VILENKIN-FOURIER SERIES IN VARIABLE LEBESGUE SPACES DAVITI ADAMADZE AND TENGIZ KOPALIANI

2025-05-06 0 0 416.51KB 15 页 10玖币

侵权投诉

VILENKIN-FOURIER SERIES IN VARIABLE LEBESGUE

SPACES

DAVITI ADAMADZE AND TENGIZ KOPALIANI

Abstract. Let Snfdenote the nth partial sum of the Vilenkin-Fourier series

of a function f∈L1(G). For 1 < p−≤p+<∞, we characterize all exponents

p(·) for which the convergence of Snfto fin Lp(·)(G) holds whenever f∈

Lp(·)(G).

1. Introduction

Let {pi}i≥0be a sequence of integers with pi≥2. Deﬁne G= Π∞

i=0Zpias the

direct product of cyclic groups of order pi, and let µbe the Haar measure on G,

normalized such that µ(G) = 1. Each element of Gcan be represented as a sequence

{xi}with 0 ≤xi< pi. Deﬁne m0= 1 and mk= Πk−1

i=0 pifor k= 1,2, . . . .

There exists a well-known and natural measure-preserving identiﬁcation between

the group Gand the closed interval [0,1]. This identiﬁcation is established by as-

sociating each sequence {xi} ∈ G, where 0 ≤xi< pi, with the point P∞

i=0 xim−1

i+1.

Disregarding the countable set of pi-rationals, this mapping is one-to-one, onto, and

measure-preserving.

For each x={xi} ∈ G, deﬁne ϕk(x) = exp(2πixk/pk), k = 0,1, .... The set

{ψn}of characters of Gconsists of all ﬁnite product of ϕk,which we enumerate

in the following manner. Express each nonnegative integer nas a ﬁnite sum n=

P∞

i=0 αkmk, with 0 ≤αk< pk,and deﬁne ψn= Π∞

i=0ϕαk

k.The functions ψnform a

complete orthonormal system on G. For the case pi= 2, i = 0,1, ..., G is the dyadic

group, ϕkare Rademacher functions and ψnare Walsh functions. In general, the

system {ψn}is a realization of the multiplicative Vilenkin system. In this paper,

there is no restriction on the orders {pi}.

For f∈L1(G), let Snf, n = 0,1, ..., be the nth partial sum of the Vilenkin-

Fourier series of f. When the orders piof cyclic groups are bounded Watari [19]

showed that for f∈Lp(G),1<p<∞,

lim

n→∞ ZG

|Snf−f|pdµ = 0.

Young [17], Schipp [14] and Simon [15] showed independently that results concern-

ing mean convergence of partial sums of the Vilenkin-Fourier series are still valid

even if the orders piare unbounded.

Date: 29.01.2022.

2020 Mathematics Subject Classiﬁcation. 42C10, 42B25,46E30.

Key words and phrases. Vilenkin-Fourier series; maximal operator; variable exponent Lebesgue

space.

This work was supported by Shota Rustaveli National Science Foundation of Georgia FR-21-

12353.

arXiv:2210.11331v4 [math.FA] 17 Feb 2025

2 DAVITI ADAMADZE AND TENGIZ KOPALIANI

Let {Gk}be the sequence of subgroups of Gdeﬁned by

G0=G, Gk= Πk−1

i=0 {0} × Π∞

i=kZpi, k = 1,2, ....

On the closed interval [0,1], cosets of Gkare intervals of the form [jm−1

k,(j+

1)m−1

k], j = 0,1, ..., mk−1.By Fwe denote the set of generalized intervals. This set

is the collection of all translations of intervals [0, jm−1

k+1], k = 0,1, ... j = 1, ..., pk.

Note that a set Ibelongs to Fif (1) for some x∈Gand k, I ⊂x+Gk,(ii) Iis a

union of cosets of Gk+1,and (iii) if we consider x+Gkas a circle, Iis an interval.

Let F−1={G}. For k= 0,1, ..., let Fkbe the collection of all I∈ F such that Iis

a proper subset of a coset of Gk, and is a union of cosets of Gk+1.The collections

Fkare disjoint, and F=∪∞

k=−1Fk.For I∈ F, we deﬁne the set 3I∈ F as follows.

If I=G, let 3I=G. For I∈ Fk, k = 0,1, ..., there is x∈Gsuch that I⊂x+Gk.

If µ(I)≥µ(Gk)

3, let 3I=x+Gk. If µ(I)<µ(Gk)

3, consider x+Gkas a circle. Then

Iis an interval in this circle. Deﬁne 3I∈ Fkto be the interval in this circle which

contains Iat its center and has measure µ(3I) = 3µ(I). In all cases, for I∈ F,

µ(3I)≤3µ(I).

We say that wis a weight function on Gif wis measurable and 0 < w(x)<∞

a.e. Gosselin [7] (case supipi<∞) and Young [18] (no restriction on the orders

pi) characterized all weight functions wsuch that if f∈Lp

w(G),1< p < ∞, Snf

converges to fin Lp

w(G). Here Lp

w(G) denotes the space of measurable functions

on Gsuch that ∥f∥p,w = (RG|f|pwdµ)1/p <∞.

Deﬁnition 1.1. (see [18]) (i) We say that wsatisﬁes Ap(G) condition, 1 <p<∞,

(1.1) [w]Ap= sup

I∈F 1

µ(I)ZI

w dµ 1

µ(I)ZI

w−1/(p−1) dµp−1

<∞.

(ii) We say that wsatisﬁes A1(G) condition if

[w]A1= sup

I∈F

µ(I)ZI

wdµ (essinfIw(x))−1<∞.

For the case where the orders of cyclic groups are bounded, Gosselin [7] deﬁned

Ap(G) condition, as the one where (1.1) condition holds for all Ithat are cosets of

Gk, k = 0,1,2, .... For this case Apconditions, deﬁned by Young and Gosselin, are

equivalent (see [18]).

Theorem 1.2. ([18]) Let wbe a weight function on G. For 1< p < ∞, the

following statements are equivalent:

(i) w∈Ap(G),

(ii) There is a constant C, depending only on wand p, such that for every

f∈Lp

w(G), we have

|Snf|pwdµ ≤CZG

|f|pwdµ,

(iii) For every f∈Lp

w(G), we have

lim

n→∞ ZG

|Snf−f|pwdµ = 0.

In this paper we characterize all exponents p(·) such that if f∈Lp(·)(G), then

partial sums Snfof the Vilenkin-Fourier series of f∈Lp(·)(G) converge to fwith

Lp(·)-norm. Now we give a deﬁnition of variable Lebesgue space. Let p(·) : G→

[1,∞) be a measurable function. The variable Lebesgue space Lp(·)(G) is the set

of all measurable functions fsuch that for some λ > 0,

ρp(·)(f/λ) = ZG

(|f(x)|/λ)p(x)dµ < ∞.

Lp(·)(G) is a Banach function space equipped with the Luxemburg norm

∥f∥p(·)= inf{λ > 0 : ρp(·)(f/λ)≤1}.

We use the notations p−(I) = essinfx∈Ip(x) and p+(I) = esssupx∈Ip(x) where

I⊂G. If I=Gwe simply use the following notation p−, p+. The function p′(·)

denotes the conjugate exponent function of p(·), i.e., 1/p(x)+1/p′(x) = 1 (x∈G).

In this paper the constants C, c are absolute constants and may be diﬀerent in

diﬀerent contexts and χAdenotes the characteristic function of set A.

Very recently the convergence of partial sums of the Walsh-Fourier series in

Lp(·)([0,1)) space was investigated by Jiao et al. [8]. We denote by Clog

dthe set of

all functions p(·) : [0,1) →[1∞), for which there exists a positive constant Csuch

that

|I|p−(I)−p+(I)≤C

for all dyadic intervals I= [k2−n,(k+ 1)2−n) (k, n ∈N,0≤k < 2n), here |I|

denotes the Lebesgue measure of I. Note that this condition may be interpreted

as a dyadic version of log-H¨older continuity condition of p(·) (or on dyadic group).

The log-H¨older condition is a very common condition for solving various problems

of harmonic analysis in Lp(·)(Rn) (see [2], [5]).

Theorem 1.3. ([8]) Let p(·)∈Clog

dwith 1< p−≤p+<∞.If f∈Lp(·)([0,1)),

then for partial sums Snfof the Walsh-Fourier series of f∈Lp(·)([0,1)) we have

sup

n∈N

∥Snf∥p(·)≤C∥f∥p(·).

Since Walsh polynomials are dense in Lp(·)([0,1)), Theorem 1.3 implies that Snf

converges to the original function in Lp(·)([0,1))-norm (for more details see [8] and

the recent book [13], chapter 9).

In order to extend techniques and results of constant exponent case to the setting

of variable Lebesgue spaces, a central problem is to determine conditions on an

exponent p(·) under which the Hardy-Littlewood maximal operator is bounded on

Lp(·)(see monographs Cruz-Uribe and Fiorenza [2] and Diening et.al. [5]). We now

deﬁne the Hardy-Littlewood maximal function that is appropriate for the study of

Vilenkin-Fourier series. For f∈L1(G), let

Mf(x) = sup

x∈I,I∈F

µ(I)ZI

|f|dµ.

This maximal function was introduced ﬁrst by P. Simon in [16]. He showed that

the maximal operator is bounded in Lp(G),1< p < ∞and is of weak type (1,1).

Young [18] obtained the following analogue of Muckenhoupt’s theorem [11].

Theorem 1.4. Let wbe a weight function on G. For 1<p<∞, the following two

statements are equivalent:

(i) w∈Ap(G),

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

VILENKIN-FOURIERSERIESINVARIABLELEBESGUESPACESDAVITIADAMADZEANDTENGIZKOPALIANIAbstract.LetSnfdenotethenthpartialsumoftheVilenkin-Fourierseriesofafunctionf∈L1(G).For1

展开>> 收起<<

VILENKIN-FOURIER SERIES IN VARIABLE LEBESGUE SPACES DAVITI ADAMADZE AND TENGIZ KOPALIANI.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

VILENKIN-FOURIER SERIES IN VARIABLE LEBESGUE SPACES DAVITI ADAMADZE AND TENGIZ KOPALIANI

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: