Неравенство типа Бочкарева Бочкарев типтi тесiздiк

ÓÄÊ 517.51

Ã.Ê. Ìóñàáàåâà

Åâðàçèéñêèé íàöèîíàëüíûé óíèâåðñèòåò èìåíè Ë.Í.Ãóìèëåâà, Ðåñïóáëèêà Êàçàõñòàí, ã. Àñòàíà

E-mail: musabaevaguliya@mail.ru Íåðàâåíñòâî òèïà Áî÷êàðåâà

 ðàáîòå èçó÷àåòñÿ çàâèñèìîñòü ñâîéñòâ ñóììèðóåìûõ ðÿäîâ Ôóðüå è èõ êîýôôèöèåíòîâ, à èìåííî, ÷òî ìîæíî ñêàçàòü î êîýôôèöèåíòàõ Ôóðüå ôóíêöèè, êîòîðûå ïðèíàäëåæàò ïðî- ñòðàíñòâó Ëîðåíöà L

₂,r

.  ñòàòüå ïîëó÷åíî íîâîå äîêàçàòåëüñòâî òåîðåìû Áî÷êàðåâà Ñ.Â., êîòîðûé ïîêàçàë, ÷òî íåðàâåíñòâà òèïà Õàðäè- Ëèòòëâóäà, êîòîðûå ïîêàçûâàþò ñâÿçü èíòå- ãðàëüíûõ ñâîéñòâ ôóíêöèé è ñâîéñòâ ñóììèðóåìîñòè åå êîýôôèöèåíòîâ Ôóðüå äëÿ òðèãî- íîìåòðè÷åñêèõ ñèñòåì èìåþò äðóãîé âèä äëÿ ôóíêöèè èç ïðîñòðàíñòâà Ëîðåíöà L

_2,r

, ÷åì äëÿ ôóíêöèè èç ïðîñòðàíñòâà L

_p,r

ïðè p ̸ = 2, â ñëó÷àå êîãäà r óäîâëåòâîðÿåò ñëåäóþùåìó óñëîâèþ 2 < r ≤ ∞. Ìåòîäû äîêàçàòåëüñòâà Áî÷êàðåâà Ñ.Â. îñíîâûâàëèñü íà ñïåöèôèêå òðèãîíîìåòðè÷åñêèõ ðÿäîâ.  ñòàòüå èñïîëüçóåòñÿ äðóãîé ïîäõîä, áàçèðóþùèéñÿ íà ýêñòðà- ïîëÿöèè ëèíåéíûõ îïåðàòîðîâ. Ýòîò ìåòîä ïîçâîëèë ïîëó÷èòü íîâîå äîêàçàòåëüñòâî íåðà- âåíñòâà òèïà Õàðäè - Ëèòòëâóäà â ñëó÷àå 2 < r ≤ ∞ è äîêàçàòü íîâûé ðåçóëüòàò â ñëó÷àå 1 < r ≤ 2 äëÿ ïðîèçâîëüíûõ îðòîíîðìèðîâàííûõ ñèñòåì â ïðîñòðàíñòâàõ Ëîðåíöà L

_2,r

. Êëþ÷åâûå ñëîâà: Ðÿäû Ôóðüå, êîýôôèöèåíòû Ôóðüå, ïðîñòðàíñòâî Ëîðåíöà, íåðàâåíñòâî òèïà Áî÷êàðåâà.

G.K.Mussabayeva Inequality type Bochkarev

CWe study the dependence of the properties of summable Fourier series and their coecients, namely, what can we say about the Fourier coecients of a function that belongs to the Lorentz L

_2,r

. In this paper a new proof of S.V. Bochkarev, which showed that the inequalities of Hardy- Littlewood, which show the relationship of integral properties of the functions and properties of summability of its Fourier coecients for trigonometric systems have another view of the Lorentz spaces L

_2,r

, in the case when r satises the condition 2 < r ≤ ∞. Methods of proof S.V. Bochkarev based on the specics of trigonometric series. The article takes a dierent approach, based on the extrapolation of linear operators. This method allowed us to obtain a new proof of an inequality of Hardy - Littlewood in the case 2 < r ≤ ∞ and prove a new result in the case 1 < r ≤ 2 for arbitrary orthonormal systems in Lorentz spaces L

_2,r

.

Key words: Fourier series, Lorentz space, Fourier coecients, inequality type Bochkarev.

Ã.‰. Ìóñàáàåâà

Áî÷êàðåâ òèïòi òå­ñiçäiê

Á³ë æ³ìûñòà Ôóðüå ©àòàðûíû­ ©îñûíäûëàó ©àñèåòòåði ìåí îëàðäû­ êîýôôèöèåíòòåðiíi­

òºóåëäiëiãi çåðòòåëåäi, ÿ¡íè L

_2,r

Ëîðåíö êå­iñòiãiíäåãi ôóíêöèÿíû­ Ôóðüå êîýôôèöèåíòòåði òóðàëû íå àéòó¡à áîëàòûíäû¡û æàéëû. Á³ë ìà©àëàäà L

_2,r

, (2 < r ≤ ∞ ) êå­iñòiãiíäåãi ôóíê- öèÿëàð ³øií Õàðäè - Ëèòòëâóä òå­ñiçäiãiíi­ ò³ði L

p,r

êå­iñòiãiíäå áàñ©à áîëóûí ê°ðñåòåòií Ñ.Â.Áî÷êàðåâ òåîðåìàñûíû­ æà­à äºëåëäåìåñi àëûíäû. Áî÷êàðåâòi­ äºëåëäåó ºäiñòåði òðè- ãîíîìåòðèÿëû© ©àòàðëàð¡à íåãiçäåëãåí. Á³ë ìà©àëàäà ñûçû©òû îïåðàòîðëàðäû ýêñòðîïîëÿ- öèÿëàó ºäiñi ©îëäàíûë¡àí. Á³ë ºäiñ àð©ûëû 2 < r ≤ ∞ æà¡äàéûíäà Õàðäè-Ëèòëëâóä òèïòi òå­ñiçäiêòi­ æà­à äºëåëäåói àëûíäû æºíå 1 < r ≤ 2 øàðòû îðûíäàë¡àí êåçäå L

_2,r

êå­iñòiãií- äåãi êåç êåëãåí îðòîíîðìàëäàí¡àí æ³éå ³øií æà­à íºòèæåãå ©îë æåòêiçäiê.

Ò³éií ñ°çäåð: Ôóðüå ©àòàðëàðû, Ôóðüå êîýôôèöèåíòòåði, Ëîðåíö êå­iñòiãi, Áî÷êàðåâ òèïòi òå­ñiçäiê.

ISSN 15630285 KazNU Bulletin. Mathematics, Mechanics, Computer Science Series 3(82) 2014

Ââåäåíèå

Îäíèì èç âàæíûõ çàäà÷ ãàðìîíè÷åñêîãî àíàëèçà ÿâëÿåòñÿ èçó÷åíèå âçàèìîñâÿçè èí- òåãðàëüíûõ ñâîéñòâ ôóíêöèé è ñâîéñòâ ñóììèðóåìîñòè åå êîýôôèöèåíòîâ Ôóðüå. Õîðî- øî èçâåñòíûå íåðàâåíñòâà Õàðäè-Ëèòòëâóäà, êîòîðûå ïîêàçûâàþò ñâÿçü èíòåãðàëüíûõ ñâîéñòâ ôóíêöèé è ñâîéñòâ ñóììèðóåìîñòè åå êîýôôèöèåíòîâ Ôóðüå, áûëè ïîëó÷åíû Õàðäè è Ëèòòëâóäîì äëÿ òðèãîíîìåòðè÷åñêèõ ñèñòåì è äëÿ ôóíêöèé èç ïðîñòðàíñòâà Ëåáåãà. Ïýëè îáîáùèë ýòè ðåçóëüòàòû íà ñëó÷àé ðàâíîìåðíî îãðàíè÷åííîé îðòîíîðìè- ðîâàííîé ñèñòåìû [1]. Àíàëîãè÷íûå ðåçóëüòàòû äëÿ ïðîñòðàíñòâ Ëîðåíöà áûëè ïîëó-

÷åíû Ñòåéíîì (ñì.[2]). Äàëüíåéøåå ðàçâèòèå äàííûå ðåçóëüòàòû ïîëó÷èëè â ðàáîòàõ [3] - [11]. Ñ.Â. Áî÷êàðåâûì áûëî ïîêàçàíî, ÷òî äëÿ ïðîñòðàíñòâà Ëîðåíöà L

2,r

ôîðìà íåðàâåíñòâà îòëè÷àåòñÿ îò êëàññè÷åñêèõ íåðàâåíñòâ òèïà Õàðäè - Ëèòòëâóäà. Èì áûëî äîêàçàíî ñëåäóþùåå óòâåðæäåíèå.

Òåîðåìà. Ïóñòü { ϕ

n

}

^∞n=1

- îðòîíîðìèðîâàííàÿ íà [0, 1] ñèñòåìà êîìïëåêñíîçíà÷íûõ ôóíêöèé,

∥ ϕ

n

∥

^∞

≤ M, n = 1, 2, ...,

è ïóñòü ôóíêöèÿ f ∈ L

2,r

, 2 < r ≤ ∞ , òîãäà ñïðàâåäëèâî ñëåäóþùåå íåðàâåíñòâî sup

n∈N

1

| n |

¹²

(log(n + 1))

^12−1r

n

∑

m=1

a

^∗_m

≤ C ∥ f ∥

^L2,r

, (1)

ãäå a

n

- êîýôôèöèåíòû Ôóðüå ïî ñèñòåìå { ϕ

n

}

^∞n=1

.

 äàííîé ðàáîòå ïðèâîäèòñÿ íîâîå äîêàçàòåëüñòâî òåîðåìû Áî÷êàðåâà, à òàêæå ïîëó-

÷àåì íåðàâåíñòâî òèïà Õàðäè è Ëèòòëâóäà â ñëó÷àå 1 < r ≤ 2.

Ëåììà. Ïóñòü 1 < q < 2, { ϕ

n

}

^∞n=1

- îðòîíîðìèðîâàííàÿ ñèñòåìà, ∥ ϕ

n

∥ ≤ M, ∀ n = 1, 2, 3, ... f ∼ ∑

m∈N

f(m)ϕ ˆ

m

, òîãäà äëÿ ëþáîãî êîíå÷íîãî ïîäìíîæåñòâà A èç N èìååò ìåñòî íåðàâåíñòâî

1

| A |

^1/q

∑

m∈A

f ˆ (m)

≤ (

1 + M

√ 2

) ( q q − 1

)

¹_q−¹₂

∥ f ∥

Lq,2

.

Äîêàçàòåëüñòâî. Âîñïîëüçóåìñÿ òåì, ÷òî îðòîíîðìèðîâàííûå ñèñòåìû îãðàíè÷åíû â ñîâîêóïíîñòè.

∑

m∈A

f ˆ (m)

≤ M | A |∥ f ∥

^L1

(2)

è èç ðàâåíñòâà Ïàðñåâàëÿ èìååì

∑

m∈A

f ˆ (m)

≤ | A |

^1/2

∥ f ∥

^L2

. (3)

Ïóñòü τ ∈ (0, 1)

f

1

(x) =

{ f(x), | f(x) | ≤ f

^∗

(τ ),

0, â îñòàëüíûõ ñëó÷àÿõ,

f

0

(x) = f (x) − f

1

(x).

Òîãäà äëÿ ïðîèçâîëüíîãî 0 < τ < 1, èìååì

f ˆ (m) = ˆ f

0

(m) + ˆ f

1

(m)

∑

m∈A

f(m) ˆ

≤

∑

m∈A

f ˆ

0

(m)

+

∑

m∈A

f ˆ

1

(m)

≤

≤ M | A |

∫

τ 0

f

^∗

(s)ds + | A |

^1/2

(∫

1

τ

(f

^∗

(s))

²

ds )

^1/2

. Ïðèìåíÿÿ íåðàâåíñòâî Ãåëüäåðà ê ïåðâîìó ñëàãàåìîìó, ïîëó÷èì

∑

m∈A

f ˆ (m)

≤ M | A | (∫

τ

0

( t

^1q

f

^∗

(t) )

2

dt t

)

1/2

(∫

τ 0

( t

^q′1

)

2

dt t

)

1/2

+

+ | A |

^1/2

(∫

∞

τ

t

^2q−1

(f

^∗

(t))

²

t

^1−2q

dt )

1/2

.

Äàëåå, äëÿ âòîðîãî ñëàãàåìîãî ó÷èòûâàÿ , ÷òî ñóïðåìóì âåëè÷èíû t

^1−2q

äîñòèãàåòñÿ ïðè t = τ , äëÿ q < 2 èìååì îöåíêó

∑

m∈A

f ˆ (m)

≤ ∥ f ∥

^Lq,2

( M

√ 2 | A | τ

^1−1q

( q

q − 1 )

1/2

+ | A |

^1/2

τ

^12−1q

)

. Ïóñòü òåïåðü τ = (

q−1 q

) / | A | , ïîëó÷èì

∑

m∈A

f ˆ (m)

≤ (

1 + M

√ 2 )

| A |

^1/q

( q

q − 1 )

¹_q−¹₂

∥ f ∥

Lq,2

.

Òåîðåìà 1 Ïóñòü { ϕ

n

}

^∞n=1

- îðòîíîðìèðîâàííàÿ ñèñòåìà, ∥ ϕ

n

∥ ≤ M, ∀ n = 1, 2, 3, ...

Òîãäà äëÿ ëþáîãî f ∈ L

2,r

[0, 1], 2 < r ≤ ∞ âûïîëíåíî íåðàâåíñòâî:

sup

N≥8

1

N

^1/2

(log

₂

(N + 1))

^1/2−1/r

N

∑

m=1

f ˆ

^∗

(m) ≤ (4 + 2 √

2M ) ∥ f ∥

L2r

.

Äîêàçàòåëüñòâî. Ïóñòü N ≥ 8. Òîãäà äëÿ ïðîèçâîëüíîãî q : 1 < q < 2 è f ∈ L

q,2

âåðíà îöåíêà: ∥ f ∥

^Lq2

≤ ∥ f ∥

^L2r

∥ 1 ∥

^Lpr′

, ãäå

¹_q

=

¹₂

+

¹_p

è

_r¹′

=

¹₂

−

¹r

,

∥ 1 ∥

^Lpr′

= (

₁

∫

0

t

^r

′ p dt

t

)

^1/r′

= (

₁

∫

0

t

^r′

(

¹q−¹₂

)

^dt

t

)

^1/r′

= (

1 r^′

(

¹q−¹₂

)

)

1/r^′

≤ (

2q 2−q

)

1/r^′

. ∥ f ∥

^Lq,2

≤ (

2q

2−q

)

1/r^′

∥ f ∥

L2,r

. Âîñïîëüçóåìñÿ ëåììîé è ïîëó÷èì ñëåäóþùåå 1

N

^1/q

N

∑

k=1

f ˆ

^∗

(k) ≤ (

1 + M

√ 2

) ( q q − 1

) (

¹q−¹₂

)

∥ f ∥

^Lq,2

≤

≤ (

1 + M

√ 2

) ( q q − 1

) (

q−₂

) ( 2q 2 − q

)

1/r^′

∥ f ∥

^L2,r

. Ó÷èòûâàÿ ïðîèçâîëüíîñòü ïàðàìåòðà q, ïîëîæèì q =

_log^{2 log2N}

2N+2

< 2,

1

q

−

¹₂

=

_log¹

2N

, N

^1/q

= 2N

^1/2

. 1

| N |

^1/2

N

∑

k=1

f ˆ

^∗

(k) ≤ 2 (

1 + M

√ 2 ) (

1 1 −

¹_q

) (

¹q−¹₂

) ( 1

1 q

−

¹₂

)

1/r^′

∥ f ∥

^L2,r

≤

≤ 2 (

1 + M

√ 2 )





1 (

1

2

−

_log¹_2N

)





1 log2N

(log

₂

N )

^1/r′

∥ f ∥

^L2,r

.

Ó÷èòûâàÿ, ÷òî N ≥ 8, ìû ïîëó÷èì ñëåäóþùóþ îöåíêó 1

| N |

^1/2

(log

₂

N )

^1/2−1/r

N

∑

k=1

f ˆ

^∗

(k) ≤ (4 + 2 √

2M ) ∥ f ∥

^L2,r

.

Âçÿâ âåðõíþþ òî÷íóþ ãðàíü ïî âñåì N èç N , ïîëó÷èì sup

N≥8

1

N

^1/2

(log

₂

(N + 1))

^1/2−1/r

N

∑

k=1

f ˆ

^∗

(k) ≤ (4 + 2 √

2M) ∥ f ∥

^L2,r

.

Òåîðåìà 2 Ïóñòü { ϕ

n

}

^∞n=1

- îðòîíîðìèðîâàííàÿ ñèñòåìà, ∥ ϕ

n

∥ ≤ M , ∀ n = 1, 2, 3, ...

f ∼ ∑

^∞

k=1

f(k)ϕ ˆ

_k

.Òîãäà äëÿ ëþáîãî f ∈ L

_2,r

[0, 1], 1 ≤ r ≤ 2 âûïîëíåíî íåðàâåíñòâî:

∥ f ∥

^L2,r

≤ C

∞

∑

k=1

f ˆ

^∗

(k)k

⁻¹²

(log

₂

(k + 1))

^1r−12

.

Äîêàçàòåëüñòâî. Èñïîëüçóÿ äâîéñòâåííîå ïðåäñòàâëåíèå íîðìû â ïðîñòðàíñòâå Ëî- ðåíöà L

2,r

, ïîëó÷èì

∥ f ∥

^L2,r

= sup

∥g∥L2,r′=1 1

∫

0

f (t)g(t)dt = sup

∥g∥L2,r′=1 N

∑

k=1

f ˆ (k)ˆ g(k).

Èñïîëüçóÿ ñâîéñòâî íåâîçðàñòàþùåé ïåðåñòàíîâêè, îöåíèì ñâåðõó ñëåäóþùóþ íîðìó

∥ f ∥

^L2,r

≤ sup

∥g∥L2,r′=1

∞

∑

k=1

f ˆ

^∗

(k)ˆ g

^∗

(k) ≤ sup

∥g∥L2,r′=1

∞

∑

k=1

( f ˆ

^∗

(k) − f ˆ

^∗

(k + 1) ) ∑

^k

m=1

ˆ

g

^∗

(m) =

= sup

∥g∥L 2,r′=1

∞

∑

k=1

( f ˆ

^∗

(k) − f ˆ

^∗

(k + 1) )

k

^1/2

(log

₂

k)

^1/2−1/r

1

k

^1/2

(log

₂

k)

^1/2−1/r

k

∑

m=1

ˆ

g

^∗

(m).

Òàê êàê

sup

∥g∥L2,r′=1

sup

k

1

k

^1/2

(log

₂

k)

^1/2−1/r

k

∑

m=1

ˆ

g

^∗

(m) ≤ C,

ïîëó÷èì

∥ f ∥

^L2,r

≤ C

∞

∑

k=1

( f ˆ

^∗

(k) − f ˆ

^∗

(k + 1) )

k

^1/2

(log

₂

k)

^1/2−1/r

≈

≈ C

∞

∑

k=1

f ˆ

^∗

(k) (

k

^1/2

(log

₂

k)

^1/2−1/r

− (k − 1)

^1/2

(log

₂

k)

^1/2−1/r

) .

Ïî òåîðåìå Ëàãðàíæà, ïîëó÷èì ñëåäóþùåå

∥ f ∥

^L2,r

≤ C

∞

∑

k=1

f ˆ

^∗

(k)k

^−1/2

(log

₂

k)

^1/2−1/r

.

Ëèòåðàòóðà

[1] Áåðã É., Ëåôñòðåì É. Èíòåðïîëÿöèîííûå ïðîñòðàíñòâà. // Ìîñêâà. 1980. 264 c. Ñ. 6068.

[2] Stein Elias M. Interpolation of linear operators // Trans. Amer. Math. Soc. 1956. 83, ðð 482 -492.

[3] Áî÷êàðåâ C.Â. Òåîðåìà Õàóñäîðôà - Þíãà - Ðèññà â ïðîñòðàíñòâàõ ëîðåíöà è ìóëüòèïëèêàòèâíûå íåðàâåíñòâà //

Òðóäû ìàòåìàòè÷åñêîãî èíñòèòóòà èì. Â.À.Ñòåêëîâà. 1997.Ò.219. C 103 -114.

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