ÚÂÓÂχı ˝ÍÒÚ‡ÔÓÎflˆËË ÂÒÚÂÒÚ‚ÂÌÌ˚Ï fl‚ËÎÒfl ·˚ ÔÂÂıÓ‰ ÓÚ ÔÓÒÚ‡ÌÒÚ‚‡ ã·„‡ Lp Í ÍÓÌÛÒ‡Ï ‚ ÔÓÒÚ‡ÌÒÚ‚‡ı ã·„‡. Ç Ì‡ÒÚÓfl˘ÂÈ ‡·ÓÚ ‰Îfl ‚‡ÊÌÂȯËı ÍÓÌÛÒÓ‚ ‚ ÔÓÒÚ‡ÌÒÚ‚‡ı ã·„‡ Ô‰·„‡ÂÚÒfl ‰Û͈Ëfl Á‡-‰‡˜Ë ÓˆÂÌÍË ÓÔÂ‡ÚÓ‡ ̇ ÍÓÌÛÒÂ Í Á‡‰‡˜Â Ó· ÓˆÂÌÍ ÓÔÂ‡ÚÓ‡ ̇ ÌÓ‚ÓÏ ÔÓÒÚ‡ÌÒÚ‚Â, ÍÓÚÓ-Ó ÒÚÓËÚÒfl ÍÓÌÒÚÛÍÚË‚ÌÓ ÔÓ ÍÓÌÛÒÛ Ë ËÒıÓ‰ÌÓ-ÏÛ ÔÓÒÚ‡ÌÒÚ‚Û. í‡Í‡fl ‰Û͈Ëfl ÔÓÁ‚ÓÎflÂÚ ÔË-ÏÂÌËÚ¸ ‚Ò˛ ‡Á‡·ÓÚ‡ÌÌÛ˛ ÚÂıÌËÍÛ ÔÓÎÛ˜ÂÌËfl ÚÓ˜Ì˚ı ÓˆÂÌÓÍ Ì‡ ‚ÂÒÓ‚˚ı ÔÓÒÚ‡ÌÒÚ‚‡ı ã·„‡ Í ÔÓÎÛ˜ÂÌ˲ ÚÓ˜Ì˚ı ÓˆÂÌÓÍ ÓÔÂ‡ÚÓÓ‚ ̇ ÍÓÌÛ-Ò‡ı. àÒÔÓθÁÛfl ‰ÛÍˆË˛, Ï˚ Ú‡ÍÊ Ô‰ÎÓÊËÎË ‰Îfl ÌÂÍÓÚÓÓ„Ó Í·ÒÒ‡ ÓÔÂ‡ÚÓÓ‚ ÌÓ‚Û˛ ÚÂÓÂ-ÏÛ ˝ÍÒÚ‡ÔÓÎflˆËË ÓÔÂ‡ÚÓÓ‚, ÓÔ‰ÂÎÂÌÌ˚ı ̇ ÍÓÌÛÒ‡ı ‚ ÔÓÒÚ‡ÌÒÚ‚‡ı ã·„‡. èÛÒÚ¸ S(µ) = S(R+, Σ, µ) (R+ = (0, +∞) – ÔÓÒÚ‡Ì-ÒÚ‚Ó ËÁÏÂËÏ˚ı ÙÛÌ͈ËÈ x: R+ → R. ç‡ÔÓÏÌËÏ, ˜ÚÓ ·‡Ì‡ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó X = (X, ||·|X||), ÒÓÒÚÓfl-k

LuLuLv 1 Lv 1 ˘Â ËÁ ËÁÏÂËÏ˚ı ÙÛÌ͈ËÈ, ̇Á˚‚‡ÂÚÒfl ˉ‡θ-Ì˚Ï [9], ÂÒÎË ËÁ yX, ËÁÏÂËÏÓÒÚË x Ë ‚˚ÔÓÎÌÂ-ÌËfl Ô.‚. ̇ R+ ÌÂ‡‚ÂÌÒÚ‚‡ |x(t)|≤|y(t)| ÒΉÛÂÚ, ˜ÚÓ xX Ë ||x|X||≤||y|X||. ä‡Í Ó·˚˜ÌÓ, ÒËÏ‚ÓÎÓÏ Lp (1 ≤ ≤ p≤∞) Ó·ÓÁ̇˜‡ÂÚÒfl Í·ÒÒ˘ÂÒÍÓ ÔÓÒÚ‡ÌÒÚ‚Ó ã·„‡. èÛÒÚ¸ w: R+ → R+ – ÔÓÎÓÊËÚÂθ̇fl ÙÛÌ͈Ëfl (‚ÂÒ). ÖÒÎË X – ˉ‡θÌÓ ÔÓÒÚ‡ÌÒÚ‚Ó, ÚÓ ÒËÏ‚Ó-ÎÓÏ Xw Ó·ÓÁ̇˜‡ÂÚÒfl ÌÓ‚Ó ˉ‡θÌÓ ÔÓÒÚ‡ÌÒÚ-‚Ó, ÌÓχ ‚ ÍÓÚÓÓÏ Á‡‰‡ÂÚÒfl ‡‚ÂÌÒÚ‚ÓÏ ||x|Xw|| = = ||wx|X||. é Ô   ‰ Â Î Â Ì Ë Â 1. èÛÒÚ¸ X – ˉ‡θÌÓ ÔÓ-ÒÚ‡ÌÒÚ‚Ó ‚ S(µ), K – ÌÂÍÓÚÓ˚È ÍÓÌÛÒ ‚ S(µ). ëËÏ-‚ÓÎÓÏ KX Ó·ÓÁ̇˜‡ÂÚÒfl, Í‡Í Ó·˚˜ÌÓ, Ô Â   -Ò Â ˜ Â Ì Ë Â Í Ó Ì Û -Ò ‡ K Ò Í Ó Ì Û Ò Ó Ï X+. é·ÓÁ̇˜ËÏ ˜ÂÂÁ K(↓) ÍÓÌÛÒ ‚ S(µ), ÒÓÒÚÓfl˘ËÈ ËÁ ÙÛÌ͈ËÈ x: R+→ R+, ͇ʉ‡fl ËÁ ÍÓÚÓ˚ı Ì ‚ÓÁ-‡ÒÚ‡ÂÚ, Ú.Â. x(t + h) x(t) ‰Îfl h 0, ˜ÂÂÁ K(↑) Ó·Ó-Á̇˜ËÏ ÍÓÌÛÒ ‚ S(µ), ÒÓÒÚÓfl˘ËÈ ËÁ ÙÛÌ͈ËÈ, ͇Ê-‰‡fl ËÁ ÍÓÚÓ˚ı Ì ۷˚‚‡ÂÚ, ‡ ˜ÂÂÁ K(↓, ↑) Ó·ÓÁ̇-˜ËÏ ÍÓÌÛÒ ‚ S(µ), ÒÓÒÚÓfl˘ËÈ ËÁ ‚Ó„ÌÛÚ˚ı ÙÛÌ͈ËÈ x: R+→ R+, ͇ʉ‡fl ËÁ ÍÓÚÓ˚ı Û‰Ó‚ÎÂÚ‚ÓflÂÚ ‰Ó-ÔÓÎÌËÚÂθÌ˚Ï ÛÒÎÓ‚ËflÏ: (t) = 0, x(t) = 0. í Â Ó  Â Ï ‡ 1. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞) Ë Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w Ú‡-͇fl, ˜ÚÓ (1) (2) èÛÒÚ¸ ÓÔÂ‡ÚÓ Q ÓÔ‰ÂÎÂÌ ‡‚ÂÌÒÚ‚ÓÏ x t→0 lim t–1 t→0 lim wp( )s ds 1 ∞

= ∞; t ∀ 0 ‚˚ÔÓÎÌÂÌÓ ÌÂ‡‚ÂÌÒÚ‚Ó wp( )s ds 0 t

<∞. > Qx t( ) x( ) ττ d . t

=

åÄíÖåÄíàäÄ

é èêÖÑëíÄÇàåéëíà çÖäéíéêõï äéçìëéÇ

Ç

à ùäëíêÄèéãüñàà éèÖêÄíéêéÇ çÄ äéçìëÄï

© 2006 „. Ö. à. ÅÂÂÊÌÓÈ, ã. å‡ÎË„‡Ì‰‡

è‰ÒÚ‡‚ÎÂÌÓ ‡Í‡‰ÂÏËÍÓÏ ë.å. çËÍÓθÒÍËÏ 30.03.2005 „. èÓÒÚÛÔËÎÓ 23.09.2005 „.

L

vp ìÑä 513.88+517.5 üÓÒ·‚ÒÍËÈ „ÓÒÛ‰‡ÒÚ‚ÂÌÌ˚È ÛÌË‚ÂÒËÚÂÚ ËÏ. è.É. ÑÂÏˉӂ‡ íÂıÌ˘ÂÒÍËÈ ÛÌË‚ÂÒËÚÂÚ „. ãÛÎÂÓ, ò‚ˆËfl

(2)

2 ÑéäãÄÑõ ÄäÄÑÖåàà çÄìä      ÚÓÏ 406      ‹ 4    2006 ÅÂÂÊÌÓÈ, å‡ÎË„‡Ì‰‡ éÔ‰ÂÎËÏ ÌÓ‚Û˛ ÙÛÌÍˆË˛ v ËÁ ‡‚ÂÌÒÚ‚‡ (3) (˜ÂÂÁ κ(D) Ó·ÓÁ̇˜‡ÂÚÒfl ı‡‡ÍÚÂËÒÚ˘ÂÒ͇fl ÙÛÌ͈Ëfl ÏÌÓÊÂÒÚ‚‡ D.) íÓ„‰‡ ÒÔ‡‚‰ÎË‚˚ ÒÓÓÚÌÓ¯ÂÌËfl: ÓÔÂ‡ÚÓ Q ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡Ì˘ÂÌ ‚ Ô‡ (4) ÒÛ˘ÂÒÚ‚ÛÂÚ ÍÓÌÒÚ‡ÌÚ‡ c > 0 ڇ͇fl, ˜ÚÓ ∀y ∈ ∈ K(↓) ∪ Ò ||y| || = 1 ̇ȉÂÚÒfl ÙÛÌ͈Ëfl x ∈ Ò ||x| || = 1, ‰Îfl ÍÓÚÓÓÈ ÔË ‚ÒÂı t ∈ (0, ∞) ‚˚ÔÓÎÌÂÌÓ ÌÂ‡‚ÂÌÒÚ‚Ó (5) éÚÏÂÚËÏ, ˜ÚÓ ÛÒÎÓ‚Ë (4) ·Û‰ÂÚ ‚˚ÔÓÎÌÂÌÓ ‚ ÒË-ÎÛ Í·ÒÒ˘ÂÒÍËı ÓˆÂÌÓÍ ÓÔÂ‡ÚÓ‡ Q ‚ ÔÓÒÚ‡ÌÒÚ-‚‡ı (ËÏÂÌÌÓ Ú‡Í Ë ‚˚·Ë‡ÎÒfl ‚ÂÒ v ‚ (3)) (ÒÏ., ̇-ÔËÏÂ, [4, 10]). ÑÎfl ÙÛÌ͈ËË y K(↓) ∩ ÙÛÌÍ-ˆËfl x ‚ (5) ÒÚÓËÚÒfl ÍÓÌÒÚÛÍÚË‚ÌÓ. á ‡ Ï Â ˜ ‡ Ì Ë Â 1. íÂÓÂχ 1 ËÏÂÂÚ ÔÓÎÌ˚È ‡Ì‡ÎÓ„ ‰Îfl ÍÓÌÛÒÓ‚ K(), K(ϕ, ↓) = {x: R+ R+: ϕ(t)x(t)} Ë K(ϕ, ↑) = {x: R+→ R+: ϕ(t)x(t)↑}. í·Û-ÂÚÒfl ÚÓθÍÓ ‚ÏÂÒÚÓ ÓÔÂ‡ÚÓ‡ Q ‡ÒÒχÚË‚‡Ú¸ ÓÔÂ‡ÚÓ˚ èÓ‰ÂÏÓÌÒÚËÛÂÏ ÔËÏÂ˚ ÔËÏÂÌÂÌËfl ÚÂÓ-ÂÏ˚ 1. í Â Ó  Â Ï ‡ 2. èÛÒÚ¸ 1 p0 < p1 < ∞ Ë Á‡‰‡Ì˚ ‰‚ ‚ÂÒÓ‚˚ ÙÛÌ͈ËË u, w, Û‰Ó‚ÎÂÚ‚Ófl˛˘Ë ÛÒ-ÎÓ‚ËflÏ (1) Ë (2). íÓ„‰‡ ÒÔ‡‚‰ÎË‚Ó ÒÓÓÚÌÓ¯ÂÌË Ú.Â. ˝ÚË ÍÓÌÛÒ˚ Ì ÒÓ‚Ô‡‰‡˛Ú ÌË ÔË Í‡ÍËı ‚ÂÒÓ-‚˚ı ÙÛÌ͈Ëflı, Û‰Ó‚ÎÂÚ‚Ófl˛˘Ëı Ô‰ÔÓÎÓÊÂ-Ì˲ ÚÂÓÂÏ˚. ÅÛ‰ÂÏ „Ó‚ÓËÚ¸, ˜ÚÓ ÓÔÂ‡ÚÓ T: S(µ) → S(µ) ÒÛ·ÎËÌÂÈÌ˚È, ÂÒÎË ‚˚ÔÓÎÌfl˛ÚÒfl ÛÒÎÓ‚Ëfl: |T(x + + y)(t)|≤ T|x|(t) + T|y|(t) Ë |T(λx)(t)|≤λ|Tx(t)| (λ≥ 0). àÁ ÚÂÓÂÏ˚ 1 Ò‡ÁÛ Ê ÔÓÎÛ˜ËÏ ÒÔ‡‚‰ÎË-‚ÓÒÚ¸ ÒÎÂ‰Û˛˘Â„Ó Ù‡ÍÚ‡. í Â Ó  Â Ï ‡ 3. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞) Ë Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w, Û‰Ó‚-κ(0 t, )w Lp κ(t,∞)1 v ---- Lp' ⋅ ≡1 Q: Lv p ( )+ K( )↓ Lw p ; ∩ → Lwp Lp w Lw p Lv p Qx ( )( )tcy t( ). Lup Lp w Px t( ) x s( )ds, Qϕx t( ) 0 t

1 ϕ( )t --- x s( )ds, t

= = Pϕx t( ) 1 ϕ( )t --- x s( )ds. 0 t

= K( ) ∩ Lu↓ p0 K( ) ∩ ↑ Lup1 , ≠ ÎÂÚ‚Ófl˛˘‡fl ÛÒÎÓ‚ËflÏ (1), (2). èÛÒÚ¸ Y – ÌÂÍÓÚÓ-Ó ˉ‡θÌÓ ·‡Ì‡ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó ‚ S(µ). ÑÎfl ÚÓ„Ó ˜ÚÓ·˚ ÒÛ·ÎËÌÂÈÌ˚È ÓÔÂ‡ÚÓ T ‰ÂÈÒÚ‚Ó‚‡Î Ë ·˚Î Ó„‡Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ ËÁ K(↓) ∩ ‚ Y, ÌÂÓ·ıÓ‰ËÏÓ Ë ‰ÓÒÚ‡ÚÓ˜ÌÓ, ˜ÚÓ·˚ ÓÔÂ‡ÚÓ ÒÛÔÂÔÓÁˈËË TQ ‰ÂÈÒÚ‚Ó‚‡Î Ë ·˚Î Ó„-‡Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ ËÁ ‚ Y. àÒÔÓθÁÛfl ÚÂıÌËÍÛ ÓˆÂÌÓÍ ÓÔÂ‡ÚÓÓ‚ L: → → Y (ÒÏ., ̇ÔËÏÂ, [2, 4, 11, 12]), ËÁ ÚÂÓÂÏ˚ 3 ÏÓÊÌÓ ÔÓÎÛ˜ËÚ¸ ‡Á΢Ì˚ ÂÁÛθڇÚ˚, ÔÓÒ‚fl-˘ÂÌÌ˚ ӈÂÌÍ‡Ï ÓÔÂ‡ÚÓÓ‚ ̇ ÍÓÌÛÒ ÏÓÌÓÚÓÌ-Ì˚ı ÙÛÌ͈ËÈ ‚ ÔÓÒÚ‡ÌÒÚ‚‡ı ã·„‡. èÂÂȉÂÏ ÚÂÔÂ¸ Í ‡ÒÒÏÓÚÂÌ˲ ÍÓÌÛÒ‡ K(↓, ↑). ÑÎfl ÌÂÓÚˈ‡ÚÂθÌ˚ı ÙÛÌ͈ËÈ Ì‡ R+ ÓÔ-‰ÂÎËÏ ‰‚‡ ÓÔÂ‡ÚÓ‡ ë ÔÓÏÓ˘¸˛ ÔÓÒÚÓ„Ó ËÌÚ„ËÓ‚‡ÌËfl ÔÓ ˜‡ÒÚflÏ Î„ÍÓ ÔÓ͇Á‡Ú¸, ˜ÚÓ ÂÒÎË ÙÛÌ͈Ëfl x K(↓, ↑) ËÏÂ-ÂÚ ‡·ÒÓβÚÌÓ ÌÂÔÂ˚‚ÌÛ˛ ÔÂ‚Û˛ ÔÓËÁ‚Ó‰-ÌÛ˛, ÚÓ ÒÔ‡‚‰ÎË‚Ó Ô‰ÒÚ‡‚ÎÂÌË „‰Â z(s) – ÌÂÓÚˈ‡ÚÂθ̇fl ÙÛÌ͈Ëfl. åÓÊÌÓ ÔÓÎÓ-ÊËÚ¸ z(s) –x"(s). í Â Ó  Â Ï ‡ 4. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞) Ë Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w Ú‡-͇fl, ˜ÚÓ ∀t ∈ R+ ‚˚ÔÓÎÌÂÌ˚ ÛÒÎÓ‚Ëfl (6) ê‡ÒÒÏÓÚËÏ ÍÓÌÛÒ K(↓, ↑) ∩ . ÑÎfl ‚ÒÂı t > 0 ÓÔ‰ÂÎËÏ ‰‚ ÌÓ‚˚ ‚ÂÒÓ‚˚ ÙÛÌ͈ËË w0, w1 ËÁ ‡‚ÂÌÒÚ‚ (7) (8) Ë ÔÓÎÓÊËÏ Lw p Lv p Lwp Q1x t( ) t x s( )ds; P1x t( ) t

sx s( )ds. 0 t

= = x t( ) z( ) ττ d s

      s = d 0 t

= = t z s( )ds t

sz s( )ds 0 t

+ = Q1z t( )+P1z t( ), min 1 s t --,            p wp( )s ds 0 ∞

<∞. Lw p κ(t,∞) 1 w0( )s --- Lp' ⋅ κ(0 t, )sw s( ) Lp ≡1, κ(0 t, ) s w1( )s --- Lp' ⋅ κ(t,∞)w s( ) Lp ≡1 v( )t = max w{ 0( )t ,w1( )t }.

(3)

ÑéäãÄÑõ ÄäÄÑÖåàà çÄìä      ÚÓÏ 406      ‹ 4     2006 é èêÖÑëíÄÇàåéëíà çÖäéíéêõï äéçìëéÇ 3 íÓ„‰‡ ÒÔ‡‚‰ÎË‚˚ ÒÓÓÚÌÓ¯ÂÌËfl: ÒÛÏχ ÓÔÂ-‡ÚÓÓ‚ Q1 + P1 ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡Ì˘Â̇ ‚ Ô‡Â: (9) ‰Îfl ÚÓ„Ó ˜ÚÓ·˚ ÒÛ˘ÂÒÚ‚Ó‚‡Î‡ ÍÓÌÒÚ‡ÌÚ‡ c > 0 ڇ͇fl, ˜ÚÓ ∀y ∈ K(↓, ↑) ∩ Ò ||y| || = 1, ̇È-‰ÂÚÒfl ÙÛÌ͈Ëfl x Ò ||x| ||= 1, ‰Îfl ÍÓÚÓÓÈ ÔË ‚ÒÂı t ∈ (0, ∞) ‚˚ÔÓÎÌÂÌÓ ÌÂ‡‚ÂÌÒÚ‚Ó (10) ÌÂÓ·ıÓ‰ËÏÓ Ë ‰ÓÒÚ‡ÚÓ˜ÌÓ, ˜ÚÓ·˚ ‚˚ÔÓÎÌflÎÓÒ¸ ÛÒÎÓ‚Ë (11) ìÒÎÓ‚Ë (6) „‡‡ÌÚËÛÂÚ ÔË̇‰ÎÂÊÌÓÒÚ¸ Í‡È-ÌËı ÙÛÌ͈ËÈ min 1, ÍÓÌÛÒ‡ K(↓, ↑) ÔÓÒÚ‡ÌÒÚ‚Û . ìÒÎÓ‚Ë (7) ‰‡ÂÚ ÌÂÓ·ıÓ‰ËÏ˚Â Ë ‰ÓÒÚ‡ÚÓ˜Ì˚ ÛÒÎÓ‚Ëfl Ó„‡Ì˘ÂÌÌÓÒÚË ÓÔÂ‡ÚÓ‡ Q1 Í‡Í ÓÔÂ‡-ÚÓ‡ ËÁ ‚ , ‡ ÛÒÎÓ‚Ë (8) ‰‡ÂÚ ÌÂÓ·ıÓ‰ËÏ˚Â Ë ‰ÓÒÚ‡ÚÓ˜Ì˚ ÛÒÎÓ‚Ëfl Ó„‡Ì˘ÂÌÌÓÒÚË ÓÔÂ‡ÚÓ-‡ P1 Í‡Í ÓÔÂ‡ÚÓ‡ ËÁ ‚ . èÓ˝ÚÓÏÛ ÔË Ú‡-ÍÓÏ ‚˚·Ó ÙÛÌ͈ËË v ÛÒÎÓ‚Ë (9) ·Û‰ÂÚ ‚˚ÔÓÎ-ÌÂÌÓ ‚Ò„‰‡. ìÒÎÓ‚Ë (11) ‚ ÚÂÓÂÏ 4 ÂÒÚ¸ ÔÓÒÚÓ ‚ÓÁÏÓÊ-ÌÓÒÚ¸ ‚˚ÔÓÎÌÂÌËfl ÛÒÎÓ‚Ëfl (10) ‰Îfl ÒÂÏÂÈÒÚ‚‡ Í‡ÈÌËı ÙÛÌ͈ËÈ min 1, ÍÓÌÛÒ‡ K(↓, ↑). á‡ÏÂÚËÏ, ˜ÚÓ ‰Îfl ÙÛÌ͈ËË y K(↓, ↑) ∩ ÙÛÌ͈Ëfl x ‚ (10) ÒÚÓËÚÒfl ÍÓÌÒÚÛÍÚË‚ÌÓ. ëΉÛÂÚ ÓÚÏÂÚËÚ¸ Ú‡ÍÊÂ, ˜ÚÓ ÚÂÓÂχ 4 ËÏÂÂÚ ‡Ì‡ÎÓ„ ‰Îfl ÍÓÌÛÒÓ‚ K(ϕ, ψ) = {x: R+ R+: x(t) · ϕ↑ & ψ(t) · x(t)↓}. ìÒÎÓ‚Ë (11) ‚˚ÔÓÎÌflÂÚÒfl Ì ‚Ò„‰‡. åÓÊÌÓ ÔË‚ÂÒÚË ‡Á΢Ì˚ ‰ÓÒÚ‡ÚÓ˜Ì˚ ÛÒÎÓ‚Ëfl ‰Îfl ‚˚ÔÓÎÌÂÌËfl (11). Ç ˜‡ÒÚÌÓÒÚË, ‚ ÒÚÂÔÂÌÌÓÈ ¯Í‡ÎÂ, Ú.Â. ‰Îfl w(t) = tα, ÛÒÎÓ‚Ëfl (11) ·Û‰ÛÚ ‚˚ÔÓÎÌÂÌ˚ ÔË α∈ – 1, . ëÂȘ‡Ò Ï˚ ÔÓ‰ÂÏÓÌÒÚËÛÂÏ ÔËÏÂÌÂÌËfl ÚÂ-ÓÂÏ˚ 4. í Â Ó  Â Ï ‡ 5. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞) Ë Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w, Q1+P1 ( ): Lv p ( )+ K(↓ ↑, ) ∩ Lw p ; → Lwp Lwp Lv p Lv p Q1+P1 ( )x ( )( )tcy t( ), κ(0 t, ) s v( )s --- Lp' t κ(t,∞) s v( )s --- Lp' +     ×    t sup × κ(t,θ)s t --w s( ) Lp + κ(0 t, )w s( ) Lp        ∞. <    s t --   Lwp Lwp0 Lwp Lwp1 Lwp    s t --   Lw p 1 p ---–   1 p ---– Û‰Ó‚ÎÂÚ‚Ófl˛˘‡fl (6). èÛÒÚ¸ ÔÓ ÙÛÌ͈ËË Ë ÔÓ-ÒÚÓÂÌ˚ ÙÛÌ͈ËË w0, w1, v, ‰Îfl ÍÓÚÓ˚ı ‚˚ÔÓÎ-ÌÂÌÓ ÛÒÎÓ‚Ë (11). èÛÒÚ¸ Y – ÌÂÍÓÚÓÓ ˉ‡θ-ÌÓ ·‡Ì‡ıÓ‚Ó ÔÓÒÚ‡ÌÒÚ‚Ó ‚ S(µ). ÑÎfl ÚÓ„Ó ˜ÚÓ·˚ ÒÛ·ÎËÌÂÈÌ˚È ÓÔÂ‡ÚÓ T ‰ÂÈÒÚ‚Ó‚‡Î Ë ·˚Î Ó„‡Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ ËÁ K(↓, ↑) ∩ ‚ Y, ÌÂÓ·ıÓ‰ËÏÓ Ë ‰ÓÒÚ‡ÚÓ˜ÌÓ, ˜ÚÓ-·˚ ÓÔÂ‡ÚÓ ÒÛÔÂÔÓÁˈËË T(Q1 + P1) ‰ÂÈÒÚ‚Ó‚‡Î Ë ·˚Î Ó„‡Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ ËÁ ‚ Y. íÂÔÂ¸ Ï˚ ÔÂÂıÓ‰ËÏ Í ÚÂÓÂÏ‡Ï ˝ÍÒÚ‡ÔÓÎfl-ˆËË ‰Îfl ÓÔÂ‡ÚÓÓ‚, ‰ÂÈÒÚ‚Û˛˘Ëı ‚ ÍÓÌÛÒ‡ı. ÑÎfl ˝ÚÓ„Ó ‚‡Ï ÔÓÚÂ·Û˛ÚÒfl ÌÂÍÓÚÓ˚ ‰ÓÔÓÎÌËÚÂθ-Ì˚ ÔÓÒÚÓÂÌËfl. èÛÒÚ¸ X0, X1 – ‰‚‡ ˉ‡θÌ˚ı ÔÓÒÚ‡ÌÒÚ‚‡ X0, X1⊂ S(µ). á‡ÙËÍÒËÛÂÏ 0 < θ < 1. çÓ‚Ó ˉ‡θÌÓ ÔÓ-ÒÚ‡ÌÒÚ‚Ó (ÍÓÌÒÚÛ͈Ëfl ä‡Î¸‰ÂÓ̇–ãÓ-Á‡ÌÓ‚ÒÍÓ„Ó) ÒÓÒÚÓËÚ ËÁ ÚÂı x S(µ), ‰Îfl ÍÓÚÓ˚ı ÍÓ̘̇ ÌÓχ (12) èÓÒÚ‡ÌÒÚ‚Ó ‚‚‰ÂÌÓ Ä.è. ä‡Î¸‰ÂÓ-ÌÓÏ [13] ÔË ËÁÛ˜ÂÌËË ÍÓÏÔÎÂÍÒÌÓ„Ó ÏÂÚÓ‰‡ ËÌ-ÚÂÔÓÎflˆËË. ÖÒÎË K – ÌÂÍÓÚÓ˚È ÍÓÌÛÒ ‚ S(µ), ÚÓ ÔÓ ‡Ì‡ÎÓ-„ËË Ò ÔÓÒÚ‡ÌÒÚ‚ÓÏ ÏÓÊÌÓ ‚‚ÂÒÚË ÌÓ‚˚È ÍÓÌÛÒ (K ∩ X0)θ(K ∩ X1)1 – θ, ‡ÒÒχÚË‚‡fl ‡ÁÎÓÊÂ-ÌËfl ‚ (12) ÚÓθÍÓ ÔÓ ˝ÎÂÏÂÌÚ‡Ï ÍÓÌÛÒ‡. ëÎÂ‰Û˛-˘‡fl ÚÂÓÂχ ÌÓÒËÚ ËÌÚÂÔÓÎflˆËÓÌÌ˚È ı‡‡ÍÚÂ. ÑÎfl ÍÓÌÛÒ‡, ÒÓÒÚÓfl˘Â„Ó ËÁ ÌÂÓÚˈ‡ÚÂθÌ˚ı ÙÛÌ͈ËÈ, Ó̇ ıÓÓ¯Ó ËÁ‚ÂÒÚ̇ (ÒÏ., ̇ÔËÏÂ, [14, 15]). í Â Ó  Â Ï ‡ 6. èÛÒÚ¸ T – ÔÓÁËÚË‚Ì˚È ÓÔÂ‡-ÚÓ, K0, K1 – ‰‚‡ ÍÓÌÛÒ‡ ‚ S(µ)+. èÛÒÚ¸ ‚ S(µ) Á‡‰‡-ÌÓ ˜ÂÚ˚ ˉ‡θÌ˚ı ·‡Ì‡ıÓ‚˚ı ÔÓÒÚ‡ÌÒÚ‚‡ X0, X1, Y0, Y1. èÛÒÚ¸ ÓÔÂ‡ÚÓ T ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡-Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ T: Xi K0→ Yi K1 (i = 0, 1). èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ θ ∈ (0, 1). íÓ„‰‡ ÓÔÂ‡ÚÓ T ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡Ì˘ÂÌ Í‡Í ÓÔÂ‡ÚÓ T: (K0 ∩ X0)θ(K0 ∩ X1)1 – θ→ (K1 ∩ Y0)θ(K1 ∩ ∩ Y1)1 – θ. á ‡ Ï Â ˜ ‡ Ì Ë Â 2. ä‡Í Ó·˚˜ÌÓ ·˚‚‡ÂÚ ‚ ÚÂÓËË ËÌÚÂÔÓÎflˆËË, ‰Îfl ÔÓËÁ‚ÓθÌÓ„Ó ÍÓÌÛÒ‡ K ‡‚ÂÌ-ÒÚ‚Ó (K ∩ )θ(K ∩ )1 – θ = K(( )θ( )1 – θ) ÒÔ‡‚‰ÎË‚Ó ‰‡ÎÂÍÓ Ì ‚Ò„‰‡ ‰‡Ê ‰Îfl ÍÓÌÛÒ‡ K(↓). í Â Ó  Â Ï ‡ 7. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞) Ë Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w, Û‰Ó‚ÎÂÚ‚Ófl˛˘‡fl ÛÒÎÓ‚ËflÏ (1) Ë (2), ÔÓ ÍÓÚÓ-ÓÈ, Òӄ·ÒÌÓ (3), ÔÓÒÚÓÂ̇ ÙÛÌ͈Ëfl v. èÓÎÓ-Lwp Lv p X0 θ X1 1–θ x X0 θ X1 1–θ inf λ 0: x t( ) λ x0( )t θ x1( )t 1–θ ⋅ ≤ > { = t ∀ ∈Ω; x0 X0≤1, x1 X1≤1}. X0 θ X1 1–θ X0 θ X1 1–θ Lv0 1 Lv1Lv0 1 Lv1

(4)

4 ÑéäãÄÑõ ÄäÄÑÖåàà çÄìä      ÚÓÏ 406      ‹ 4    2006 ÅÂÂÊÌÓÈ, å‡ÎË„‡Ì‰‡ ÊËÏ θ = . èÛÒÚ¸ Á‡‰‡Ì ÎËÌÂÈÌ˚È ÔÓÁËÚË‚Ì˚È ÓÔÂ‡ÚÓ T, ‰ÂÈÒÚ‚Û˛˘ËÈ Ë Ó„‡Ì˘ÂÌÌ˚È ‚ Ô‡ íÓ„‰‡ ̇ȉÛÚÒfl ÙÛÌ͈ËË v0, v1, u0, u1 Ú‡ÍËÂ, ˜ÚÓ ‚˚ÔÓÎÌÂÌ˚ ÒÓÓÚÌÓ¯ÂÌËfl (13) ÓÔÂ‡ÚÓ TQ ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡Ì˘ÂÌ, ÂÒÎË Â„Ó ‡ÒÒχÚË‚‡Ú¸ ‚ Ô‡‡ı: é·˙‰ËÌÂÌË ÚÂÓÂÏ 6 Ë 7 ‰‡ÂÚ ‚‡ˇÌÚ ÚÂÓÂÏ˚ ˝ÍÒÚ‡ÔÓÎflˆËË ‰Îfl ÓÔÂ‡ÚÓÓ‚ ̇ ÍÓÌÛÒ K(↓). í Â Ó  Â Ï ‡ 8. èÛÒÚ¸ ÙËÍÒËÓ‚‡ÌÓ ˜ËÒÎÓ p ∈ ∈(1, ∞), Á‡ÙËÍÒËÓ‚‡Ì‡ ‚ÂÒÓ‚‡fl ÙÛÌ͈Ëfl w, Û‰Ó‚-ÎÂÚ‚Ófl˛˘‡fl ÛÒÎӂ˲ (6), ÔÓ ÍÓÚÓÓÈ ÔÓÒÚÓÂ-Ì˚ ÙÛÌ͈ËË w0, w1, v, ÔÛÒÚ¸ ‚˚ÔÓÎÌÂÌÓ ÛÒÎÓ‚Ë (11). èÓÎÓÊËÏ θ = . èÛÒÚ¸ Á‡‰‡Ì ÎËÌÂÈÌ˚È ÔÓ-ÁËÚË‚Ì˚È ÓÔÂ‡ÚÓ T, ‰ÂÈÒÚ‚Û˛˘ËÈ Ë Ó„‡ÌË-˜ÂÌÌ˚È ‚ Ô‡ íÓ„‰‡ ̇ȉÛÚÒfl ÙÛÌ͈ËË v0, v1, u0, u1 Ú‡ÍËÂ, ˜ÚÓ ‚˚ÔÓÎÌÂÌ˚ ÒÓÓÚÌÓ¯ÂÌËfl (13) Ë ÓÔÂ‡ÚÓ T(Q1 + P1) ‰ÂÈÒÚ‚ÛÂÚ Ë Ó„‡Ì˘ÂÌ, ÂÒÎË Â„Ó ‡ÒÒχÚË-‚‡Ú¸ ‚ Ô‡‡ı: é·˙‰ËÌÂÌË ÚÂÓÂÏ 6 Ë 7 ‰‡ÂÚ ‚‡ˇÌÚ ÚÂÓÂÏ˚ ˝ÍÒÚ‡ÔÓÎflˆËË ‰Îfl ÓÔÂ‡ÚÓÓ‚ ̇ ÍÓÌÛÒ K(↓, ↑). ꇷÓÚ‡ ·˚· ÔÓ‰‰Âʇ̇ „‡ÌÚÓÏ ò‚‰ÒÍÓÈ äÓÓ΂ÒÍÓÈ Äç ‰Îfl ÒÓÚÛ‰Ì˘ÂÒÚ‚‡ Ò êÓÒÒËÂÈ (ÔÓÂÍÚ 35160). èÂ‚˚È ‡‚ÚÓ ÔÓθÁÓ‚‡ÎÒfl ÔÓ‰-‰ÂÊÍÓÈ êÓÒÒËÈÒÍÓ„Ó ÙÓ̉‡ ÙÛ̉‡ÏÂÌڇθÌ˚ı ËÒÒΉӂ‡ÌËÈ (ÔÓÂÍÚ 05–01–00206). ëèàëéä ãàíÖêÄíìêõ

1. Sawyer E.T. // Stud. math. 1990. V. 96. P. 145–158. 2. ÅÂÂÊÌÓÈ Ö.à. // í. å‡Ú. ËÌ-Ú‡ êÄç. 1993. í. 204.

ë. 3–36.

3. Heinig H., Maligranda L. // Stud. math. 1995. V. 116. P. 133–165.

4. Kufner A., Persson L.-E. Weighted Inequalities of Hardy Type. Singapore: World Sci., 2003.

5. äÓÓÚÍÓ‚ Ç.Å. àÌÚ„‡Î¸Ì˚ ÓÔÂ‡ÚÓ˚. çÓ‚Ó-ÒË·ËÒÍ: ç‡Û͇, 1983.

6. Garcia-Cuerva J., Rubio de Francia J. Weighted Norm Inequalities and Related Topics. Amsterdam: North Hol-land, 1985. 7. ÅÂÂÊÌÓÈ Ö.à. // ÑÄç. 1995. í. 344. ‹ 6. ë. 727– 730. 8. ÅÂÂÊÌÓÈ Ö.à., å‡ÎË„‡Ì‰‡ ã. // ÑÄç. 2003. í. 393. ‹ 5. ë. 583–586. 9. äÂÈÌ ë.É., èÂÚÛÌËÌ û.à., ëÂÏÂÌÓ‚ Ö.å. àÌÚÂ-ÔÓÎflˆËfl ÎËÌÂÈÌ˚ı ÓÔÂ‡ÚÓÓ‚. å.: ç‡Û͇, 1978. 10. Maz’ja V.G. Sobolev Spaces. B.: Springer, 1985. 11. ÅÂÂÊÌÓÈ Ö.à. // í. 凯. ËÌ-Ú‡ Äç ëëëê. 1991.

í. 201. ë. 26–42.

12. Berezhnoi E.I. // Proc. AMS. 1999. T. 127. ‹ 1. P. 79–87. 13. ä‡Î¸‰ÂÓÌ Ä.è. // å‡ÚÂχÚË͇. 1965. í. 9. ‹ 3.

ë. 56–129.

14. ÅÂÂÊÌÓÈ Ö.à. Ç Ò·.: 䇘ÂÒÚ‚ÂÌÌ˚Â Ë ÔË·ÎËÊÂÌ-Ì˚ ÏÂÚÓ‰˚ ËÒÒΉӂ‡ÌËfl ÓÔÂ‡ÚÓÌ˚ı Û‡‚ÌÂ-ÌËÈ. üÓÒ·‚θ: üÓÒ·‚. „ÓÒ. ÛÌ-Ú. 1981. ë. 3–12. 15. Maligranda L. Orlicz Spaces and Interpolation.

Campi-nas, 1989. 1 p ---T : K( )↓ ∩LwpLwp. v0θ( )tv11–θ( )t v( )t , u0 θ t ( ) u1 1–θ t ( ) ⋅ u t( ); ≡ ≡ TQ: Lv0 1 Lu0 1 , TQ: Lv1Lu1 ∞ . → → 1 p ---T : K(↓ ↑, ) Lw pLu p . → T Q( 1+P1): Lv0 1 Lu10, T Q( 1+P1): Lv1Luv1. → →

References

Related documents

Orlicz Spaces and Interpolation // Semi- nars in Math. Classical Banach

The particle filter algorithm developed incorporates measurements from both a 2 DOF laser seam tracker and the robot forward kinematics under an assumed external force..

In this way, a self-consistent equation is established for the determination of the equilibrium density distribution, in which the beam frequency spectrum depends on the bunch shape

Kohei Watanabe attributes the persistence of these old patterns of inequality in the representation of the world in the age of the news aggregators to the continuous predominant

In Section 5.1 we define the Eisenstein series and in Section 6.4 we define an inner product on the space of modular forms, called the Petersson product, which is defined when one

In the second part of the chapter I will raise the question: ‘What kind of methodological steps do we as narrative researchers have to take when dealing with sensitive issues?’ Since

The presence of perianal disease is considered to be a risk factor with a relative risk of both clinical and surgical recurrence after surgery of 1,4 in the Stockholm

The evidence suggests that in the countries of Eastern Europe (post-socialist countries) have relatively high income-related health inequality levels, in particular