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What is the Hardy and Littlewood theorem?

. The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform . The theorem was proved in 1914 by G. H.

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What are the applications of Hardy Littlewood maximal function?

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Is Hardy Littlewood maximal function measurable?

The averages are jointly continuous in x and r, so the maximal function Mf, being the supremum over r > 0, is measurable . It is not obvious that Mf is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality.

How do you argue a function is continuous?

Key Concepts For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. Discontinuities may be classified as removable, jump, or infinite. More items... Dec 20, 2020

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Some Typical Continuous Functions Trigonometric Functions in certain periodic intervals (sin x, cos x, tan x etc.) Polynomial Functions (x 2 +x +1, x 4 + 2…. etc.) Exponential Functions (e 2x , 5e x etc.) Logarithmic Functions in their domain (log 10 x, ln x 2 etc.)

Are all holomorphic functions continuous?

If there exists a holomorphic function F defined on Ω such that F0 = f, we say that F is a primitive of f. If f is holomorphic in all of C then f is said to be entire. Like in real variable theory we find that f is continuous on an open set Ω if it is holomorphic on Ω .

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sin(x) , cos(x) , arctan(x)=tan−1(x) , 11+x2 , and 11+ex are all commonly used examples of bounded functions.

How do you know if a function is bounded?

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What functions are not bounded?

Unbounded function : A function is unbounded if there exists a real number , such that lim x → a − | f ( x ) | = ∞ , or lim x → a + | f ( x ) | = ∞ , or lim x → − ∞ | f ( x ) | = ∞ or lim x → ∞ | f ( x ) | = ∞ .

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 $g=\left(\sum_{n=1}^{N}g_{n}\right)^{1/2}$

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

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

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

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.

2.1. Noncommutative $L_{p}$ spaces

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