A Direct Proof for Operator Hardy-Littlewood Maximal Inequality (2024)

\stackMath

Chian Yeong ChuahDepartment of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210-1174, USA
ORCid:0000-0003-3776-6555chuah.21@osu.edu,Zhen-Chuan LiuDepartamento de Matemáticas
Universidad Autónoma de Madrid
C/ Francisco Tomás y Valiente, 7 Facultad de Ciencias, módulo 17, 28049 Madrid,Spain.
ORCid: 0000-0002-6092-5473liu.zhenchuan@uam.esand Tao MeiDepartment of Mathematics
Baylor University
1301 S University Parks Dr, Waco, TX 76798, USA.
ORCid: 0000-0001-6191-6184tao_mei@baylor.edu

Abstract.

We give a direct proof of the operator valued Hardy-Littlewood maximal inequality for $2\leq p<\infty$ , which was first proved in [7].

Key words and phrases:

Maximal Inequality, Schatten $p$ -class, von Neumann Algebra, $L_{p}$ -Space

2010 Mathematics Subject Classification:

Primary: 46B28, 46L52. Secondary: 42A45.

1. Introduction

The operator valued Hardy-Littlewood maximal inequality ([7]) has become a basic tool in the study of noncommutative analysis. See e.g. [2, 4, 8] for the recent works which apply this inequality.

The original proof of the inequality contained in [7] reduces the problem to the martingale case where the noncommutative Doob’s maximal inequality due to M. Junge ([5]) is applied. G. Hong reproved this maximal inequality in [3]. He follows Stein’s idea of dominating the Hardy-Littlewood maximal function by maximal averages of heat semigroup operators and applies the noncommutative maximal ergodic theory due to M. Junge and Q. Xu ([6]). Both proofs are indirect which prevent researchers, who are not familiar with the terminology of noncommutative analysis from a good understanding of the theorem. The purpose of this article is to provide a direct and more understandable proof of this inequality for the case $p\geq 2$ .

Recall that, for a locally integrable function $f$ , the Hardy-Littlewood maximal function is defined as

Mf(x)=\sup_{t}\frac{1}{2t}\int_{I(x,t)}|f(y)|dy,

where $I(x,t)$ is the interval centered at $x$ with length $2t.$ The classical Hardy-Littlewood maximal inequality states that

(1.1)

\displaystyle\|Mf\|_{L^{p}}\leq c\frac{p}{p-1}\|f\|_{L^{p}},

for all $f\in L^{p}(\mathbb{R}),1<p\leq\infty$ .

Let $X$ be a Banach space, for X-valued functions $f$ , their maximal function $Mf$ can be defined by considering the maximal function of the norm of $f$ ,

(1.2)

\displaystyle Mf(x)=\sup_{t}\frac{1}{2t}\int_{I(x,t)}\|f(y)\|_{X}dy.

Apply the classical Hardy-Littlewood maximal inequality to the $L_{p}$ function $\|f\|_{X}$ , one obtains that

(1.3)

\displaystyle\|Mf\|_{L^{p}(X)}\leq c\frac{p}{p-1}\|f\|_{L^{p}(X)}=(\int_{%\mathbb{R}}\|f\|_{X}^{p})^{\frac{1}{p}},

for all $f\in L^{p}(\mathbb{R},X),1<p\leq\infty$ . A shortcoming is that this type of maximal function is scalar-valued and may lose a lot of information of $X$ that $f$ originally carried.

When $X=L^{p}(\Omega)$ with $\Omega$ a measurable space, one may define a $X$ -valued maximal function $F$ as

(1.4)

\displaystyle F(x,\omega)=\frac{1}{2t}\int_{I(x,t)}|f(y,\omega)|dy,

and deduce from (1.3) that

(1.5)

\displaystyle\|F\|_{L^{p}(\mathbb{R},L^{p}(\Omega))}\leq c\frac{p}{p-1}\|f\|_{%L^{p}(\mathbb{R},L^{p}(\Omega))}.

Note that in (1.4), $F$ is a $X$ -valued function that dominates the average of $|f|$ pointwisely, while in (1.2), $Mf$ is merely a scalar valued function that carries much less information. Can similar results like (1.4) and (1.5) hold for $X$ -valued functions when the Banach space $X$ is not equipped with a total order but still has a reasonable partial order, e.g. for $X=$ the Schatten $p$ classes?Based on G. Pisier’s work on operator spaces and M. Junge and Junge/Xu’s work on the theory of noncommutative martingales, T. Mei ([7]) proved a maximal inequality like (1.5) for $X$ being the noncommutative $L^{p}$ -spaces. In the case that $X=$ the Schatten $p$ -class $S_{p}$ and $f\in L_{p}({S_{p}})$ for some $1<p\leq\infty$ , this operator-maximal inequality says that there exists a $S_{p}$ -valued function $F$ such that

(i) $\frac{1}{2t}\int_{x-t}^{x+t}|f(y)|dy\leq F(x)$ as operators for all $t>0$ , i.e. $F-\frac{1}{2t}\int_{x-t}^{x+t}|f(y)|dy\in S^{p}$ is a self adjoint positive definite operator almost everywhere.

(ii) There exists an absolute constant $c$ such that

(1.6)

\displaystyle\|F\|_{L^{p}(S^{p})}\leq c\frac{p^{2}}{(p-1)^{2}}\|f\|_{L^{p}(S^{%p})}.

A main obstacle for the proof of (1.6) is the lack of a total order. This was overcome by M. Junge using G. Pisier’s duality result for the operator spaces $L_{p}(\ell_{\infty})$ and $L_{p}(\ell_{1})$ . This article aims togive a direct proof for (1.6) and a more understandable proof of Pisier’s duality result (Lemma 2.2) for analysts who are not familiar with operator spaces.

2. Preliminary

Let $H$ be a separable Hilbert space. We denote the space of bounded linear operators on $H$ by $B(H)$ . For $x\in B(H)$ , we denote by $x^{*}$ the adjoint operator of $x$ , and define $|x|^{p}=(x^{*}x)^{\frac{p}{2}}$ by the functional calculus for $0<p<\infty$ . We say that $x\in B(H)$ is self-adjoint if $x=x^{*}$ . We say a self-adjoint operator $x$ is positive, denoted by $x\geq 0$ if

(2.1)

\displaystyle\langle xe,e\rangle\geq 0,

for all $e\in H$ . This is equivalent to saying that $x=y^{*}y$ for some $y\in B(H)$ . For two self-adjoint $x,y\in B(H)$ , we write $x\leq y$ if $y-x\geq 0$ .The Schatten $p$ -classes $S_{p}$ , $0<p<\infty$ are the spaces of $x\in B(H)$ so that

\|x\|_{p}=(tr|x|^{p})^{\frac{1}{p}}<\infty.

Here, $tr$ is the usual trace $tr(x)=\sum_{k}\langle xe_{k},e_{k}\rangle$ for $x\geq 0\in B(H)$ . TheSchatten $p$ classes share many properties with the $\ell_{p}$ spaces of sequences. In particular, $S_{p}$ are Banach spaces for $p\geq 1$ and $B(H)$ (resp. $S_{q}$ ) is the dual space of $S_{1}$ (resp. $S_{p}$ for $1<p,q<\infty,\frac{1}{p}+\frac{1}{q}$ ), via the isometric isomorphism,

x\mapsto\phi_{x}:\phi_{x}(y)=tr(xy)

for $y\in S_{1}$ (resp. $S_{p}$ ).

2.1. Noncommutative $L_{p}$ spaces

A von Neumann algebra, by definition, is a weak- $*$ closed subalgebras $\mathcal{M}$ of $B(H)$ .The completeness according to the weak $*$ -topology of $\mathcal{M}$ ensures that it contains the spectral projections of its self-adjoint elements. $\ell_{\infty}({\mathbb{N}})$ which is isometrically isomorphic to the subalgebras of the diagonal operators and $B(H)$ itself are two basic examples of von Neumann algebras.The usual trace $\tau=tr$ on $B(H)$ is a linear functional on the weak- $*$ dense subspace $S_{1}$ satisfying the following properties,

i)
Tracial: $\tau(xy)=\tau(xy)$ ,
ii)
Faithful: if $x\geq 0$ and $\tau(x)=0$ then $x=0$ ,
iii)
Lower semi-continuous: $\tau(\sup x_{i})=\sup\tau(x_{i})$ when $x_{i}\geq 0$ is increasing,
iv)
Semifinite: for any $x\geq 0$ , there exists $0\leq y\leq x$ such that $\tau(y)<\infty$ .

This leads to defining semifinite von Neumann algebras $\mathcal{M}$ as those equipped with a trace $\tau$ , which is an unbounded linear functional satisfying (i)-(iv) for $x,y$ belonging to a weak $*$ dense subspace. Note the restriction of the usual trace of $B(H)$ on $\mathcal{M}$ may not be semifinite, and not every von Neumann algebras is semifinite. Given such a pair $(\mathcal{M},\tau)$ (which is usually viewed as a noncommutative $L_{\infty}$ - space) the noncommutative $L_{p}$ spaces associated to it are the completion of $f\in{\mathcal{M}}$ with finite quasi norm

\|f\|_{p}\,=\,\left[\tau\left(|f|^{p}\right)\right]^{\frac{1}{p}}\quad\mbox{%for}\quad 0<p<\infty,

where $|f|^{p}=(f^{*}f)^{p/2}$ is constructed via functional calculus. We set $L_{\infty}(\mathcal{M})=\mathcal{M}$ .

The Schatten $p$ -classes $S_{p}$ and the $L_{p}$ spaces on a semifinite measure space $(\Omega,\mu)$ are examples of noncommutative $L_{p}$ spaces associated with $\mathcal{M}=B(H)$ and ${\mathcal{M}}=L_{\infty}(\Omega,\mu)$ respectively. Another basic example arises from group von Neumann algebras. Every commutative semifinite von Neumann algebra is isometrically isomorphic to the space $L_{\infty}(\Omega,\mu)$ of essentially bounded functions on some semifinite measure space $(\Omega,\mu)$ .Many basic properties of $L_{p}(\Omega,\mu)$ extend to $L_{p}(\mathcal{M})$ . In particular, one has the Hölder’s inequality which states that

\displaystyle\|xy\|_{p}\leq\|x\|_{r}\|y\|_{q},

for $x\in L_{r}(\mathcal{M}),y\in L_{q}(\mathcal{M}),0\leq p,q,r,\leq\infty,\frac{1%}{q}+\frac{1}{r}=\frac{1}{p}$ . The interpolation properties of $L_{p}(\Omega,\mu)$ extend to $L_{p}(\mathcal{M})$ as well, and the duality properties of $L_{p}(\Omega,\mu)$ extend to $L_{p}(\mathcal{M})$ for the range $p\geq 1$ . The elements in $L_{p}(\mathcal{M})$ may not belong to $\mathcal{M}$ or $B(H)$ in general. They can be understood as unbounded operators affiliated with $\mathcal{M}$ .For self-adjoint elements $x,y$ in $L_{p}(\mathcal{M})$ , we say $x$ is positive, denoted by $x\geq 0$ if (2.1) holds (the quantity may be $\infty$ though). We write $x\leq y$ if $y-x\geq 0$ .

Define $\ell_{\infty}(L_{p}(\mathcal{M}))$ as the space of all sequences $x=(x_{n})_{n\geq 1}$ in $L_{p}(\mathcal{M})$ such that

(2.2)

\displaystyle\|x\|_{\ell_{\infty}(L_{p}(\mathcal{M}))}=\sup_{n}\|x_{n}\|_{p}<\infty.

Define $\ell_{1}(L_{p}(\mathcal{M}))$ as the space of all sequences $x=(x_{n})_{n\geq 1}$ in $L_{p}(\mathcal{M})$ such that

(2.3)

\displaystyle\|x\|_{\ell_{1}(L_{p}(\mathcal{M}))}=\sum_{n}\|x_{n}\|_{p}<\infty.

We have the duality that

\displaystyle\ell_{\infty}(L_{q}(\mathcal{M}))=(\ell_{1}(L_{p}(\mathcal{M})))^%{*}

for $1\leq p<\infty,\frac{1}{p}+\frac{1}{q}=1$ , via the isometric isomorphism

x\mapsto\varphi_{x};\varphi_{x}(y)=\sum_{n}\tau(x_{n}y_{n}).

On the other hand, $\ell_{1}(L_{q}(\mathcal{M}))$ isometrically embeds into the dual of $\ell_{\infty}(L_{p}(\mathcal{M}))$ as a weak $*$ -dense subspace via the same isomorphism for the same range and relation of $p,q$ . We refer to the survey paper [11] for more information on noncommutative $L_{p}$ spaces.

2.2. Pisier’s $L_{p}(\mathcal{M};\ell_{\infty})$ norm

Given two positive operators $x,y\in L_{p}(\mathcal{M})$ e.g. $x,y\in S_{p}$ , the expression $\sup(x,y)$ does not make sense unless $x,y$ commutes so that the least upper element exists.Nevertheless, the following is a reasonable expression for $\|\sup_{n}x_{n}\|_{L_{p}(\mathcal{M})}$ for sequences of positive elements $x_{n}$ in $L_{p}(\mathcal{M})$ ,

(2.4)

\displaystyle\||(x_{n})\|_{L_{p}(\mathcal{M};\ell_{\infty}^{+})}=\inf\{\|a\|_{%L_{p}};x_{n}\leq a,a\in L_{p}(\mathcal{M})\}.

For sequences of positive elements $y_{n}$ in $L_{p}(\mathcal{M}),1\leq p<\infty$ such that $\sum_{n=1}^{N}y_{n}$ converges in $L_{p}(\mathcal{M})$ , let

(2.5)

\displaystyle\|(y_{n})\|_{L_{p}(\mathcal{M};\ell_{1}^{+})}=\left\|\sum_{n=1}^{%\infty}y_{n}\right\|_{L_{p}(\mathcal{M})}.

Here the right hand side is an increasing sequence. G. Pisier ([9] and M. Junge ([5] proved that the expressions (2.4) and (2.5) extend to Banach space norms.¹¹1The obvious extension $\|\sum_{n}|x_{n}|\|_{p}$ for sequences $x_{n}\in L_{p}(\mathcal{M})$ fails the triangle inequality. Furthermore, they proved a duality result for these two norms (see Lemma 2.2), which is the key for Junge’s proof of the noncommutative Doob’s maximal inequality.

Define $L_{p}(\mathcal{M};\ell_{\infty})$ as the space of all sequences $x=(x_{n})_{n\geq 1}$ in $L_{p}(\mathcal{M})$ which admits a factorization

(2.6)

x_{n}=az_{n}b,\quad\forall n\geq 1,

where $a,b\in L_{2p}(\mathcal{M})$ and $z=(z_{n})$ belongs to the unit ball of $\mathcal{M}$ .Given $x\in L_{p}(\mathcal{M};\ell_{\infty})$ , define

(2.7)

\displaystyle\|x\|_{L_{p}(\mathcal{M};\ell_{\infty})}=\inf\left\{\|a\|_{2p}%\sup_{n\geq 1}\|z_{n}\|_{\infty}\|b\|_{2p}\right\},

where the infimum is taken over all factorizations of $x$ as in (2.6). We denote the subset of $L_{p}(\mathcal{M};\ell_{\infty})$ consisting of all positive sequences by $L_{p}(\mathcal{M};\ell^{+}_{\infty})$ . Next, we denote the subspace of $L_{p}(\mathcal{M};\ell_{\infty})$ consisting of all sequences $x=(x_{n})_{n}$ such that $x_{n}=0$ for $n>N$ by $L_{p}(\mathcal{M};\ell^{N}_{\infty})$ .

Define $L_{p}(\ell_{1})$ as the space of all sequences $x=(x_{n})_{n\geq 1}$ in $L_{p}(\mathcal{M})$ which admits a factorization

(2.8)

x_{n}=a_{n}b_{n},\quad\text{ for all }n\geq 1,

such that the series $\sum_{n}a_{n}a_{n}^{*}$ and $\sum_{n}b_{n}^{*}b_{n}$ converges in $L_{p}(\mathcal{M}).$ Given $x\in L_{p}(\mathcal{M};\ell_{1})$ , define

(2.9)

\|x\|_{L_{p}(\mathcal{M};\ell_{1})}=\inf\left\{\left\|\left(\sum a_{j}a_{j}^{*%}\right)^{\frac{1}{2}}\right\|_{2p}\cdot\left\|\left(\sum b_{j}^{*}b_{j}\right%)^{\frac{1}{2}}\right\|_{2p}\right\},

where the infimum is taken over all possible factorizations (2.8).²²2There are slightly different definition of the space $L_{p}(\mathcal{M};\ell_{1})$ in the literature. Here, we use the definition given in [10]. We denote the subset of $L_{p}(\mathcal{M};\ell_{1})$ consisting of all positive sequences by $L_{p}(\mathcal{M};\ell^{+}_{1})$ .Define $L_{p}(\ell_{1}^{N})$ to be the space of all sequences $x=(x_{n})_{n\geq 1}\in L_{p}(\ell_{1})$ with $x_{n}=0$ for $n>N$ .G. Pisier ([9]) proved that (2.7) and (2.9) are norms extending (2.4) and (2.5).

Lemma 2.1 (Pisier [9]).

For sequences $x_{n}\geq 0\in L_{p}(\mathcal{M})$ , we have

(2.10)

\displaystyle\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{1})}=\|(x_{n})\|_{L_{p}(%\mathcal{M};\ell_{1}^{+})}

in the sense that both sides are equally finite or both sides are infinite, and

(2.11)

\displaystyle\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{\infty})}\leq\|(x_{n})\|_{L_%{p}(\mathcal{M};\ell_{\infty}^{+})}\leq 4\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{%\infty})}

Proof.

We prove (2.10) first. The left hand side is obviously smaller because we may choose $a_{n}=b_{n}=(x_{n})^{\frac{1}{2}}$ for $x_{n}\geq 0$ . To prove that the right hand side is smaller, we assume $\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{1})}=1$ and assume that there exists a factorization that $x_{n}=a_{n}b_{n}$ such that

\|\sum_{n}a_{n}a_{n}^{*}\|_{p},\|\sum_{n}b_{n}b_{n}^{*}\|_{p}\leq 1+\varepsilon.

Then, $2x_{n}=a_{n}b_{n}+b_{n}^{*}a_{n}^{*}\leq a_{n}a_{n}^{*}+b_{n}^{*}b_{n}$ because $(a_{n}^{*}-b_{n})^{*}(a_{n}^{*}-b_{n})\geq 0$ . So, $\sum_{n=1}^{N}x_{n}$ converges in $L_{p}({\mathcal{M}})$ because $\sum_{n}^{N}a_{n}a_{n}^{*}$ and $\sum_{n}^{N}b_{n}^{*}b_{n}$ do. Moreover,

2\|\sum_{n=1}^{N}x_{n}\|_{p}\leq\|\sum_{n=1}^{N}a_{n}a_{n}^{*}+b_{n}b_{n}^{*}%\|_{p}\leq 2+2\varepsilon.

We conclude by taking $N\rightarrow\infty,\varepsilon\rightarrow 0.$

For (2.11), assuming $x_{n}\leq a\in L_{p}(\mathcal{M})$ , we denote by $p$ the projection onto the kernel of $a$ . Then, $p\in\mathcal{M}$ . Let $z_{n}=(p+a^{\frac{1}{2}})^{-1}x_{n}(p+a^{\frac{1}{2}})^{-1}$ . Then, $z_{n}$ belongs to the unit ball of $\mathcal{M}$ and $x_{n}=a^{\frac{1}{2}}z_{n}a^{\frac{1}{2}}$ .We see that the first inequality holds. Next, we show the second inequality. We assume that $\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{\infty})}=1$ and $x_{n}$ has a factorization $x_{n}=az_{n}b$ with $\|z_{n}\|=1$ and $\|a\|_{2p},\|b\|_{2p}\leq 1+\varepsilon$ .We write $z_{n}=\sum_{k=0}^{3}i^{k}z_{n,k}$ with contractions $z_{n,k}\geq 0$ , and consider the new decomposition $x_{n}=\sum_{k}a_{k}z_{n,k}b$ with $a_{k}=i^{k}a$ . Noting that $(a_{k}^{*}-b)^{*}z_{n,k}(a_{k}^{*}-b)\geq 0$ , we have

x_{n}=\frac{1}{2}\sum_{k=0}^{3}(a_{k}z_{n,k}b+b^{*}z_{n,k}a_{k}^{*})\leq\frac{%1}{2}\sum_{k=1}^{3}(a_{k}^{*}z_{n_{k}}a_{k}+bz_{n,k}b^{*})\leq 2(a^{*}a+bb^{*}),

with $\|2(a^{*}a+bb^{*})\|_{p}\leq(2+2\varepsilon)^{2}.$ Taking $\varepsilon\rightarrow 0$ , we conclude (2.11).∎

The following lemma is another key to understanding the proof of the operator Hardy-Littlewood maximal inequality.The result was proved by G. Pisier ([9, 5]). We include an argument for the case of finite sequences below.

Lemma 2.2 ([9, 5]).

The norms (2.4) and (2.5) are in duality. More precisely, for $1\leq p<\infty,\frac{1}{p}+\frac{1}{q}=1$ ,

(i)

For any $N$ -tuple $(y_{1},\dots,y_{N})$ in $L_{p}(\mathcal{M})$ and $y_{k}\geq 0,k=1,\dots,N$ , we have

(2.12)

\|(y_{n})\|_{L_{p}(\mathcal{M};\ell_{1}^{N})}=\sup\left\{|\tau\left(\sum_{j=1}%^{N}x_{j}y_{j}\right)|:\|(x_{j})\|_{L_{q}(\mathcal{M};\ell_{\infty}^{+})}\leq 1%\right\}.

(ii)

For any bounded sequence $(x_{n})$ in $L_{q}(\mathcal{M})$ with $x_{n}\geq 0$ , we have

(2.13)

\|(x_{n})\|_{L_{q}(\mathcal{M};\ell_{\infty})}=\sup_{N}\left\{|\tau\left(\sum_%{j=1}^{N}x_{j}y_{j}\right)|:y_{j}\geq 0,\|(y_{j})\|_{L_{p}(\mathcal{M};\ell_{1%}^{N})}\leq 1\right\}.

(iii)

$L_{q}(\mathcal{M};\ell_{\infty}^{+})$ embeds isometrically into the dual space of $L_{p}(\mathcal{M};\ell_{1})$ for $1\leq p<\infty$ via the isomorphism

x\mapsto\varphi_{x}:\varphi_{x}(y)=\sum_{n}x_{n}y_{n}.

For any $\varphi$ in $L_{p}(\mathcal{M};\ell_{1})^{*}$ such that $\varphi((y_{n}))\geq 0$ for finite positive sequences $(y_{n})\in L_{p}(\mathcal{M})$ , there is a (unique) positive sequence $(x_{n})$ in $L_{q}(\mathcal{M})$ with

\|x\|_{L_{q}(\mathcal{M};\ell_{\infty})}=\|\varphi\|_{(L_{p}(\mathcal{M};\ell_%{1}))^{*}}

such that for any $N\geq 1$ and any $y=(y_{1},\dots,y_{N})\in L_{p}(\mathcal{M};\ell_{1}^{N})$ ,

\varphi(y)=\tau\left(\sum_{j=1}^{N}x_{j}y_{j}\right).

Proof.

(i). Let $x=(x_{n})\in L_{q}(\mathcal{M};\ell_{\infty}),y=(y_{n})\in L_{p}(\mathcal{M};%\ell_{1})$ . First, we prove that

(2.14)

\left|\sum\tau(x_{j}y_{j})\right|\leq\|(x_{j})\|_{L_{q}(\mathcal{M};\ell_{%\infty})}\|y\|_{L_{p}(\mathcal{M};\ell_{1}^{N})}.

Consider a factorization $x_{j}=az_{j}b$ where $a,b\in L_{2q}(\mathcal{M})$ and $(z_{j})$ belongs to the unit ball of $\mathcal{M}$ .Also consider a factorization of $y_{j}=u_{j}v_{j}$ where $u_{j},v_{j}\in L_{2p}(\mathcal{M})$ .Then, by Hölder’s inequality and the Cauchy-Schwarz inequality,

	$\displaystyle\left\|\sum_{j}\tau(az_{j}bu_{j}v_{j})\right\|$	$\displaystyle=\left\|\sum_{j}\tau(z_{j}bu_{j}v_{j}a)\right\|\leq\sum_{j}\\|bu_{j}%v_{j}a\\|_{1}$
		$\displaystyle\leq\sum_{j}\\|bu_{j}\\|_{2}\\|v_{j}a\\|_{2}$
		$\displaystyle\leq\left(\sum_{j}\\|bu_{j}\\|_{2}^{2}\right)^{\frac{1}{2}}\left(%\sum_{j}\\|v_{j}a\\|_{2}^{2}\right)^{\frac{1}{2}}$
		$\displaystyle=\left(\tau(b\sum_{j}u_{j}u_{j}^{}b^{})\right)^{\frac{1}{2}}%\left(\tau(a^{}\sum_{j}v_{j}v_{j}^{}a)\right)^{\frac{1}{2}}$
		$\displaystyle\leq\\|b^{}b\\|^{\frac{1}{2}}_{q}\\|\sum_{j}u_{j}u_{j}^{}\\|^{\frac%{1}{2}}_{p}\\|aa^{}\\|^{\frac{1}{2}}_{q}\\|\sum_{j}v_{j}v_{j}^{}\\|^{\frac{1}{2}%}_{p}$
		$\displaystyle\leq\\|b\\|_{2q}\\|(\sum_{j}u_{j}u_{j}^{})^{\frac{1}{2}}\\|_{p}\\|a\\|%_{2q}\\|(\sum_{j}v_{j}v_{j}^{})^{\frac{1}{2}}\\|_{p}$

Hence, we proved the one side inequality of (i) and (ii) and the first half of (iii).

Now, suppose $y_{n}\geq 0,y=(y_{n})_{n}\in L_{p}(\mathcal{M};\ell_{1}^{N})$ .Choose $x=(x_{n})_{n=1}^{N}$ with $x_{n}=(\sum_{k=1}^{N}y_{k})^{p-1}$ for all $1\leq n\leq N$ . Note

\|(x_{n})\|_{L_{q}(\mathcal{M};\ell_{\infty})}=\left\|\left(\sum_{k}y_{k}%\right)^{p-1}\right\|_{q}=\|\sum_{k=1}^{N}~{}y_{k}\|_{p}^{p-1}

Thus,

\tau\left(\sum_{k}y_{k}x_{k}\right)=\tau\left(\sum_{k=1}^{N}~{}y_{k}\left(\sum%_{k=1}^{N}~{}y_{k}\right)^{p-1}\right)=\|\sum_{k=1}^{N}~{}y_{k}\|_{p}.

Therefore, we proved the other direction of (i).

We now prove the other direction of (ii). By the Hahn-Banach theorem, for any $(x_{n})\in L_{p}(\mathcal{M};\ell_{\infty})$ , there exists $\varphi\in L_{p}(\mathcal{M};\ell_{\infty})^{*}$ , such that $\|\varphi\|=1$ and $\varphi((x_{n}))=\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{\infty})}$ .Since $L_{p}(\mathcal{M};\ell_{\infty})$ is a subspace of $\ell_{\infty}(L_{p}(\mathcal{M}))$ , there exists $\tilde{\varphi}\in(\ell_{\infty}(L_{p}(\mathcal{M})))^{*}$ such that $\varphi(x)=\tilde{\varphi}(x)$ . Since the unit ball of $\ell_{1}^{N}(L_{p}(\mathcal{M}))$ is weak $*$ -dense in the unit ball of $(\ell_{\infty}(L_{p}(\mathcal{M})))^{*}$ and $x_{n}\geq 0$ , we conclude that for any $\varepsilon>0$ , there exists a $\varphi_{\varepsilon}\in(\ell_{\infty}(L_{p}(\mathcal{M})))^{*}$ in the form

\varphi_{\varepsilon}((x_{n}))=\sum_{n=1}\tau(x_{n}y_{n}),

such that $\varphi(x)=\tilde{\varphi}(x)\leq\varphi_{\varepsilon}(x)+\varepsilon$ and $(y_{n})_{n}\in\ell_{1}^{N}(L_{p}(\mathcal{M}))$ with $y_{n}\geq 0$ . On the other hand, we know from (i) that,

$\displaystyle\\|(y_{n})\\|_{L_{p}(\mathcal{M};\ell_{1})}$	$\displaystyle=$	$\displaystyle\sup_{\\|x\\|_{L_{p}(\mathcal{M};\ell_{\infty})}\leq 1}\|\tau(x_{n}y%_{n})\|$
	$\displaystyle\leq$	$\displaystyle\sup_{\\|x\\|_{\ell_{\infty}(L_{p})}\leq 1}\|\tau(x_{n}y_{n})\|$
	$\displaystyle\leq$	$\displaystyle\\|\varphi_{\varepsilon}\\|=1.$

We obtain

(2.15)

\|(x_{n})\|_{L_{p}(\mathcal{M};\ell_{\infty})}=\varphi(x)\leq\varphi_{%\varepsilon}(x)+\varepsilon\leq\sup\left\{|\tau\left(\sum x_{n}y_{n}\right)|:y%_{j}\geq 0,\|(y_{n})\|_{L_{p^{{}^{\prime}}}(\mathcal{M};\ell_{1}^{N})}\leq 1%\right\}+\varepsilon.

We then conclude (ii) by letting $\varepsilon\rightarrow 0$ .

We now prove the other direction of (iii).Note that $\ell_{1}(L_{p}(\mathcal{M}))$ is a sub-linear vector space of $L_{p}(\mathcal{M};\ell_{1})$ equipped with a larger norm. So, for any bounded linear functional $\varphi\in(L_{p}(\mathcal{M};\ell_{1}))^{*}$ , its restriction on $\ell_{1}(L_{p}(\mathcal{M}))$ defines a bounded linear functional $\tilde{\varphi}\in(\ell_{1}(L_{p}(\mathcal{M})))^{*}=\ell_{\infty}(L_{q}(%\mathcal{M}))$ . We conclude that there exists $x_{n}\geq 0,(x_{n})\in\ell_{\infty}(L_{q}(\mathcal{M}))$ such that

(2.16)

\displaystyle\varphi((y_{n}))=\sum_{n=1}^{\infty}\tau(x_{n}y_{n}),

for all $(y_{n})\in\ell_{1}(L_{p}(\mathcal{M}))\subset L_{p}(\mathcal{M};\ell_{1})$ . In particular, the expression (2.16) holds for any finite sequences $(y_{n})\in L_{p}(\mathcal{M};\ell_{1})$ . By (ii), we have

	$\displaystyle\\|(x_{n})\\|_{L_{q}(\mathcal{M};\ell_{\infty})}$	$\displaystyle=$	$\displaystyle\sup\left\{\tau\left(\sum_{n=1}x_{n}y_{n}\right);{\rm finite\ %sequences}(y_{n}),\\|y_{n}\\|_{L_{p}(\mathcal{M};\ell_{1})}\leq 1\right\}$
		$\displaystyle=$	$\displaystyle\sup\left\{\varphi((y_{n}));{\rm finite\ sequences}\ (y_{n}),\\|y_%{n}\\|_{L_{p}(\mathcal{M};\ell_{1})}\leq 1\right\}\leq\\|\varphi\\|_{(L_{p}(%\mathcal{M};\ell_{1}))^{*}}.$

∎

3. Operator-Maximal Inequality

Let $\mathcal{M}$ be a semifinite von Neumann algebra, e.g. $L_{\infty}(\Omega)$ or $B(H)$ . Let $L_{p}(\mathcal{M})$ be the associated noncommutative $L_{p}$ spaces, e.g. $L_{p}(\Omega)$ or the Schatten classes $S_{p}$ .Let $L_{p}(\mathbb{R},\mathcal{M}),1\leq p<\infty$ be the space of all $L_{p}(\mathcal{M})$ -valued Bochner-measurable functions $f$ on the real line such that

\|f\|_{L^{p}(\mathbb{R},L_{p}(\mathcal{M}))}=\left(\int_{\mathbb{R}}\|f(x)\|_{%p}^{p}dx\right)^{\frac{1}{p}}=\left(\int_{\mathbb{R}}\tau\left[|f(x)|^{p}%\right]dx\right)^{\frac{1}{p}}<\infty.

We prove the following operator Hardy-Littlewood maximal inequality for $f\in L_{p}(\mathbb{R},L_{p}(\mathcal{M}))$ where $2\leq p<\infty$ . The corresponding result for $p=\infty$ is trivial.

Theorem 3.1.

Given $f\in L_{p}(\mathbb{R},L_{p}(\mathcal{M}))$ for some $2\leq p<\infty$ , there exists a $L_{p}(\mathcal{M})$ -valued Bochner-measurable function $F$ such that

(i) $\frac{1}{2t}\int_{x-t}^{x+t}|f(y)|dy\leq F(x)$ as operators for all $t>0$ , i.e. $F-\frac{1}{2t}\int_{x-t}^{x+t}|f(y)|dy\geq 0$ almost everywhere.

(ii) There exists an absolute constant $c$ such that

(3.1)

\displaystyle\|F\|_{L^{p}(\mathbb{R},L_{p}(\mathcal{M}))}\leq c\|f\|_{L^{p}(%\mathbb{R},L_{p}(\mathcal{M}))}.

In order to prove main theorem, we prove the dual form of Theorem 3.1. Let $\mathcal{N}$ be the von Neumann algebra tensor product $L_{\infty}(\mathbb{R})\otimes\mathcal{M}$ equipped with the semifinite trace $\nu=\int\otimes\tau$ . Then, $L_{p}(\mathbb{R},\mathcal{M})$ coincides with the noncommutative $L_{p}$ spaces $L_{p}({\mathbb{N}})$ associated with the pair $({\mathbb{N}},\nu)$ for $1\leq p<\infty$ . Let $T_{n},n>0$ be the averaging operator on $L_{p}(\mathbb{R})\otimes L_{p}(\mathcal{M})$ defined by

\displaystyle(T_{n}f)(x)=\frac{1}{2^{n+1}}\int_{x-2^{n}}^{x+2^{n}}f(t)\,dt

It is easy to verify that $\{T_{n}\}_{n\in\mathbb{Z}}$ is a family of operators from $L_{2}(\mathcal{N})\to L_{2}(\mathcal{N})$ satisfying

•
$T_{n}=T_{n}^{*}$ ;
•
$T_{n}g\geq 0$ if $g\geq 0$ ;
•
$T_{n}T_{m}\leq 2T_{\sigma(m)}$ for $\sigma(m)=m+1$ and any $n\leq m$ .

Lemma 3.2.

For any finite sequence $g_{n}\in L_{2}(\mathcal{N})$ with all $g_{n}\geq 0$ , we have

(3.2)

\|(T_{n}g_{n})\|_{L_{2}(\mathcal{N};\ell_{1})}\leq 4\|(g_{n})\|_{L_{2}(%\mathcal{N};\ell_{1})}.

Proof.

Given a positive sequence $(g_{n})_{n}\in L_{2}({\mathbb{N}})$ with only finitely many non-zero terms and a bijection $\alpha$ on $\mathbb{Z}$ , we have that for $\sigma(m)=m+1$ ,

	$\displaystyle\left\\|\sum_{n}T_{n}g_{\alpha(n)}\right\\|_{L_{2}(\mathcal{N})}^{2}$	$\displaystyle=\nu\left(\sum_{n,m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)$
		$\displaystyle=\nu\left(\sum_{n<m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)+%\nu\left(\sum_{n\geq m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)$
	$\displaystyle({\rm by\ the\ tracial\ property\ of}\ \nu)$	$\displaystyle=\nu\left(\sum_{n<m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)+%\nu\left(\sum_{n\geq m}T_{m}g_{\alpha(m)}T_{n}g_{\alpha(n)}\right)$
		$\displaystyle=\nu\left(\sum_{n<m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)+%\nu\left(\sum_{n\leq m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)$

Note that $\nu\left(\sum_{n}T_{n}g_{\alpha(n)}T_{n}g_{\alpha(n)}\right)\geq 0$ . So, we have that

	$\displaystyle\left\\|\sum_{n}T_{n}g_{\alpha(n)}\right\\|_{L_{2}(\mathcal{N})}^{2}$	$\displaystyle\leq 2\nu\left(\sum_{n\leq m}T_{n}g_{\alpha(n)}T_{m}g_{\alpha(m)}\right)$
		$\displaystyle=2\nu\left(\sum_{n\leq m}g_{\alpha(n)}T_{n}T_{m}g_{\alpha(m)}\right)$
		$\displaystyle\leq 4~{}\nu\left(\sum_{n\leq m}g_{\alpha(n)}T_{\sigma(m)}g_{%\alpha(m)}\right).$

By the tracial property of $\nu$ , we have that $\nu(ab)=\nu(b^{\frac{1}{2}}ab^{\frac{1}{2}})\geq 0$ for any $a,b\geq 0$ . So

	$\displaystyle\left\\|\sum_{n}T_{n}g_{\alpha(n)}\right\\|_{L_{2}(\mathcal{N})}^{2}$	$\displaystyle\leq 4~{}\nu\left(\sum_{n,m}g_{\alpha(n)}T_{\sigma(m)}g_{\alpha(m%)}\right)$
		$\displaystyle\leq 4~{}\nu\left(\left(\sum_{n}g_{\alpha(n)}\right)\left(\sum_{m%}T_{\sigma(m)}g_{\alpha(m)}\right)\right)$
		$\displaystyle=4~{}\nu\left(\left(\sum_{n}g_{n})\right)\left(\sum_{m}T_{m}g_{%\sigma^{-1}\alpha(m)}\right)\right)$
		$\displaystyle\leq 4~{}\left\\|\sum_{n}g_{n}\right\\|_{L_{2}(\mathcal{N})}~{}%\left\\|\sum_{m}T_{m}g_{\sigma^{-1}\alpha(m)}\right\\|_{L_{2}(\mathcal{N})}.$

Now, taking the supremum over all bijections $\alpha$ on both sides, we get

	$\displaystyle\sup_{\alpha}\left\\|\sum_{n}T_{n}g_{\alpha(n)}\right\\|_{L_{2}(%\mathcal{N})}^{2}$	$\displaystyle\leq 4\left\\|\sum_{n}g_{n}\right\\|_{L_{2}(\mathcal{N})}\sup_{%\alpha}\left\\|\sum_{m}T_{m}g_{\sigma^{-1}\alpha(m)}\right\\|_{L_{2}(\mathcal{N})}$
		$\displaystyle=4\left\\|\sum_{n}g_{n}\right\\|_{L_{2}(\mathcal{N})}\sup_{\alpha}%\left\\|\sum_{m}T_{m}g_{\alpha(m)}\right\\|_{L_{2}(\mathcal{N})}$

By dividing the finite number $\sup_{\alpha}\|\sum_{m}T_{m}g_{\alpha(m)}\|_{L_{2}(\mathcal{N})}$ on both sides, we get (3.2).∎

Note that $\{T_{n}\}_{n\in\mathbb{Z}}$ is a family of positive-preserving contractions on $L_{1}(\mathcal{N})$ . Lemma 3.2 holds trivially if we replace $L_{2}({\mathbb{N}})$ with $L_{1}({\mathbb{N}})$ . We show that this remains true if we replace $L_{2}({\mathbb{N}})$ with $L_{p}({\mathbb{N}})$ for all $1<p<2$ .We need the following Cauchy-Schwartz inequality. We include a proof for completeness.

Lemma 3.3.

Suppose $a_{n}\in L_{q}(\mathcal{N},\nu),b_{n}\in L_{r}(\mathcal{N},\nu)$ . Then,we have

(3.3)

\left\|\sum_{n=1}^{N}T_{n}(a_{n}^{*}b_{n})\right\|_{p}\leq\left\|\sum_{n=1}^{N%}T_{n}(a_{n}^{*}a_{n})\right\|_{q}~{}\left\|\sum_{n=1}^{N}T_{n}(b_{n}^{*}b_{n}%)\right\|_{r}.

In particular,

(3.4)

\left\|\sum_{n=1}^{N}T_{n}(a_{n}^{*}b_{n})\right\|_{p}\leq\left\|\sum_{n=1}^{N%}T_{n}(a_{n}^{*}a_{n})\right\|_{p}~{}\left\|\sum_{n=1}^{N}T_{n}(b_{n}^{*}b_{n}%)\right\|_{p}.

Proof.

Let $X_{n}=\begin{pmatrix}a_{n}&b_{n}\\0&0\end{pmatrix}$ . Then we have

X_{n}^{*}X_{n}=\begin{pmatrix}a_{n}^{*}a_{n}&a_{n}^{*}b_{n}\\b_{n}^{*}a_{n}&b_{n}^{*}b_{n}\end{pmatrix}.

This implies that

\begin{pmatrix}\sum_{n=1}^{N}T_{n}(a_{n}^{*}a_{n})&\sum_{n=1}^{N}T_{n}(a_{n}^{%*}b_{n})\\\sum_{n=1}^{N}T_{n}(b^{*}a_{n})&\sum_{n=1}^{N}T_{n}(b_{n}^{*}b_{n})\end{%pmatrix}=\begin{pmatrix}\alpha&\gamma\\\gamma^{*}&\beta\end{pmatrix}\geq 0.

Then, by [1, Prop. 1.3.2], there exists a contraction $y$ such that $\gamma=\alpha^{\frac{1}{2}}y\beta^{\frac{1}{2}}$ .Thus by Hölder’s inequality,

\left\|\sum_{n=1}^{N}T_{n}(a_{n}^{*}b_{n})\right\|_{p}=\|\gamma\|_{p}=\|\alpha%^{\frac{1}{2}}y\beta^{\frac{1}{2}}\|_{p}\leq\|\alpha^{\frac{1}{2}}\|_{q}\|%\beta^{\frac{1}{2}}\|_{r}.

The lemma is proved.∎

Lemma 3.4.

Under the same assumption of Lemma 3.2, we have that,

(3.5)

\|(T_{n}g_{n})\|_{L_{p}(\mathcal{N};\ell_{1})}\leq 4^{2-\frac{2}{p}}\|(g_{n})%\|_{L_{p}(\mathcal{N};\ell_{1})}.

for all finite sequences $(g_{n})\in L_{p}(\mathcal{N};\ell_{1})$ , $g_{n}\geq 0$ , $1\leq p\leq 2$ .

Proof.

Assume that $\|\sum_{n=1}^{N}~{}g_{n}\|_{p}=1$ . Let $g=\left(\sum_{n=1}^{N}g_{n}\right)^{1/2}$ . We then have $\|g\|_{2p}=1.$ By approximation, we may assume $g$ is invertible.Let $h_{n}=g^{-1}g_{n}g^{-1}\geq 0$ . Then, $\|\sum_{n=1}^{N}h_{n}\|_{\infty}\leq 1$ . Let $\theta$ be defined by $\frac{1}{p}=\frac{1-\theta}{1}+\frac{\theta}{2}$ . Let

F(z)=g^{(1-z)p+zp/2}h_{n}g^{(1-z)p+zp/2}

and $U_{t}=g^{-ipt/2}$ . Note that $F(\theta)=g_{n}$ and $F(it)=U_{t}g^{p}h_{n}g^{p}U_{t}$ , $F(1+it)=U_{t}g^{p/2}h_{n}g^{p/2}U_{t}$ .Therefore,

	$\displaystyle\left\\|\sum_{n=1}^{N}T_{n}F(it)\right\\|_{1}$	$\displaystyle\leq\sum_{n=1}^{N}\left\\|T_{n}F(it)\right\\|_{1}\leq\sum_{n=1}^{N}%\left\\|F(it)\right\\|_{1}\leq\sum_{n=1}^{N}\\|g^{p}h_{n}g^{p}\\|_{1}$
		$\displaystyle=\nu\left(g^{p}\sum_{n=1}^{N}h_{n}~{}g^{p}\right)$
(3.6)			$\displaystyle\leq\nu(g^{2p})=1.$

On the other hand, applying Lemma 3.3 to $a_{n}^{*}=U_{t}g^{\frac{p}{2}}h_{n}^{\frac{1}{2}},b_{n}=h_{n}^{\frac{1}{2}}g^{%\frac{p}{2}}U_{t}$

	$\displaystyle\left\\|\sum_{n=1}^{N}T_{n}F(1+it)\right\\|_{2}=\left\\|\sum_{n=1}^{%N}~{}T_{n}\left(U_{t}g^{\frac{p}{2}}h_{n}^{\frac{1}{2}}h_{n}^{\frac{1}{2}}a^{%\frac{p}{2}}U_{t}\right)\right\\|_{2}$
	$\displaystyle\leq\left\\|\sum_{n=1}^{N}T_{n}\left(U_{t}g^{p/2}h_{n}^{1/2}h_{n}^%{1/2}g^{g/2}U_{t}^{}\right)\right\\|_{2}^{\frac{1}{2}}\left\\|\sum_{n=1}^{N}T_{%n}\left(U_{t}^{}a^{p/2}h_{n}^{1/2}h_{n}^{\frac{1}{2}g^{p/2}}U_{t}\right)%\right\\|_{2}^{\frac{1}{2}}$
	$\displaystyle\leq 4\left\\|U_{t}\left(g^{p/2}\sum_{n=1}^{N}~{}h_{n}~{}g^{p/2}%\right)U_{t}^{}\right\\|_{2}^{\frac{1}{2}}\left\\|U_{t}^{}\left(g^{p/2}\sum_{n%=1}^{N}~{}h_{n}~{}g^{p/2}\right)U_{t}\right\\|_{2}^{\frac{1}{2}}$
(3.7)		$\displaystyle\leq 4\\|g^{p}\\|_{2}=4.$

Then, by the three line lemma, we have

\left\|\sum_{n=1}^{N}T_{n}g_{n}\right\|_{p}=\left\|\sum_{n=1}^{N}~{}T_{n}F(%\theta)\right\|_{p}\leq 4^{\theta}.

We complete the proof by applying the hom*ogeneity property.∎

Finally, we return to the proof of Theorem 3.1 by duality.

Proof.

For $f\in L_{p}({\mathbb{N}})$ , we have that $|f|\in L_{p}({\mathbb{N}})$ and $|f|$ has the same norm with $f$ by definition.We apply (2.13) to the positive sequence in $L_{p}({\mathbb{N}})$ $T_{n}(|f|)$ , and obtain

	$\displaystyle\\|(T_{n}(\|f\|))\\|_{L_{p}(\mathcal{N};\ell_{\infty})}$	$\displaystyle=\sup\left\{\nu\left(\sum T_{n}(\|f\|)y_{n}\right):y_{n}\geq 0,\\|(y%_{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 1\right\}$
		$\displaystyle=\sup\left\{\nu\left(\sum\|f\|T_{n}(y_{n})\right):y_{n}\geq 0,\\|(y_%{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 1\right\}$
		$\displaystyle\leq\sup\left\{\nu\left(\|f\|\sum T_{n}y_{n}\right):z_{n}\geq 0,\\|(%z_{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 4^{2-\frac{2}{q}}\right\}$
		$\displaystyle\leq 4^{\frac{2}{p}}\\|f\\|_{L_{p}(\mathcal{N})},$

for all $2\leq p<\infty$ . By (2.11), $\|(T_{n}(|f|))\|_{L_{p}(\mathcal{N};\ell_{\infty})}\leq 4^{1+\frac{2}{p}}\|f\|%_{L_{p}(\mathcal{N})}$ . By definition (2.4), This means that there exists $F\in L_{p}({\mathbb{N}})$ such that $\|F\|_{L_{p}({\mathbb{N}})}\leq 4^{1+\frac{2}{p}}\|f\|_{L_{p}(\mathcal{N})}$ and $T_{n}|f|\leq F$ . Theorem 3.1 follows since

\frac{1}{2t}\int_{x-t}^{x+t}|f(y)|dy\leq 2T_{n}|f|

for every $2^{n-1}\leq t<2^{n},n\in\mathbb{Z}$ .∎

References

[1] R. Bhatia, Positive definite matrices,Princeton Ser. Appl. Math.Princeton University Press, Princeton, NJ, 2007, x+254 pp.
[2]Z. Chen, Q. Xu, Z. Yin, Harmonic analysis on quantum tori,Comm. Math. Phys. 322 (2013), no. 3, 755-805.
[3]G. Hong, The behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^{n}$ for large $n$ ,Illinois J. Math. 57 (2013), no. 3, 855-869.
[4]G. Hong, B. Liao, S. Wang, Noncommutative maximal ergodic inequalities associated with doubling conditionsDuke Math. J. 170 (2021), no. 2, 205-246.
[5] M. Junge, Doob’s Inequality for noncommutative Martingales, J. Reine Angew. Math. 549 (2002), 149-190. MR1916654
[6] M. Junge, Q. Xu, Noncommutative maximal ergodic theorems.J. Amer. Math. Soc.20(2007), no.2, 385-439.
[7]T. Mei, Operator valued Hardy spaces. Mem. Amer. Math. Soc. 188 (2007), 64 pp.
[8] Fourier multipliers in $SL_{n}(\mathbb{R})$ ,J. Parcet, E. Ricard, M. de la Salle, Duke Math. J. 171 (2022), no. 6, 1235-1297.
[9] G. Pisier, noncommutative Vector Valued $L_{p}$ -Spaces and Completely p-Summing Maps, Soc. Math. France Astérisque (1998) 237. MR1648908
[10] G. Pisier, Martingales in Non-Commutative $L_{p}$ -spaces, unpublished manuscript.
[11] G. Pisier, Q. Xu, Non-commutative $L_{p}$ -spaces. In Handbook of the geometry of Banach spaces, Vol. 2, pages 1459–1517. North-Holland, Amsterdam, 2003. MR1999201
[12] E. M. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, New Jersey, 1993. MR1232192

A Direct Proof for Operator Hardy-Littlewood Maximal Inequality (2024)

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	$\displaystyle\left\|\sum_{j}\tau(az_{j}bu_{j}v_{j})\right\|$	$\displaystyle=\left\|\sum_{j}\tau(z_{j}bu_{j}v_{j}a)\right\|\leq\sum_{j}\\|bu_{j}%v_{j}a\\|_{1}$
		$\displaystyle\leq\sum_{j}\\|bu_{j}\\|_{2}\\|v_{j}a\\|_{2}$
		$\displaystyle\leq\left(\sum_{j}\\|bu_{j}\\|_{2}^{2}\right)^{\frac{1}{2}}\left(%\sum_{j}\\|v_{j}a\\|_{2}^{2}\right)^{\frac{1}{2}}$
		$\displaystyle=\left(\tau(b\sum_{j}u_{j}u_{j}^{}b^{})\right)^{\frac{1}{2}}%\left(\tau(a^{}\sum_{j}v_{j}v_{j}^{}a)\right)^{\frac{1}{2}}$
		$\displaystyle\leq\\|b^{}b\\|^{\frac{1}{2}}_{q}\\|\sum_{j}u_{j}u_{j}^{}\\|^{\frac%{1}{2}}_{p}\\|aa^{}\\|^{\frac{1}{2}}_{q}\\|\sum_{j}v_{j}v_{j}^{}\\|^{\frac{1}{2}%}_{p}$
		$\displaystyle\leq\\|b\\|_{2q}\\|(\sum_{j}u_{j}u_{j}^{})^{\frac{1}{2}}\\|_{p}\\|a\\|%_{2q}\\|(\sum_{j}v_{j}v_{j}^{})^{\frac{1}{2}}\\|_{p}$

$\displaystyle\\|(y_{n})\\|_{L_{p}(\mathcal{M};\ell_{1})}$	$\displaystyle=$	$\displaystyle\sup_{\\|x\\|_{L_{p}(\mathcal{M};\ell_{\infty})}\leq 1}\|\tau(x_{n}y%_{n})\|$
	$\displaystyle\leq$	$\displaystyle\sup_{\\|x\\|_{\ell_{\infty}(L_{p})}\leq 1}\|\tau(x_{n}y_{n})\|$
	$\displaystyle\leq$	$\displaystyle\\|\varphi_{\varepsilon}\\|=1.$

	$\displaystyle\\|(x_{n})\\|_{L_{q}(\mathcal{M};\ell_{\infty})}$	$\displaystyle=$	$\displaystyle\sup\left\{\tau\left(\sum_{n=1}x_{n}y_{n}\right);{\rm finite\ %sequences}(y_{n}),\\|y_{n}\\|_{L_{p}(\mathcal{M};\ell_{1})}\leq 1\right\}$
		$\displaystyle=$	$\displaystyle\sup\left\{\varphi((y_{n}));{\rm finite\ sequences}\ (y_{n}),\\|y_%{n}\\|_{L_{p}(\mathcal{M};\ell_{1})}\leq 1\right\}\leq\\|\varphi\\|_{(L_{p}(%\mathcal{M};\ell_{1}))^{*}}.$

	$\displaystyle\left\\|\sum_{n=1}^{N}T_{n}F(1+it)\right\\|_{2}=\left\\|\sum_{n=1}^{%N}~{}T_{n}\left(U_{t}g^{\frac{p}{2}}h_{n}^{\frac{1}{2}}h_{n}^{\frac{1}{2}}a^{%\frac{p}{2}}U_{t}\right)\right\\|_{2}$
	$\displaystyle\leq\left\\|\sum_{n=1}^{N}T_{n}\left(U_{t}g^{p/2}h_{n}^{1/2}h_{n}^%{1/2}g^{g/2}U_{t}^{}\right)\right\\|_{2}^{\frac{1}{2}}\left\\|\sum_{n=1}^{N}T_{%n}\left(U_{t}^{}a^{p/2}h_{n}^{1/2}h_{n}^{\frac{1}{2}g^{p/2}}U_{t}\right)%\right\\|_{2}^{\frac{1}{2}}$
	$\displaystyle\leq 4\left\\|U_{t}\left(g^{p/2}\sum_{n=1}^{N}~{}h_{n}~{}g^{p/2}%\right)U_{t}^{}\right\\|_{2}^{\frac{1}{2}}\left\\|U_{t}^{}\left(g^{p/2}\sum_{n%=1}^{N}~{}h_{n}~{}g^{p/2}\right)U_{t}\right\\|_{2}^{\frac{1}{2}}$
(3.7)		$\displaystyle\leq 4\\|g^{p}\\|_{2}=4.$

	$\displaystyle\\|(T_{n}(\|f\|))\\|_{L_{p}(\mathcal{N};\ell_{\infty})}$	$\displaystyle=\sup\left\{\nu\left(\sum T_{n}(\|f\|)y_{n}\right):y_{n}\geq 0,\\|(y%_{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 1\right\}$
		$\displaystyle=\sup\left\{\nu\left(\sum\|f\|T_{n}(y_{n})\right):y_{n}\geq 0,\\|(y_%{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 1\right\}$
		$\displaystyle\leq\sup\left\{\nu\left(\|f\|\sum T_{n}y_{n}\right):z_{n}\geq 0,\\|(%z_{n})\\|_{L_{q}(\mathcal{N};\ell_{1}^{N})}\leq 4^{2-\frac{2}{q}}\right\}$
		$\displaystyle\leq 4^{\frac{2}{p}}\\|f\\|_{L_{p}(\mathcal{N})},$

A Direct Proof for Operator Hardy-Littlewood Maximal Inequality (2024)

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

2. Preliminary

2.1. Noncommutative Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT spaces

2.2. Pisier’s Lp⁢(ℳ;ℓ∞)subscript𝐿𝑝ℳsubscriptℓL_{p}(\mathcal{M};\ell_{\infty})italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( caligraphic_M ; roman_ℓ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ) norm

Lemma 2.1 (Pisier [9]).

Proof.

Lemma 2.2 ([9, 5]).

Proof.

3. Operator-Maximal Inequality

Theorem 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Proof.

References

FAQs

What is the best constant for the centered Hardy-Littlewood maximal inequality? ›

What are the famous continuous functions? ›

2.1. Noncommutative $L_{p}$ spaces

2.2. Pisier’s $L_{p}(\mathcal{M};\ell_{\infty})$ norm