On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup
We study nonuniform ergodic averages of the Kozlov – Treshchev type for operator semigroups and obtain estimates for the corresponding maximal functions.
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| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508893942120448 |
|---|---|
| author | Korolev, A. V. Королев, А. В. Королев, А. В. |
| author_facet | Korolev, A. V. Королев, А. В. Королев, А. В. |
| author_sort | Korolev, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:51Z |
| description | We study nonuniform ergodic averages of the Kozlov – Treshchev type for operator semigroups and obtain estimates for the corresponding maximal functions. |
| first_indexed | 2026-03-24T02:32:27Z |
| format | Article |
| fulltext |
UDK 519.21
A. V. Korolev (Mosk. un-t ym. M. V. Lomonosova, Rossyq)
OB ∏RHODYÇESKOJ TEOREME V FORME
KOZLOVA – TREWEVA DLQ POLUHRUPPÁ OPERATOROV
*
We study nonuniform ergodic averagings of the Kozlov – Treshchev type for operator semigroups. We
obtain estimates for the corresponding maximal functions.
Vyvçagt\sq nerivnomirni erhodyçni userednennq typu Kozlova – Treweva dlq operatornyx piv-
hrup. Otrymano ocinky dlq vidpovidnyx maksymal\nyx funkcij.
V rabotax [1, 2] naçato rassmotrenye neravnomern¥x usrednenyj obweho vyda
F f x f T x dst t s( ) ( ) ( )=
∞
∫ ν
0
dlq polupotokov Tt{ } s ynvaryantnoj meroj µ . Dlq absolgtno neprer¥vn¥x
veroqtnostn¥x mer ν na poluosy y ohranyçenn¥x funkcyj f b¥lo dokazano,
çto pry t → + ∞ velyçyn¥ F f xt ( ) pry µ-poçty vsex x ymegt tot Ωe predel,
çto y v sluçae klassyçeskoho ravnomernoho usrednenyq v teoreme Byrkhofa –
Xynçyna. ∏to yssledovanye b¥lo prodolΩeno v rabote [3] (sm. takΩe [ 4 ],
hl.610), hde b¥lo v¥qsneno, çto dlq neohranyçenn¥x funkcyj f πto utverΩde-
nye terqet sylu, odnako pry nekotor¥x sootnoßenyqx meΩdu xarakteramy yn-
tehryruemosty f y plotnosty mer¥ ν ymegtsq poloΩytel\n¥e rezul\tat¥. V
dannoj rabote prodolΩaetsq yssledovanye srednyx F f xt ( ) . V çastnosty, oka-
z¥vaetsq, çto dlq sxodymosty ukazann¥x srednyx v Lp
na okruΩnosty s meroj
Lebeha moΩno otkazat\sq ot absolgtnoj neprer¥vnosty mer¥ ν. Krome toho,
nekotor¥e rezul\tat¥, poluçenn¥e v [3], obobwagtsq na operatorn¥j sluçaj,
kohda poluhruppa dejstvuet v L1( )µ .
Teorema 1. Pust\ ( , )X λ — proyzvedenye n okruΩnostej s normyrovan-
noj meroj Lebeha, a Ts s, ,…{ } — n-parametryçeskaq poluhruppa sdvyhov na X,
t.6e. Ts s, ,… x : = ( , , )T x T xs s n1 … , hde Ts — povorot i-j okruΩnosty na
uhol −si . Pust\ ν — proyzvol\naq veroqtnostnaq mera na R+
n
, ymegwaq
preobrazovanye Fur\e ν̂ . Ravenstvo
lim ( ) ( ) ( ) (
min ( , , )
, ,
t t
ts tsf T x d s f x d
… → +∞ …( ) −ν λ xx
X L p
)
( )
∫∫ =
R λ
0 (1)
v¥polneno dlq lgboj funkcyy f Lp∈ ( )λ pry p ∈ +∞[ ]1, v toçnosty tohda,
kohda pry t → +∞ v¥polneno sootnoßenye ˆ( )ν t → 0.
Dokazatel\stvo. PredpoloΩym, çto ˆ( )ν t → 0 pry t → +∞ . Polo-
Ωym Tts : = Tts ts, ,… pry t = ( , , )t tn1 … y s = ( , , )s sn1 … . Tohda dlq lgboj
funkcyy g C X∈ ( ) dlq kaΩdoho x X∈ v¥polneno ravenstvo
lim ( ) ( )
min ( , , )t t tsg T x ds
… → +∞
( )∫ ν
R
= g x dx
X
( ) ( )λ∫ .
*
V¥polnena pry podderΩke Rossyjskoho fonda fundamental\n¥x yssledovanyj (proekt¥ 07-
01-00536, 08-01-90431-Ukr.).
© A. V. KOROLEV, 2010
702 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
OB ∏RHODYÇESKOJ TEOREME V FORME KOZLOVA – TREWEVA … 703
Yz teorem¥ Lebeha sleduet, çto πtot predel suwestvuet y v Lp ( )λ . Pust\ te-
per\ f Lp∈ ( )λ . Voz\mem posledovatel\nost\ neprer¥vn¥x funkcyj gk na X
takyx, çto f gk L− ( )λ → 0 pry k → + ∞. Pust\ ε > 0. V¥berem takoe N > 0,
çto pry vsex k > N v¥polneno neravenstvo f gk L− ( )λ < ε. Najdetsq t0 0≥
takoe, çto ukazannaq v (1) norma v¥raΩenyq dlq gk vmesto f budet men\ße ε
pry vsex t = ( , , )t tn1 … , dlq kotor¥x t ti > 0 , i = 1, … , n. Tohda dlq takyx k,
t ymeem
f T x d s f x dx
ts
X L p
( ) ( ) ( ) ( )
( )
( ) − ∫∫ ν λ
λR
≤ ( ) ( ) ( )
( )
f g T x d sk ts
L p
− ( )∫ ν
λR
+
+ g T x d s g x dxk ts k
X L p
( ) ( ) ( ) ( )
( )
( ) − ∫∫ ν λ
λR
+ g x f x dxk
X L p
( ) ( ) ( )
( )
−( )∫ λ
λ
≤ 3ε,
tak kak
( ) ( ) ( )
( )
f g T x d sk ts
L p
− ( )∫ ν
λR
≤ ( ) ( ) ( ) ( )f g T x dx dsk ts
p
X
− ( )∫∫ λ ν
R
=
= f x g x dx dsk
p
X
( ) ( ) ( ) ( )−∫∫ λ ν
R
≤ ε.
PredpoloΩym teper\, çto ˆ( )ν t ne stremytsq k 0 pry t → +∞ . Tohda dlq
funkcyy f x xn( , , )1 … : = exp (ix1 + … + ixn ) ymeem
f T x d s f x dx
ts
X L p
( ) ( ) ( ) ( )
( )
( ) − ∫∫ ν λ
λR
= ˆ ( )ν t p
,
no poslednee v¥raΩenye ne stremytsq k 0 pry min ( , , )t tn1 … → + ∞.
Teorema dokazana.
Pust\ ( , )X λ — edynyçnaq okruΩnost\ s meroj Lebeha, Tt — povorot ok-
ruΩnosty na uhol −t . Yzvestno, çto suwestvuet takaq bezatomyçeskaq synhu-
lqrnaq borelevskaq veroqtnostnaq mera ν na 0 1,[ ] , çto ee preobrazovanye
Fur\e ν̂ stremytsq k nulg na beskoneçnosty (sm. [5, s. 35]). Tohda dlq lgboj
funkcyy f Lp∈ ( )λ srednye F f xt ( ) sxodqtsq v Lp ( )λ . Odnako ostaetsq ot-
kr¥t¥m vopros o suwestvovanyy synhulqrn¥x mer, dlq kotor¥x ymeetsq sxody-
most\ srednyx poçty vsgdu.
Rassmotrym teper\ bolee obwug sytuacyg, kohda Tt{ } — poluhruppa polo-
Ωytel\n¥x operatorov na L1( )µ , hde µ — veroqtnostnaq mera na yzmerymom
prostranstve ( , )X A . Sluçaj klassyçeskyx ravnomern¥x usrednenyj b¥l ras-
smotren v [6, 7]. Pust\ Tt{ } — syl\no yzmerymaq poluhruppa. V stat\e [7] po-
kazano, çto yz uslovyj Tt 1 ≤ 1 y Tt ∞ ≤ 1 sleduet neravenstvo
T f x s dss ( ) ( ) ( )β
0
∞
∫ ≤ f x s ds∗
∞
∫( ) ( )β
0
,
hde
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
704 A. V. KOROLEV
f x
t
T f x ds
t
s
t
∗ = ∫( ) : sup ( ) ( )
1
0
dlq lgboj funkcyy f Lp∈ ( )µ y lgboj poloΩytel\noj y nevozrastagwej
funkcyy β na R+ .
Zameçanye 1. Pust\ ν = ρ ds — veroqtnostnaq mera na 0, +∞[ ) , pryçem
suwestvuet takaq nevozrastagwaq funkcyq β ∈ L1( , )R+ λ , çto ρ ≤ β na
0, +∞[ ) . Tohda
( ) ( ) ( )T f x s dsts ρ
0
∞
∫ ≤ f x s ds∗
∞
∫( ) ( )β
0
.
Esly p > 1, to suwestvuet postoqnnaq C, ne zavysqwaq ot f, takaq, çto
sup ( ) ( ) ( )
( )
t
s
L
T f x s ds
p
β
µ0
∞
∫ ≤ C f
L p( )µ .
Çtob¥ proveryt\ pervoe neravenstvo, dostatoçno v¥polnyt\ zamenu u : = t s.
Vtoroe neravenstvo neposredstvenno sleduet yz teorem¥ ob ocenke norm¥ f ∗
(sm. [6, s. 735], teorema 7).
Sledugwee utverΩdenye rasprostranqet ocenku, poluçennug v [6, s. 735]
(teorema 7), na sluçaj usrednenyj s plotnost\g.
Teorema 2. Pust\ Tt — syl\no yzmerymaq poluhruppa poloΩytel\n¥x ope-
ratorov na L1( )µ , pryçem Tt 1 ≤ 1 y Tt ∞ ≤ 1. Pust\ ρ λ∈ Lq ( ) , hde λ
— mera Lebeha na R+ , ρ ≥ 0 y suwestvuet nevozrastagwaq funkcyq β ∈
∈ L1( , )R+ λ takaq, çto dlq nekotoroho t0 0≥ ymeem ρ( )s ≤ β( )s pry s ∈
∈ t0, +∞[ ) . Tohda dlq lgboj funkcyy f Lp∈ ( )µ , p−1 + q−1 < 1, p, q ∈ 1, +∞( ] ,
v¥polneno neravenstvo
sup ( ) ( )
( )
t
ts
L
T f x s ds
p
( )
+∞
∫ ρ
µ0
≤ C p t fq q
L t L Lq p( ) ( )
, , ( ) ( )0
1
0
−
( ) +
ρ βλ λ µ ,
hde C p( ) : = 2
1
1
p
p
p
−
/
.
Dokazatel\stvo. Sluçaj p = + ∞ sleduet yz neravenstva T ∞ ≤ 1.
Pust\ p < + ∞. Najdetsq ε > 0 takoe, çto ( )p − −ε 1 + q−1 = 1. PoloΩym σ : =
: = ρI t0,[ ] y τ : = ρI t , +∞[ ) . Tohda v¥polneno neravenstvo
sup ( ) ( )
( )
t
ts
L
T f x s ds
p
( )
+∞
∫ ρ
µ0
≤ sup ( ) ( )
( )
t
ts
L
T f x s ds
p
( )
+∞
∫ σ
µ0
+
+ sup ( ) ( )
( )
t
ts
L
T f x s ds
p
( )
+∞
∫ τ
µ0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
OB ∏RHODYÇESKOJ TEOREME V FORME KOZLOVA – TREWEVA … 705
Snaçala ocenym pervoe slahaemoe. Po uslovyg Tt 1 ≤ 1 y Tt ∞ ≤ 1, otkuda
pry kaΩdom t ymeem
T f x T f xt
p
t
p( ) ( )− −≤ε ε
dlq µ-poçty vsex x . Yz teorem¥ Fubyny sleduet, çto dlq µ-poçty vsex x su-
westvugt takye mnoΩestva Bx ⊂ +R , çto λ( \ )R+ Bx = 0 y pry t Bx∈ v¥pol-
neno ukazannoe neravenstvo. Poluçaem
T f x s dsts ( ) ( )σ
0
+∞
∫ ≤ σ λ
ε
ε
L ts
p
t p
q T f x ds
( )
( )
( )
−
−
∫
−
0
1
≤
≤ σ λ
ε
ε
L s
p
t p
q
t
T f x ds
( )
( )
( )
1
0
1
−
−
∫
−
≤
≤ t
t t
T f x dsq q
L s
p
t p
q0
1
0 0
1( )
( )
(
( )− −
−
∫
σ λ
ε
εε)−1
dlq µ-poçty vsex x X∈ , pryçem suwestvovanye yntehralov v prav¥x çastqx
neravenstv sleduet yz πrhodyçeskoj teorem¥, prymenennoj k poslednemu ynteh-
ralu. Tohda
sup ( ) ( )
t
tsT f x s dsσ
0
+∞
∫ ≤ t
t
T f x dsq q
L
t
s
p
t
q0
1
0
1( )
( )
(
sup ( )− −∫
σ λ
ε
pp − ε)
dlq µ-poçty vsex x X∈ . Poskol\ku f Lp∈ ( )µ , to f p − ε ∈ Lp p/( )( )− ε µ y,
sohlasno teoreme 7 [6, s. 735], v¥polneno neravenstvo
sup ( )
( )
( )
( )
t
s
p
t p
L
t
T f x ds
p
1
0
1
−
−
∫
−
ε
ε
µ
≤ C p f
p
L
p
p p
( )
( )
( )
( )
/ ( )
− −
−
−
ε
µ
ε
ε
1
= C p f
Lp( )
( )µ .
Prymenqq poslednee neravenstvo k ocenkam, poluçenn¥m v¥ße, okonçatel\no
poluçaem
sup ( ) ( )
( )
t
ts
L
T f x s ds
p
σ
µ0
+∞
∫ ≤
≤ t
t
T f x dsq q
L
t
s
p
t
q0
1
0
1( )
( )
(
sup ( )− −∫
σ λ
ε
pp
L p
− −ε
µ
)
( )
1
≤
≤ t C p fq q
L Lq p0
1( )
( ) ( )
( )− σ λ µ .
Teper\ ocenym yntehral s plotnost\g τ. MoΩno sçytat\, çto β ≥ 0. Ymeem
T f x s dsts ( ) ( )τ
0
+∞
∫ ≤ T f x s dsts ( ) ( )β
0
+∞
∫ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
706 A. V. KOROLEV
≤ β µL
t
s
t
t
T f x ds1
1
0
( )
sup ( )∫
dlq µ-poçty vsex x X∈ , çto sleduet yz zameçanyq 1. Snova prymenqq teoremu
yz [6, s. 735], naxodym
sup ( ) ( )
( )
t
ts
L
T f x s ds
p
τ
µ0
+∞
∫ ≤ C p f
L L p( )
( ) ( )
β λ µ1 .
Okonçatel\no
sup ( ) ( )
( )
t
ts
L
T f x s ds
p
( )
+∞
∫ ρ
µ0
≤ C p f t
L
q q
L t Lp q( )
( )
( )
, , ( )µ λ λρ β0
1
0 1
−
( ) +
.
Teorema dokazana.
Rassmotrym sxodymost\ srednyx Ft dlq operatornoho sluçaq.
Teorema 3. Pust\ Tt — syl\no yzmerymaq poluhruppa poloΩytel\n¥x ope-
ratorov na L1( )µ , pryçem T 1 ≤ 1 y T ∞ ≤ 1. Pust\ f Lp∈ ( )µ y ρ ∈
∈ Lq ( )λ — veroqtnostnaq plotnost\, hde λ — mera Lebeha na R+ , p−1 +
+ q−1 = 1 y p, q ∈ +∞[ ]1, . PredpoloΩym, çto v¥polneno odno yz uslovyj:
i) plotnost\ ρ ymeet ohranyçenn¥j nosytel\ v otrezke a b,[ ] ;
ii) p > 1 y suwestvuet nevozrastagwaq funkcyq β na 0, +∞[ ) , dlq ko-
toroj β ≥ 0, β ∈6 Lq 0, +∞[ ) y ρ( )t ≤ β( )t na t0, ∞[ ) dlq nekotoroho t0 .
Tohda dlq µ-poçty vsex x X∈ v¥polneno ravenstvo
lim ( ) ( )
t
tsT f x s ds
→ +∞
+∞
∫ ρ
0
= ET f x( ) , (2)
hde ET f — uslovnoe matematyçeskoe oΩydanye f otnosytel\no T — σ -
alhebr¥ Tt -ynvaryantn¥x mnoΩestv.
Dokazatel\stvo. MoΩno sçytat\, çto f ≥ 0. Pust\ nosytel\ ρ leΩyt v
a b,[ ] . PoloΩym fN = min ( , )f N , gN = f – fN . Dlq ohranyçenn¥x funkcyj
fN dokaz¥vaemoe utverΩdenye verno, poπtomu dlq µ-poçty vsex x pry vsex N
spravedlyvo ravenstvo (2) dlq fN vmesto f. Pust\ t ≥ 0 y h st x N, , ( ) =
= T g xts N ( )( ) . Po neravenstvu Hel\dera
T g x s dsts N
a
b
( ) ( )( )∫ ρ ≤ ht x N L Lp q, , ( ) ( )λ λρ .
V sylu µ-yntehryruemosty funkcyy x � T g xts N
p( )( ) yz πrhodyçeskoj teore-
m¥ sleduet, çto
lim ( )
t
ts N
p
a
b
T g x ds
→ ∞ ( )∫ = ( ) ( )b a g xN
p− ET
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
OB ∏RHODYÇESKOJ TEOREME V FORME KOZLOVA – TREWEVA … 707
dlq µ-poçty vsex x . Pust\ ε > 0 fyksyrovano. Poskol\ku µ-poçty vsgdu v¥-
polneno ravenstvo lim ( )
N
N
pg x
→ ∞
ET = 0, suwestvugt N0 y E ∈A s µ( )E >
> 1 – ε takye, çto pry N > N0 spravedlyva ocenka
( ) ( )b a g xN
p− ET < ε, x E∈ .
Dlq µ-poçty lgboho x E∈ suwestvuet takoe çyslo T x( , )ε , çto pry vsex t ≥
≥ T x( , )ε y N > N0 v¥polnen¥ neravenstva
T g x dsts N
p
a
b
( )( ) ≤∫ ε ,
T f x s ds f xts N
a
b
N( ) ( ) ( )( ) − <∫ ρ εET
.
Yz neravenstv Tt 1 ≤ 1 y Tt ∞ ≤ 1 sleduet, çto dlq µ-poçty vsex x X∈
pry vsex t ≥ 0 ymeet mesto neravenstvo
T f x dsts
p
a
b
( )∫ ≤ T f x dsts
p
a
b
( )∫ .
Tohda dlq µ-poçty vsex x E∈ pry t ≥ T x( , )ε y N > N0 ymeem
T g x s dsts N
a
b
( ) ( )( )∫ ρ ≤ ε ρ λLq ( )
.
Okonçatel\no poluçaem
T f x s ds f xts
a
b
( ) ( ) ( )( ) −∫ ρ ET < 2ε ε ρ λ+
Lq ( )
.
Teper\ dostatoçno rassmotret\ sluçaj, kohda ρ — ohranyçennaq funkcyq y
p > 1. MoΩno sçytat\, çto t0 0= . Ostalos\ prymenyt\ pred¥duwug teoremu
y vospol\zovat\sq tem, çto dokaz¥vaemoe ravenstvo v¥polneno dlq vsex ohrany-
çenn¥x yntehryruem¥x funkcyj f.
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6. Danford N., Ívarc DΩ. Lynejn¥e operator¥. I. Obwaq teoryq. – M.: Yzd-vo ynostr. lyt.,
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Poluçeno 30.11.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
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| id | umjimathkievua-article-2901 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:32:27Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/88/8d59a1029bd1a11f4219ff7543b63388.pdf |
| spelling | umjimathkievua-article-29012020-03-18T19:39:51Z On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup Об эргодической теореме в форме Козлова - Трещева для полугруппы операторов Korolev, A. V. Королев, А. В. Королев, А. В. We study nonuniform ergodic averages of the Kozlov – Treshchev type for operator semigroups and obtain estimates for the corresponding maximal functions. Вивчаються нерівномірні ергодичні усереднення типу Козлова - Трещева для операторних пів-груп. Отримано оцінки для відповідних максимальних функцій. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2901 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 702–707 Український математичний журнал; Том 62 № 5 (2010); 702–707 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2901/2553 https://umj.imath.kiev.ua/index.php/umj/article/view/2901/2554 Copyright (c) 2010 Korolev A. V. |
| spellingShingle | Korolev, A. V. Королев, А. В. Королев, А. В. On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title | On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title_alt | Об эргодической теореме в форме Козлова - Трещева для полугруппы операторов |
| title_full | On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title_fullStr | On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title_full_unstemmed | On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title_short | On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup |
| title_sort | on the ergodic theorem in the kozlov–treshchev form for an operator semigroup |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2901 |
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