Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space
The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the $L_2$-norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.
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| Date: | 2009 |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509132322242560 |
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| author | Bilichenko, R. O. Babenko, V. F. Биличенко, Р. О. Бабенко, В. Ф. Биличенко, Р. О. Бабенко, В. Ф. |
| author_facet | Bilichenko, R. O. Babenko, V. F. Биличенко, Р. О. Бабенко, В. Ф. Биличенко, Р. О. Бабенко, В. Ф. |
| author_sort | Bilichenko, R. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:45:28Z |
| description | The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the $L_2$-norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space. |
| first_indexed | 2026-03-24T02:36:14Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t; Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
R. O. Bylyçenko (Dnepropetr. nac. un-t)
UTOÇNENYE NERAVENSTVA
TYPA XARDY – LYTTLVUDA – POLYA
DLQ STEPENEJ SAMOSOPRQÛENNÁX
OPERATOROV V HYL|BERTOVOM PROSTRANSTVE
The well-known Taikov refinements of the Hardy – Littlewood – Polya inequality for L2 -norms of
intermediate derivatives of a function defined on the real line are extended to the case of powers of self-
adjoint operators in the Hilbert space.
Vidomi utoçnennq Tajkova nerivnosti Xardi – Littlvuda – Polia dlq L2 -norm promiΩnyx po-
xidnyx funkci], zadano] na dijsnij osi, uzahal\neno na vypadok stepeniv samosprqΩenyx operato-
riv u hil\bertovomu prostori.
Oboznaçym çerez Lr
2 2, ( )R , r ∈ N , prostranstvo vsex funkcyj x L∈ 2( )R , (r –
– 1)-q proyzvodnaq kotor¥x lokal\no absolgtno neprer¥vna y r-q proyzvodnaq
prynadleΩyt prostranstvu L2( )R . Dlq x Lr∈ 2 2, ( )R yzvestno neravenstvo
Xardy – Lyttlvuda – Polya (sm., naprymer, [1], [2], § 1.6):
x x xk k r r k r( ) / ( ) /
2 2
1
2
≤ −
, 0 < k < r,
yly πkvyvalentnoe emu neravenstvo
x
r k
r
h x
k
r
h xk k r k r( ) ( )
2 2 2
≤
−
+− −
, h > 0. (1)
V 1991 h. L. V. Tajkov [3] predloΩyl sledugwee utoçnenye neravenstva (1).
Esly r, k ∈ N , k < r (pryçem r > 2 pry k = 1) y h > 0, to dlq lgboj funkcyy
x Lr∈ 2 2, ( )R v¥polnqetsq neravenstvo
x
r k
r
h x
k
r
h xk k
h
r k r( ) ( )
2 2 2
1
2
≤
−
+− −∆π . (2)
V sluçae k = 1, r = 2 ymeet mesto neravenstvo
′ ≤ + ′′x
h
x h xh2 2 2
1 1 1
2π π∆ .
Zdes\ ∆hx t x t( ) : ( )= – x t h( )+ dlq x Lr∈ 2 2, ( )R y h > 0.
© V. F. BABENKO, R. O. BYLYÇENKO, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10 1299
1300 V. F. BABENKO, R. O. BYLYÇENKO
V 1992 h. L. V. Tajkov [4] rasprostranyl neravenstvo (2) na proyzvol\n¥j
porqdok n, n ∈ N , n < k < r, raznosty funkcyy x :
x
r k
r
h x
k
r
h xk k
n h
n r k r( ) ( )
2 2 2
1
2
≤
−
+− −∆π . (3)
Zdes\ ∆h
nx t( ) = ∆ ∆h h
n x t−( )1 ( ) dlq x Lr∈ 2 2, ( )R y h > 0.
Pust\ H — hyl\bertovo prostranstvo so skalqrn¥m proyzvedenyem (x, y) y
normoj x = ( , ) /x x 1 2
, A — lynejn¥j, neohranyçenn¥j, samosoprqΩenn¥j
operator v H, D A( ) — oblast\ eho opredelenyq.
Dlq k, r ∈ N , k < r, y dlq lgboho x D Ar∈ ( ) v¥polnqetsq neravenstvo
(sm., naprymer, [2], § 5.1)
A x x A xk k r r k r
≤ −1 / /
, (4)
predstavlqgwee soboj neravenstvo typa Xardy – Lyttlvuda – Polya dlq ste-
penej samosoprqΩennoho operatora, yly πkvyvalentnoe emu (dlq lgboho h > 0 )
neravenstvo
A x
r k
r
h x
k
r
h A xk k r k r≤
−
+− −
. (5)
V dannoj rabote m¥ poluçym estestvenn¥j analoh neravenstv (2) y (3) dlq
stepenej samosoprqΩennoho operatora A, utoçnqgwyj neravenstvo (5).
Pryvedem nekotor¥e svedenyq yz spektral\noj teoryy samosoprqΩenn¥x
operatorov, kotor¥e moΩno najty, naprymer, v [5] (§ 67, 75, 88).
RazloΩenyem edynyc¥ naz¥vaetsq odnoparametryçeskoe semejstvo proekty-
rugwyx operatorov Et , zadannoe v koneçnom yly beskoneçnom yntervale α[ ,
β] (esly otrezok α β,[ ] beskoneçen, to, po opredelenyg, prynymaetsq E−∞ =
= lim
t
tE
→ −∞
, E E
t
t∞ → ∞
= lim v sm¥sle syl\noj sxodymosty) y udovletvorqgwee
sledugwym uslovyqm:
a) E E E uu sv = ∀ , v ∈[ ]α β, , hde s u= { }min , v ,
b) v sm¥sle syl\noj sxodymosty
E Et t− =0 , α < t < β,
v) Eα = 0 , E Iβ = (I — toΩdestvenn¥j operator: I x x x H= ∀ ∈ ).
Polahaem Et = 0 pry t ≤ α y E It = pry t ≥ β.
Yz opredelenyq sleduet, çto dlq lgboho x H∈ funkcyq
σ( ) ( , )t E x xt= , – ∞ < t < ∞,
qvlqetsq neprer¥vnoj sleva, neub¥vagwej funkcyej ohranyçennoj varyacyy,
dlq kotoroj
σ α( ) = 0 , σ β( ) ( , )= x x .
Sohlasno spektral\noj teoreme kaΩdomu samosoprqΩennomu operatoru A
sootvetstvuet razloΩenye edynyc¥ Et , t ∈ R , takoe, çto vektor x prynadle-
Ωyt D A( ) tohda y tol\ko tohda, kohda
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
UTOÇNENYE NERAVENSTVA TYPA XARDY – LYTTLVUDA – POLYA… 1301
t d E x xt
2 ( , ) < ∞
−∞
∞
∫ ,
y esly x D A∈ ( ) , to
Ax tdE xt=
−∞
∞
∫ .
Pryvedenn¥j zdes\ yntehral — πto operatorn¥j yntehral Stylt\esa (sm., na-
prymer, [5], § 72). Pry πtom
Ax t d E x xt
2 2= < ∞
−∞
∞
∫ ( , ) .
Funkcyej ϕ( )A ot operatora A naz¥vaetsq operator, opredelqem¥j
formuloj
ϕ ϕ( ) ( )A x t dE xt=
−∞
∞
∫
na vsex tex vektorax x H∈ , dlq kotor¥x v¥polneno sootnoßenye
ϕ( ) ( , )t d E x xt
2
−∞
∞
∫ < ∞ .
Pry πtom
ϕ ϕ( ) ( ) ( , )A x t d E x xt
2 2=
−∞
∞
∫ .
V çastnosty, dlq x D Ak∈ ( ) , k ∈ N ,
A x t dE xk k
t=
−∞
∞
∫
y
A x t d E x xk k
t
2 2=
−∞
∞
∫ ( , ) .
Sledugwye svedenyq, kasagwyesq spektral\noj teoryy unytarn¥x operato-
rov, moΩno najty v [5] (§ 73).
Lynejn¥j operator U : H → H naz¥vaetsq unytarn¥m, esly:
a) ( , ) ( , )U x U y x y= ∀ x , y H∈ ,
b) oblast\ znaçenyj operatora U sovpadaet s H.
Pust\ dano semejstvo unytarn¥x operatorov Us , zavysqwyx ot odnoho pa-
rametra s, – ∞ < s < ∞, y udovletvorqgwee sledugwym uslovyqm:
1) U U Us t s t= + ,
2) U I0 = ,
3) ( , )U x yt — neprer¥vnaq funkcyq ot t pry lgb¥x x, y H∈ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1302 V. F. BABENKO, R. O. BYLYÇENKO
Yz uslovyj 1 y 2 sleduet, çto U Ut t
−
−=1
. A tak kak U Ut t
∗ −= 1
(sm., napry-
mer, [5], § 40), to U Ut t
∗
−= .
Oçevydno, çto rassmatryvaemoe semejstvo predstavlqet soboj neprer¥vnug
abelevu hruppu.
Lgbomu razloΩenyg edynyc¥ Es , – ∞ < s < ∞ , v hyl\bertovom prostranst-
ve sootvetstvuet hruppa unytarn¥x operatorov Et , – ∞ < t < ∞, tak çto dlq lg-
boho x H∈
U x e dE xt
ist
s=
−∞
∞
∫ . (6)
Y naoborot, kaΩdoj hruppe unytarn¥x operatorov Ut sootvetstvuet nekoto-
roe razloΩenye edynyc¥ Es , tak çto dlq lgboho x H∈ spravedlyvo ra-
venstvo (6).
Sledugwaq teorema qvlqetsq obobwenyem neravenstv (2) y (3).
Teorema. Pust\ A — samosoprqΩenn¥j, v obwem sluçae neohranyçenn¥j
operator v hyl\bertovom prostranstve H , Et — razloΩenye edynyc¥, pry-
nadleΩawee operatoru A, Us — hruppa unytarn¥x operatorov, kotoraq
sootvetstvuet razloΩenyg edynyc¥ Et . Pust\ takΩe n , k , r ∈ N , pryçem
n = k = 1 y r > 2 yly n < k < r. Tohda dlq lgboho h > 0 y lgboho x D Ar∈ ( )
v¥polnqetsq neravenstvo
A x
r k
r
h U I x
k
r
h A xk k
n h
n r k r≤
−
− +− −1
2
( )π . (7)
Esly operator A takov, çto dlq lgb¥x dejstvytel\n¥x u, z, – ∞ ≤ u <
< z ≤ ∞,
( )E E D Az u
r− ( ) ≠ { }2 θ , (8)
to dlq lgboho h > 0 neravenstvo (7) qvlqetsq toçn¥m v tom sm¥sle, çto
ono perestanet b¥t\ vern¥m dlq nekotoroho x D Ar∈ ( ) , esly xotq b¥ odnu
yz konstant v pravoj çasty (7) zamenyt\ men\ßej.
Dokazatel\stvo. Poskol\ku Et — razloΩenye edynyc¥, sootvetstvug-
wee operatoru A, to
A t dEt=
−∞
∞
∫ .
Rassmotrym funkcyg ot operatora ϕ( )A , hde
ϕ( )
,
/(
t
t
k
r
h t t t
r
k
k r k r r k
r k
=
− { } ≤
− −
−
sign
1 ))
/( )
,
.
1
11
h
t
r
k h
r k
0, ≥
−
(9)
V¥berem x D Ar∈ ( ) . Spravedlyva ocenka
A x A x A x A xk k≤ + −ϕ ϕ( ) ( ) . (10)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
UTOÇNENYE NERAVENSTVA TYPA XARDY – LYTTLVUDA – POLYA… 1303
Ocenym velyçynu vtoroho slahaemoho v pravoj çasty (10):
A x A x t dE x t dE x t tk k
t t
k− = − = −( )
−∞
∞
−∞
∞
∫∫ϕ ϕ ϕ( ) ( ) ( ) ddE xt
−∞
∞
∫ =
= t t d E x x
t t
t
k
t
k
−
=
−
−∞
∞
∫ ϕ
ϕ
( ) ( , )
( )
/
2
1 2 2
22
2
1 2
r
r
tt d E x x( , )
/
−∞
∞
∫
≤
≤ max
( )
( , )
/
t
k
r
r
t
t t
t
t d E x x
k−
=
−∞
∞
∫
ϕ 2
1 2
rr
h A xr k r− . (11)
Ravenstvo
max
( )
t
k
r
r kt t
t
k
r
h
−
= −ϕ
proverqetsq neposredstvenn¥m v¥çyslenyem.
Ocenyvaq slahaemoe ϕ( )A x , vospol\zuemsq spektral\n¥m razloΩenyem
funkcyy ot operatora ϕ( )A y ravenstvom ϕ( )t = 0 pry t
r
k h
r k
≥
−1 1/( )
.
Polahaq τ = τ (r, k, h) =
r
k h
r k
−1 1/( )
, ymeem
ϕ ϕ
τ
( ) ( )A x t dE x
t
t=
≤
∫ =
=
t
k
r
h t t
e
e
k r k r r k
i ht n
t
i ht
− { }
−
−
− −
≤
∫
sign
(1 )
(1π
τ
π ))n
tdE x ,
ϕ π
τ
( )A x
t
k
r
h t t
e
k r k r r k
i ht n
t
=
− { }
−
− −
≤
∫
sign
(1 )
1
2
−−
e d E x xi ht n
t
π 2
1 2
( , )
/
≤
≤ max
t
k r k r r k
i ht n
i h
t
k
r
h t t
e
e
≤
− −− { }
−
−
τ π
π
sign
(1 )
1 tt n
td E x x
2
1 2
( , )
/
−∞
∞
∫
=
= max ( )
0 < ≤
−−
−
−
t
k r k r
i ht n h
n
t
k
r
h t
e
U x x
τ π
π
1
.
Zdes\
U x e dE xh
i ht
tπ
π=
−∞
∞
∫ ,
ysxodq yz yntehral\noho predstavlenyq (6).
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
1304 V. F. BABENKO, R. O. BYLYÇENKO
Uçyt¥vaq ustanovlennoe v [4] dlq sluçaq n < k < r y v [3] dlq sluçaq n =
= k = 1 pry r > 2 neravenstvo
max
0
1
2
1
< ≤
−−
−
≤
−
t
k r k r
i ht n n k
t
k
r
h t
e
r k
r hτ π1
,
ymeem ocenku
ϕ π( ) ( )A x
r k
r
h U x xk n
h
n≤
−
−− −2 . (12)
Obæedynqq (11) y (12), poluçaem trebuemoe neravenstvo (7).
PredpoloΩym, çto neravenstvo (7) ne qvlqetsq toçn¥m, t.Ke. dlq nekotoroho
h > 0 y dlq nekotoroho δ > 0 pry vsex x D Ar∈ ( ) v¥polnqetsq neravenstvo
A x
r k
r
h U I x
k
r
h A xk k
n h
n r k r≤ −
−
− +− −( ) ( )1
1
2
δ π . (13)
V¥berem ε > 0 yz uslovyq
( )1 1
2
+ < +h
k
rε
δ
. (14)
Yspol\zuq (8), dlq dann¥x h y ε v¥berem πlement x D Ah
r
, ( )ε ∈ 2
:
x dE xh t
h
h
,
/
/
ε
ε
=
+
∫
1
1
.
Tohda
A x t dE xk
h
k
t h
h
h
, ,
/
/
ε ε
ε
=
+
∫
1
1
y
A x t d E x xk
h
k
t h h
h
h
, , ,
/
/
( , )ε ε ε
ε
=
+
∫ 2
1
1 11 2
1
/
,≥
h
x
k h ε . (15)
Analohyçno moΩno pokazat\, çto
A x
h
xr
h
r
h, ,ε εε≤ +
1
. (16)
Krome toho, ymeet mesto ocenka
( ) ( , ), , ,
/
/
U I x e d E x xh
n
h
it h n
t h h
h
h
π ε
π
ε ε− = −
+
1
2
1
1 εε
∫
1 2/
≤
≤ max
,
, ,
1 1
1 1 2
h h
it h n
h
i n
h
ne x e x x
+
− = − =
ε
π
ε
π
ε hh, ε . (17)
Podstavlqq (15) – (17) v neravenstvo (13), ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
UTOÇNENYE NERAVENSTVA TYPA XARDY – LYTTLVUDA – POLYA… 1305
1
1
1
2
2
1
h
x
r k
r
h x
k
r
h
hk h
k
n
n
h
r k
, ,( )ε εδ ε≤ −
−
⋅ + +
− −
r
hx , ε .
Uçyt¥vaq (14), poluçaem neravenstvo
r k≤ +
1
2
,
a πto protyvoreçyt tomu, çto k, r ∈ N , k < r. Poluçennoe protyvoreçye poka-
z¥vaet, çto koπffycyent pry pervom slahaemom v pravoj çasty neravenstva (7)
umen\ßyt\ nel\zq. Analohyçno moΩno pokazat\, çto nel\zq umen\ßyt\ koπf-
fycyent pry vtorom slahaemom v pravoj çasty. Sledovatel\no, neravenstvo (7)
qvlqetsq toçn¥m.
Teorema dokazana.
1. Xardy H. H., Lyttlvud D. E., Polya H. Neravenstva. – M.: Komknyha, 2006. – 456 s.
2. Babenko V. F., Kornejçuk N. P., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x
y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 590 s.
3. Tajkov L. V. Utoçnenye neravenstva Xardy, soderΩaweho ocenku velyçyn¥ promeΩutoçnoj
proyzvodnoj funkcyy // Mat. zametky. – 1991. – 50, # 4. – S. 114 – 122.
4. Tajkov L. V. O neravenstvax Xardy // Tam Ωe. – 1992. – 52, # 4. – S. 106 – 111.
5. Axyezer N. Y., Hlazman Y. M. Teoryq lynejn¥x operatorov v hyl\bertovom prostranstve. –
M.: Nauka, 1966. – 544 s.
Poluçeno 02.03.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 10
|
| id | umjimathkievua-article-3101 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:36:14Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/27/a69b8cc98481b4d4e696e58875d2af27.pdf |
| spelling | umjimathkievua-article-31012020-03-18T19:45:28Z Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space Уточнение неравенства типа Харди - Литтлвуда - Полиа для степеней самосопряженных операторов в гильбертовом пространстве Bilichenko, R. O. Babenko, V. F. Биличенко, Р. О. Бабенко, В. Ф. Биличенко, Р. О. Бабенко, В. Ф. The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the $L_2$-norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space. Відомі уточнення Тайкова нерівності Харді - Літтлвуда - Поліа для $L_2$-норм проміжних похідних функції, заданої на дійсній осі, узагальнено на випадок степенів самоспряжених операторів у гільбертовому просторі. Institute of Mathematics, NAS of Ukraine 2009-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3101 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 10 (2009); 1299-1305 Український математичний журнал; Том 61 № 10 (2009); 1299-1305 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3101/2950 https://umj.imath.kiev.ua/index.php/umj/article/view/3101/2951 Copyright (c) 2009 Bilichenko R. O.; Babenko V. F. |
| spellingShingle | Bilichenko, R. O. Babenko, V. F. Биличенко, Р. О. Бабенко, В. Ф. Биличенко, Р. О. Бабенко, В. Ф. Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title | Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title_alt | Уточнение неравенства типа Харди - Литтлвуда - Полиа для степеней самосопряженных операторов в гильбертовом пространстве |
| title_full | Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title_fullStr | Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title_full_unstemmed | Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title_short | Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space |
| title_sort | refinement of a hardy–littlewood–pólya-type inequality for powers of self-adjoint operators in a hilbert space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3101 |
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