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Hardy-Littlewood maximal function on noncommutative Lorentz spaces

Abstract

This paper is mainly devoted to the study of the Hardy-Littlewood maximal function on noncommutative Lorentz spaces and to obtaining (p,q)-(p,q)-type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces.

MSC: 47A30, 47L05.

1 Introduction

In [1] Nelson defined the measure topology of τ-measurable operators affiliated with a semi-finite von Neumann algebra. Fack and Kosaki [2] investigated generalized s-numbers of τ-measurable operators and proved dominated convergence theorems for a gage and convexity (or concavity) inequality.

As for noncommutative maximal inequalities, a version of ergodic theory was given by Junge [3] and Junge, Xu [4]. In 2007, Mei [5] presented a version of noncommutative Hardy-Littlewood maximal inequality for an operator-valued function. In this paper, we study another version of Hardy-Littlewood maximal inequality introduced by Bekjan [6]. In [6], Bekjan defined the Hardy-Littlewood maximal function for τ-measurable operators and, among other things, obtained weak (1,1)-type and (p,p)-type inequalities for the Hardy-Littlewood maximal function. In [6], for an operator T affiliated with a semi-finite von Neumann algebra, the Hardy-Littlewood maximal function of T is defined by

MT(x)= sup r > 0 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T | E [ x r , x + r ] ( | T | ) ) .

The classical Hardy-Littlewood maximal function of a Lebesgue measurable function f:RR denoted by Mf(x) is defined as

Mf(x)= sup r > 0 1 m ( [ x r , x + r ] ) [ x r , x + r ] | f ( t ) | dt,

where m is a Lebesgue measure on (,+) (cf. [7]). Moreover, a natural generalization of this is the case f:RR and μ, a Borel measure on (,+), where

M μ f(x)= sup r > 0 1 μ ( [ x r , x + r ] ) [ x r , x + r ] | f ( t ) | dμ(t).

As discussed by Bekjan in [6], let μ(A)=τ( E A (|T|)), where A is a Borel subset of (,+). Then μ is a Borel measure and

MT(x)= sup r > 0 1 μ ( [ x r , x + r ] ) [ x r , x + r ] tdμ(t),

i.e., MT(x) is the Hardy-Littlewood maximal function M μ f(x) of f:RR defined by

f(t)={ t , t σ ( | T | ) , 0 , t σ ( | T | ) ,
(1.1)

with respect to μ.

In view of spectral theory, |T| is represented as

|T|= σ ( | T | ) td E t ,
(1.2)

and MT(|T|) is represented as MT(x). Thus, for T, MT(|T|) is considered as the operator analogue of the Hardy-Littlewood maximal function in the classical case. Therefore, roughly speaking, MT(|T|) stands in relation to T as Mf(x) stands in relation to f in classical analysis.

In this paper, we study the Hardy-Littlewood maximal function on noncommutative Lorentz spaces. By primarily adapting the techniques in [8], we obtain the (p,q)-(p,q)-type inequality for the Hardy-Littlewood maximal function on noncommutative Lorentz spaces.

The remainder of this paper is organized as follows. Section 2 consists of some notations and preliminaries, including the noncommutative Lorentz spaces and their properties. In Section 3, we present the main result of this paper.

2 Preliminaries

Throughout the paper, let be a finite von Neumann algebra acting on the Hilbert space with a normal faithful tracial state τ, and C will be a numerical constant not necessarily the same in each instance. The identity in is denoted by 1, and we denote by M proj the lattice of (orthogonal) projections in . A linear operator T:dom(T)H, with domain dom(T)H, is said to be affiliated with if uT=Tu for all unitary u in the commutant M of . The closed densely defined linear operator T affiliated with is called τ-measurable if for every ϵ>0 there exists an orthogonal projection P M proj such that P(H)dom(T) and τ(1P)<ϵ. The collection of all τ-measurable operators is denoted by M ˜ . With the sum and product defined as the respective closures of the algebraic sum and product, M ˜ is an -algebra. For a positive self-adjoint operator T affiliated with , we set

τ(T)= sup n τ ( 0 n λ d E λ ) = 0 λdτ( E λ ),

where T= 0 λd E λ is the spectral decomposition of T.

Let T be a τ-measurable operator and t>0. The ‘t th singular number (or generalized s-number) of T’ is defined by

μ t (T)=inf { TE : E M proj , τ ( 1 E ) t } .

By Proposition 2.2 of [2], we have

μ t (T)=inf { s 0 : λ s ( T ) t } (t>0),

where λ s (T)=τ( E ( s , ) (|T|)) (s0) and E ( s , ) (|T|) is the spectral projection of |T| corresponding to the interval (s,). The reader is referred to [2] for basic properties and detailed information on generalized s-numbers and the distribution function of τ-measurable operators.

Definition 2.1 (See, e.g., [9])

Let T be a τ-measurable operator affiliated with a finite von Neumann algebra , and let 0<p,q. Define

T L p , q ( M ) ={ ( 0 ( t 1 p μ t ( T ) ) q d t t ) 1 q if  q < , sup t > 0 t 1 p μ t ( T ) if  q = .
(2.1)

The set of all T M ˜ with T L p , q ( M ) < is called the noncommutative Lorentz space, denoted by L p , q (M) with indices p and q.

For convenience, we need the following Hardy inequalities in [10].

Lemma 2.2 If q1, r>0 and f0, then

( 0 [ 0 t f ( y ) d y ] q t r 1 d t ) 1 q q r ( 0 [ y f ( y ) ] q y r 1 ) 1 q

and

( 0 [ t f ( y ) d y ] q t r 1 d t ) 1 q q r ( 0 [ y f ( y ) ] q y r 1 ) 1 q .

Lemma 2.3 Let 0< r 2 <p< and 0<q,s<, then

L p , q (M) L r 2 , s (M).

Let L loc (M;τ) be the set of all τ-measurable operators such that

τ ( | T | E I ( | T | ) ) <+

for all bounded intervals I[0,+).

Definition 2.4 (See, e.g., [6])

Let T L loc (M;τ), the maximal function of T is defined by

MT(x)= sup r > 0 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T | E [ x r , x + r ] ( | T | ) )

(let 0 0 =0). M is called the Hardy-Littlewood maximal operator.

Remark 2.5 By the introduction of [6], we know that MT(|T|) is represented as MT(x). Hence, for T L loc (M;τ), MT(|T|) is considered as the operator analogue of the Hardy-Littlewood maximal function in the classical case. Therefore, roughly speaking, MT(|T|) stands in relation to T as Mf(x) stands in relation to f in classical analysis. Also, in [6], it was proved that MT(|T|) defined in Definition 2.4 was weak (1,1)-type and (p,p)-type. We refer the readers to [6] for more details and basic properties of MT(|T|).

3 Main result

Lemma 3.1 Let 0<q<, 1p, p 0 , p 1 < and p 0 p 1 such that

1 p = 1 θ p 0 + θ p 1 for some 0<θ<1.

Assume that has no minimal projection, then there exists a constant C such that T L p , q (M) we have

MT p , q C T p , q .
(3.1)

Proof We assume that p 0 < p 1 . Theorem 2 of [6] and Lemma 2.3 imply that

MT p 0 , MT p 0 C T p 0 C T p 0 , m
(3.2)

and

MT p 1 , MT p 1 C T p 1 C T p 1 , m ,
(3.3)

where m= 1 2 min(1,q).

By Lemma 1.8 of [11], for all t(0,1), we can take P M proj such that P|T|=|T|P and τ(P)=t. Set T 1 =|T|P, T 2 =|T| T 1 , it is easy to check that T 1 L p 0 , m (M) and T 2 L p 1 , m (M). Indeed, we see that μ v ( T 1 )= μ v (|T|P)= μ v (T) χ [ 0 , t ] and μ v ( T 2 )= μ v (|T| P )= μ v + t (T). Thus we obtain

T 1 p 0 , m m = 0 v m p 0 1 μ v ( | T | P ) m d v = 0 t v m p 0 1 μ v ( T ) m d v = 0 t ( v 1 p μ v ( T ) ) m v m p 0 m p 1 d v T p , m 0 t v m p 0 m p 1 d v ( q p ) m q T p , q m 1 m p 0 m p t m p 0 m p <

and

T 2 p 1 , m m = 0 v m p 1 1 μ v ( | T | P ) m d v = 0 v m p 1 1 μ v + t ( T ) m d v 0 t v m p 1 1 μ t ( T ) m d v + t v m p 1 1 μ v ( T ) m d v p 1 m t m p 1 μ t ( T ) m + sup v > t ( v 1 p μ v ( T ) ) m t v m p 1 m p 1 d v = p 1 m t m p 1 μ t ( T ) m + T p , m 1 m p m p 1 t m p 1 m p p 1 m t m p 1 m p ( sup t > 0 t 1 p μ t ( T ) ) m + ( q p ) m q 1 m p m p 1 t m p 1 m p T p , q m = [ ( q p ) m q t m p 1 m p ( p 1 m + 1 m p m p 1 ) ] T p , q m < .

Since

1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T | E [ x r , x + r ] ( | T | ) ) 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T | P E [ x r , x + r ] ( | T | ) ) + 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T | P E [ x r , x + r ] ( | T | ) ) = 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T 1 | E [ x r , x + r ] ( | T 1 | ) ) + 1 τ ( E [ x r , x + r ] ( | T | ) ) τ ( | T 2 | E [ x r , x + r ] ( | T 2 | ) ) 1 τ ( E [ x r , x + r ] ( | T 1 | ) ) τ ( | T 1 | E [ x r , x + r ] ( | T 1 | ) ) + 1 τ ( E [ x r , x + r ] ( | T 2 | ) ) τ ( | T 2 | E [ x r , x + r ] ( | T 2 | ) ) ,

taking supremum, we get

MT(x) MT 1 (x)+ MT 2 (x),

which implies that

MT p , q C ( MT 1 p , q + MT 2 p , q ) .

We estimate each term separately. For the first term, using (3.2) we get

MT 1 p , q = { 0 t q p ( μ t ( MT 1 ) ) q d t t } 1 q = { 0 t q p q p 0 ( t 1 p 0 μ t ( MT 1 ) ) q d t t } 1 q C { 0 t q ( 1 p 1 p 0 ) T 1 p 0 , m q d t t } 1 q = C { [ 0 t q ( 1 p 0 1 p ) 1 ( 0 t v m p 0 1 μ v ( T ) m d v ) q m d t ] m q } 1 m .

After replacing r and q respectively with q( 1 p 0 1 p ) and q m in the first inequality in Lemma 2.2, we see that the last expression is estimated as follows:

C ( m p 0 m p 1 ) 1 m { 0 [ v v m p 0 1 μ v ( T ) m ] q m v q ( 1 p 0 1 p ) 1 d v } 1 q = C ( 0 v q p 1 μ v ( T ) q d v ) 1 q = C T p , q ,

i.e., MT 1 p , q C T p , q . To estimate the second term, by applying (3.3) we obtain

MT 2 p , q = { 0 t q p ( μ t ( MT 2 ) ) q d t t } 1 q = { 0 t q p q p 1 ( t 1 p 1 μ t ( MT 2 ) ) q d t t } 1 q C { 0 t q ( 1 p 1 p 1 ) T 2 p 1 , m q d t t } 1 q = C { [ 0 t q ( 1 p 1 p 1 ) 1 ( 0 v m p 1 1 μ v ( T 2 ) m d v ) q m d t ] m q } 1 m C { 0 t q ( 1 p 1 p 1 ) 1 ( 0 t v m p 1 1 μ t ( T ) m d v ) q m d t } 1 q + C { [ 0 t q ( 1 p 1 p 1 ) 1 ( t v m p 1 1 μ v ( T ) m d v ) q m d t ] m q } 1 m .

For the second term { [ 0 t q ( 1 p 1 p 1 ) 1 ( t v m p 1 1 μ v ( T ) m d v ) q m d t ] m q } 1 m , replace r and q respectively with q( 1 p 1 p 1 ) and q m in the second inequality in Lemma 2.2, and we estimate the last expression as follows:

C { p 1 m μ t ( T ) q 0 t q p 1 d t } 1 q + C { 0 [ v v m p 1 1 μ v ( T ) m ] q m v q ( 1 p 1 p 1 ) 1 d v } 1 q = C { p 1 m μ t ( T ) q 0 t q p 1 d t } 1 q + C T p , q C T p , q ,

i.e., MT 2 p , q C T p , q .

For the case of p 0 > p 1 , we may simply reverse the roles of p 0 and p 1 in the above proof.

We have now shown that

MT p , q C T p , q .

 □

Theorem 3.2 Let 0<q<, 1p, p 0 , p 1 < and p 0 p 1 be such that

1 p = 1 θ p 0 + θ p 1 for some 0<θ<1.

Assume that has minimal projections, then there exists a constant C such that for all T L p , q (M) we have

MT p , q C T p , q .

Proof Since has minimal projections, we consider the von Neumann algebra tensor product M ¯ L ([0,1];dm) denoted by M ¯ , equipped with the tensor product trace τdm, where dm is the Lebesgue measure on [0,1], then M ¯ has no minimal projection.

Let |T|= σ ( | T | ) λd E λ (|T|) be the spectral decomposition of T. Since

σ ( | T | ) =σ ( | T | 1 ) ,

we have

|T1|=|T|1= σ ( | T | ) λd ( E λ ( | T | ) 1 ) = σ ( | T | 1 ) λd ( E λ ( | T | ) 1 ) .

It is easy to check that E λ (|T|)1 is a spectral series for each λ0. Hence, for any interval

Iσ ( | T | ) =σ ( | T 1 | ) =σ ( | T | 1 ) ,

by the uniqueness of the spectral decomposition, we see that

E I ( | T 1 | ) = E I ( | T | ) 1.

For r>0, since

τ ( E [ x r , x + r ] ( | T | ) ) = 0 1 τ ( E [ x r , x + r ] ( | T | ) ) dm=τdm ( E [ x r , x + r ] ( | T | ) 1 )

and

τ d m ( | T 1 | E [ x r , x + r ] ( | T | ) 1 ) = τ d m { ( | T 1 | E [ x r , x + r ] ( | T | ) 1 ) ( | T 1 | E [ x r , x + r ] ( | T | ) 1 ) } 1 2 = τ d m { ( E [ x r , x + r ] ( | T | ) 1 ) | T 1 | ( | T 1 | E [ x r , x + r ] ( | T | ) 1 ) } 1 2 = τ d m { ( E [ x r , x + r ] ( | T | ) 1 ) | T 1 | 2 E [ x r , x + r ] ( | T | ) 1 } 1 2 = τ d m { E [ x r , x + r ] ( | T | ) 1 ( | T | 2 1 ) E [ x r , x + r ] ( | T | ) 1 } 1 2 = τ d m { ( E [ x r , x + r ] ( | T | ) 1 ) ( | T | 1 ) ( | T | 1 ) ( E [ x r , x + r ] ( | T | ) 1 ) } 1 2 = τ d m ( | T | E [ x r , x + r ] ( | T | ) 1 ) = τ ( | T | E [ x r , x + r ] ( | T | ) ) ,

which implies that

M(T1)(x)=MT(x).

By an adaptation of the proof of Lemma 3.1, we deduce that

M ( T 1 ) ( T 1 ) p , q C T 1 p , q .

With the trivial fact μ t (T)= μ t (T1), we know

T 1 p , q = T p , q .

Combing this result with M(T1)(|T1|)=MT(|T|), we infer that

MT p , q C T p , q .

 □

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Acknowledgements

The author would like to thank the editor and anonymous referees for their helpful comments and suggestions on the quality improvement of the manuscript. This research is supported by the National Natural Science Foundation of China (No. 11071204) and XJUBSCX-2012002.

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Shao, J. Hardy-Littlewood maximal function on noncommutative Lorentz spaces. J Inequal Appl 2013, 384 (2013). https://doi.org/10.1186/1029-242X-2013-384

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Keywords

  • noncommutative Lorentz spaces
  • Hardy-Littlewood maximal function
  • von Neumann algebra