On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$
We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class $A_1$.
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2010
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| author | Leonchik, E. Yu. Леончик, Е. Ю. Леончик, Е. Ю. |
| author_facet | Leonchik, E. Yu. Леончик, Е. Ю. Леончик, Е. Ю. |
| author_sort | Leonchik, E. Yu. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:40:46Z |
| description | We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class $A_1$. |
| first_indexed | 2026-03-24T02:33:16Z |
| format | Article |
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UDK 517.5
E. G. Leonçyk (Odes. nac. un-t ym. Y. Y. Meçnykova)
OB OCENKE PERESTANOVKY FUNKCYY
YZ KLASSA MAKENXAUPTA A1
An exact estimate is obtained for a nonincreasing equimeasurable rearrangement of a function of two
variables from the Muckenhoupt class A1 .
OderΩano toçnu ocinku nezrostagço] rivnomirno] perestanovky funkci] dvox zminnyx iz klasu
Makenxaupta A1 .
Pust\ poloΩytel\naq summyruemaq funkcyq dvux peremenn¥x f otdelena ot
nulq na kvadrate Q0 , t.,e. ess inf ( )x Q f x∈ 0
> 0. Zdes\ y v dal\nejßem rassmat-
ryvagtsq tol\ko takye kvadrat¥, storon¥ kotor¥x parallel\n¥ koordynatn¥m
osqm. Hovorqt, çto f udovletvorqet uslovyg Makenxaupta A1 pry nekotorom
C > 1 f A C∈( )1( ) , esly dlq vsex podkvadratov Q Q⊂ 0 v¥polneno neravenstvo
[1]
f
Q
f y dyQ
Q
≡ ∫
1
( ) ≤ C f x
x Q
ess inf ( )
∈
, (1)
hde çerez ⋅ oboznaçena mera Lebeha. Dannoe uslovye na ves, v çastnosty,
qvlqetsq neobxodym¥m y dostatoçn¥m dlq prynadleΩnosty rqda operatorov
slabomu vesovomu prostranstvu L1 (sm., naprymer, [1, 2]).
Narqdu s klassyçeskym uslovyem Makenxaupta rassmatryvaetsq y eho mody-
fykacyq, kohda v¥polnenye neravenstva (1) trebuetsq na vsex prqmouhol\nykax
yz Q0 , a ne tol\ko na kvadratax (sm., naprymer, [3, s. 145; 4]). V πtom sluçae
budem pysat\ f A∈ ′1 . Qsno, çto ′ ⊂A C A C1 1( ) ( ) .
Nevozrastagwaq perestanovka f ∗
, ravnoyzmerymaq s funkcyej f, moΩet
b¥t\ opredelena ravenstvom [5, s. 332]
x Q f x∈ >{ }0 : ( ) λ = t Q f t∈[ ] >{ }∗0 0; : ( ) λ , λ > 0 .
Dlq yzuçenyq svojstv nekotor¥x klassov funkcyj bol\ßoe znaçenye ymegt
ocenky perestanovok funkcyj yz πtyx klassov (sm., naprymer, [4, 6 – 8]).
Yzvestno, çto dlq ravnoyzmerymoj perestanovky funkcyy f A C∈ 1( ) takΩe
v¥polneno uslovye Makenxaupta, no s nekotoroj, voobwe hovorq, druhoj po-
stoqnnoj. Yspol\zuq lemmu Kal\derona – Zyhmunda o pokr¥tyy [9] y metod do-
kazatel\stva yz [4], moΩno harantyrovat\, çto f ∗ ∈ A C1 4( ) . Znaçenye po-
stoqnnoj 4 obuslovleno prymenenyem πtoj lemm¥. V rabote [6] v odnomernom
sluçae ustanovleno, çto uslovye f ∈ A C1( ) vleçet f ∗ ∈ A C1( ) . Zametym, çto
dlq klassov ′A1 analohyçn¥j rezul\tat ymeet mesto y v mnohomernom sluçae
[4]. Dlq funkcyj odnoj peremennoj pry dokazatel\stve yspol\zovalas\ mody-
fykacyq lemm¥ Kal\derona – Zyhmunda, tak naz¥vaemaq lemma Ryssa „o vo-
sxodqwem solnce”, a dlq sluçaq funkcyj neskol\kyx peremenn¥x prymenqlsq
ee varyant po mnohomern¥m sehmentam [10].
S druhoj storon¥, v [6] postroen prymer, pokaz¥vagwyj, çto dlq lgboho
B C< 2 najdetsq takaq funkcyq dvux peremenn¥x f0 ∈ A C1( ) , dlq kotoroj
© E. G. LEONÇYK, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8 1145
1146 E. G. LEONÇYK
f A B0 1
∗ ∉ ( ) . Osnovn¥m rezul\tatom dannoj rabot¥ qvlqetsq sledugwee ut-
verΩdenye.
Teorema. Pust\ funkcyq dvux peremenn¥x f prynadleΩyt A C1( ) . Tohda
f ∗
prynadleΩyt A C1 2( ) .
Dokazatel\stvo. Zafyksyruem λ ≥ ess inf ( )x Q f x∈ 0
y oboznaçym Eλ ≡
≡ x Q f x∈ >{ }0 : ( ) λ . Teoremu lehko poluçyt\ yz neravenstva
α ≡
1
2
fEλ ≤ Cλ . (2)
Esly α ≤ fQ0
, to πto neravenstvo neposredstvenno sleduet yz uslovyq f ∈
∈ A C1( ) .
Dlq dokazatel\stva (2) pry α > fQ0
zametym, çto v πtom sluçae Eλ <
<
1
2
0Q . Razob\em kvadrat Q0 na çet¥re podkvadrata Qk k
1
1
4{ } =
, razdelyv
kaΩdug storonu na dve ravn¥e çasty. VozmoΩn¥ dva varyanta:
i) fQk
1 2≥ α pry nekotorom k (vozmoΩno, ne edynstvennom);
ii) fQk
1 2< α dlq vsex k .
Pust\ v sluçae i) dlq opredelennosty k = 1. Tohda
f Q2 1
1 ≥
1
2 1
1
1
1Q
f x dx
Q
( )∫ =
1
2 1
1fQ ≥ α ,
hde çerez 2 1
1Q oboznaçen takoj kvadrat, çto Q1
1 ⊂ 2 1
1Q ⊂ Q0 y
2 1
1Q = 2 1
1Q . Poskol\ku 2 1
1Q =
1
2
0Q > Eλ , to 2 1
1Q E\ λ > 0 y,
znaçyt, ess inf ( )x Q f x∈ 2 1
1 ≤ λ. Otsgda
α ≤ f C f xQ x Q2 21
1
1
1≤ ∈ess inf ( ) ≤ C λ. (3)
V sluçae ii) prymenym tak naz¥vaemug texnyku momentov ostanovok (sm., na-
prymer, [11, s. 232]). Esly fQk
1 ≥ α, to takoj kvadrat pomestym v sovokupnost\,
kotorug oboznaçym Si{ } . V sluçae fQk
1 < α snova rassmotrym dva varyanta:
a) esly E Qkλ ∩ 1 ≥
1
2
1Qk , to otklad¥vaem takoj kvadrat v sovokupnost\
Ri{ } y dlq neho naxodym
1
1
1E Q
f x dx
k E Qk
λ
λ
∩ ∩
( )∫ ≤
1
1
1E Q
f x dx
k Qk
λ ∩
( )∫ =
=
Q
E Q
f
k
k
Qk
1
1 1
λ ∩
< 2α; (4)
b) esly E Qkλ ∩ 1 <
1
2
1Qk , to kvadrat Qk
1
delym na çet¥re podkvadrata
Qk k
2
1
4{ } =
podobno tomu, kak b¥l podelen Q0 ; dalee povtorqem rassmotrennug
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
OB OCENKE PERESTANOVKY FUNKCYY … 1147
v¥ße sxemu s zafyksyrovann¥m ranee α =
1
2
fEλ , zamenqq kvadrat Q0 na Qk
1
,
a mnoΩestvo Eλ na E Qkλ ∩ 1
.
V sluçae i), kohda fQk
2 2≥ α pry nekotorom k, ymeem
f Qk2 2 ≥
1
2 2
2Q
f x dx
k Qk
( )∫ =
1
2
2fQk
≥ α .
Krome toho, 2 2Qk =
1
2
1Qk > E Qkλ ∩ 1
, y, sledovatel\no,
ess inf ( )x Qk
f x∈ 2 2 ≤ λ. Otsgda analohyçno (3) poluçaem (2).
V sluçae ii), esly fQk
2 2< α dlq vsex k, to lybo pomewaem sootvetstvug-
wyj kvadrat v sovokupnost\ Si{ } yly Ri{ } , lybo snova razbyvaem eho na çet¥-
re podkvadrata.
ProdolΩaq πtot process delenyq y otklad¥vanyq kvadratov, yly na kakom-
lybo ßahe pryxodym k sluçag i) y tohda ymeem (2), yly poluçaem dve sovokup-
nosty kvadratov Si{ } y Ri{ } s takymy svojstvamy:
1) vnutrennosty kvadratov yz Si{ } y Ri{ } poparno ne peresekagtsq, tak
kak pomewenn¥j v sovokupnost\ kvadrat v dal\nejßem ne razbyvalsq;
2) dlq vsex Si spravedlyvo α ≤ fSi
< 2α;
3) dlq vsex Ri v¥polneno (4);
4) f x( ) ≤ α dlq poçty vsex x ∈ Q0 \ ( )S R∪ , hde S = ∪ i iS y R = ∪ i iR ; πto
neposredstvenno sleduet yz teorem¥ Lebeha o dyfferencyrovanyy yntehralov
(sm., naprymer, [12]), poskol\ku kaΩd¥j takoj x soderΩytsq v stqhyvagwej
posledovatel\nosty vloΩenn¥x kvadratov so srednym znaçenyem ne bolee
çem,α.
Teper\ dokaΩem, çto sredy Si{ } najdetsq po krajnej mere odyn kvadrat
SN , ne soderΩawyjsq celykom v Eλ , t.,e. S EN \ λ > 0. Dlq πtoho predpolo-
Ωym protyvnoe, t.,e. çto S E⊂ λ s toçnost\g do mnoΩestva mer¥ nul\ (zdes\
soderΩytsq y sluçaj, kohda sovokupnost\ Si{ } qvlqetsq pustoj). Ymeem
f x dx
E
( )
λ
∫ = f x dx
E S R
( )
\( )λ ∪
∫ + f x dx
E R
( )
λ ∩
∫ + f x dx
S
( )∫ ≤
≤ α λE S R\( )∪ + f x dx
E Ri i
( )
λ ∩
∫∑ + f x dx
Si i
( )∫∑ <
< 2α λE S R\( )∪ + 2α λE Ri
i
∩∑ + 2α Si
i
∑ = 2α λE .
Strohyj znak neravenstva obespeçyvaetsq tem, çto xotq b¥ odno yz mnoΩestv
Eλ \ ( )S R∪ , E Riλ ∩ yly Si ymeet poloΩytel\nug meru. Poluçyly proty-
voreçye s tem, çto fEλ α= 2 . Znaçyt, S EN \ λ > 0 pry nekotorom N y, sledo-
vatel\no, ess inf ( )x SN
f x∈ ≤ λ. Otsgda
α ≤ fSN
≤ C f xx SN
ess inf ( )∈ ≤ C λ.
Takym obrazom, neravenstvo (2) dokazano.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
1148 E. G. LEONÇYK
Teper\ pust\ proyzvol\noe b ∈ 0 0; Q( ] y λ = f b∗( ) ≥ ess inf ( )x Q f x∈ 0
.
Tohda v sylu ravnoyzmerymosty funkcyj f y f ∗
ymeem Eλ ≤ b. Okonça-
tel\no, yspol\zuq monotonnost\ f ∗
, naxodym
1
b a
f u du
a
b
−
∗∫ ( ) ≤
1
0b
f u du
b
∗∫ ( ) ≤ fEλ ≤ 2 Cλ = 2 C f b∗( )
dlq lgboho otrezka a b;[ ] ⊂ 0 0; Q[ ] .
Teorema dokazana.
Zameçanye. Yz upomqnutoho v¥ße prymera v rabote [6] neposredstvenno
sleduet, çto znaçenye postoqnnoj 2 v teoreme, voobwe hovorq, umen\ßyt\
nel\zq.
Yzvestno, çto esly funkcyq odnoj peremennoj f prynadleΩyt A C1( ) , to
f p
budet summyruemoj pry vsex p <
C
C − 1
[6]. Takym obrazom, s uçetom rav-
noyzmerymosty f y f ∗
, yz teorem¥ neposredstvenno v¥tekaet takoe sledstvye.
Sledstvye. Pust\ funkcyq f prynadleΩyt A C1( ) na kvadrate Q0 .
Tohda f p
summyruema pry vsex p <
2
2 1
C
C −
.
1. Muckenhoupt B. Weighted norm inequalities for the Hardy maximal function // Trans. Amer.
Math. Soc. – 1972. – 165. – P. 207 – 226.
2. Hunt R., Muckenhoupt B., Wheeden R. L. Weighted norm inequalities for the conjugate function
and Hilbert transform // Ibid. – 1973. – 176. – P. 227 – 251.
3. Korenovskii A. Mean oscillations and equimeasurable rearrangements of functions // Lect. Notes
Unione Mat. Ital. – 2007. – # 4. – 188 p.
4. Leonçyk E. G. Ob uslovyy Makenxaupta v mnohomernom sluçae // Visn. Odes. nac. un-tu. –
2007. – 12, vyp. 7. – S. 80 – 84.
5. Xardy H. H., Lyttl\vud D. E., Polya H. Neravenstva. – M.: Yzd-vo ynostr. lyt., 1948. –
456,s.
6. Bojarski B., Sbordone C., Wik I. The Muckenhoupt class A1( )R // Stud. Math. – 1992. – 101 (2).
– P. 155 – 163.
7. Klemes I. A mean oscillation inequality // Proc. Amer. Math. Soc. – 1985. – 93, # 3. – P. 497 –
500.
8. Leonçyk E. G., Malaksyano N. A. Toçn¥e pokazately summyruemosty dlq funkcyj yz
klassov A∞ // Yzv. vuzov. Matematyka. – 2007. – # 2. – S. 17 – 26.
9. Calderon A. P., Zygmund A. On the existence of certain singular integrals // Acta Math. – 1952. –
88. – P. 85 – 139.
10. Korenovskyy A. A., Lerner A. K., Stokolos A. M. On a multidimensional form of F. Riesz’s „rising
sun” lemma // Proc. Amer. Math. Soc. – 2005. – 133, # 5. – P. 1437 – 1440.
11. Harnett DΩ. Ohranyçenn¥e analytyçeskye funkcyy. – M.: Myr, 1984. – 469 s.
12. Stejn Y. M. Synhulqrn¥e yntehral¥ y dyfferencyal\n¥e svojstva funkcyj. – M.: Myr,
1973. – 342 s.
Poluçeno 28.08.09,
posle dorabotky — 07.05.10
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 8
|
| id | umjimathkievua-article-2944 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:33:16Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/da/6265cc1e392d08891d8441605ba101da.pdf |
| spelling | umjimathkievua-article-29442020-03-18T19:40:46Z On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ Об оценке перестановки функции из класса Макенхаупта $A_1$ Leonchik, E. Yu. Леончик, Е. Ю. Леончик, Е. Ю. We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class $A_1$. Одержано точну оцінку пезростаючої рівномірної перестановки функції двох змінних із класу Макенхаупта $A_1$. Institute of Mathematics, NAS of Ukraine 2010-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2944 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 8 (2010); 1145–1148 Український математичний журнал; Том 62 № 8 (2010); 1145–1148 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2944/2638 https://umj.imath.kiev.ua/index.php/umj/article/view/2944/2639 Copyright (c) 2010 Leonchik E. Yu. |
| spellingShingle | Leonchik, E. Yu. Леончик, Е. Ю. Леончик, Е. Ю. On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title | On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title_alt | Об оценке перестановки функции из класса Макенхаупта $A_1$ |
| title_full | On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title_fullStr | On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title_full_unstemmed | On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title_short | On an estimate for the rearrangement of a function from the Muckenhoupt class $A_1$ |
| title_sort | on an estimate for the rearrangement of a function from the muckenhoupt class $a_1$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2944 |
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