Vector-valued $q$-variational inequalities for averaging operators and Hilbert transform

TL;DR

This paper investigates the conditions under which vector-valued $q$-variational inequalities hold for averaging operators and the Hilbert transform, establishing necessity of martingale cotype $q$ and characterizing UMD and cotype properties.

Contribution

It proves that martingale cotype $q$ is necessary for vector-valued $q$-variational inequalities and characterizes UMD and cotype properties via these inequalities.

Findings

Martingale cotype $q$ is necessary for inequalities.

Characterization of UMD and cotype properties.

Extension of previous $L^p$-boundedness results.

Abstract

Recently, in \cite{GXHTM}, the authors established $L^p$-boundedness of vector-valued $q$-variational inequalities for averaging operators which take values in the Banach space satisfying martingale cotype $q$ property. In this paper, we prove that martingale cotype $q$ property is also necessary for the vector-valued $q$-variational inequalities, which is a question left open. Moreover, we characterize UMD property and martingale cotype $q$ property in terms of vector valued $q$-variational inequalities for Hilbert transform.

Equations

$$V_{q}(\mathfrak{a}_{t}(x):t\in Z)=\sup_{\begin{subarray}{c}0<t_{0}<\cdots<t_{J}\\ t_{j}\in Z\end{subarray}}\bigg{(}\sum_{j=0}^{J}\|\mathfrak{a}_{t_{j+1}}(x)-\mathfrak{a}_{t_{j}}(x)\|^{q}\bigg{)}^{1/q},$$

$$\sum _{n\ge 0}{\Vert {\mathbb{E}}_{n}f-{\mathbb{E}}_{n-1}f\Vert }_{L^q(\Omega ;X)}^q\le C_q{\Vert f\Vert }_{L^q(\Omega ;X)}^q,$$

$$\Vert V_q({\mathbb{E}}_{k}f:k\in \mathbb{N}){\Vert }_{L^p}\le C_{p,q,X}\Vert f{\Vert }_{L^p},\forall f\in L^p(\Omega ;X);$$

$$A_{t}f(x)=\frac{1}{|B_t|}{\int }_{B_t}f(x-y)dy=\frac{1}{|B_t|}{\int }_{{\mathbb{R}}^d}f(y)\text{mathds}{1}_{B_t}(x-y)dy,x\in \mathbb{R},$$

$${\Vert V_q(A_{t}f:t>0)\Vert }_{L^p}\le C_{p,q}{\Vert f\Vert }_{L^p},\forall f\in L^p(\mathbb{R};X);$$

$${\Vert V_q(A_{t}f:t>0)\Vert }_{L^p}\le C_{p,q}{\Vert f\Vert }_{L^p},\forall f\in L^p(\mathbb{R};X),$$

$$\Vert \sum _{n=1}^N{r}_{n}d{f}_n){\Vert }_{L^p(\Omega ;X)}\le C_{p,X}\Vert \sum _{n=1}^{N}d{f}_n{\Vert }_{L^p(\Omega ;X)}.$$

$$H_{ϵ}f(x)=\frac{1}{\pi }{\int }_{|y|>ϵ}\frac{f(x-y)}{y}dy,$$

$${\Vert V_q(H_{ϵ}f:ϵ>0)\Vert }_{L^p}\le C_{p,q}{\Vert f\Vert }_{L^p},\forall f\in L^p(\mathbb{R};X);$$

$${\Vert V_q(H_{ϵ}f:ϵ>0)\Vert }_{L^p}\le C_{p,q}{\Vert f\Vert }_{L^p},\forall f\in L^p(\mathbb{R};X),$$

$$\mathbb{P}_{t}f(\xi)=\mathbb{P}_{t}\ast f(\xi)=\int_{0}^{1}f(e^{2\pi i\theta})\bigg{(}\sum_{n}e^{-t|n|}e^{2\pi in(\xi-\theta)}\bigg{)}d\theta.$$

$$a_k$$

$$=\sum _{|m_1|\le N_1}\cdots \sum _{|m_{k-1}|\le N_{k-1}}{x}_{m_1,\cdots ,m_{k-1}}{e}^{2\pi i{m}_1{\theta }_1}\cdots e^{2\pi i{m}_{k-1}{\theta }_{k-1}},$$

$$b_k(e^{2\pi i{\theta }_k})=\sum _{1\le |j|\le M_k}{y}_j{e}^{2\pi ij{\theta }_k}.$$

$$a_{k,(n)}(\theta )$$

$$b_{k,(n)}(\theta )$$

$$f(\theta )$$

$$=\sum _{k=1}^m{f}_{k,(n)}(\theta ).$$

$$\begin{split}f_{k,(n)}(\theta)&=\sum_{|m_{1}|\leq N_{1},\cdots,|m_{k-1}|\leq N_{k-1}}\sum_{1\leq|j|\leq M_{k}}\big{(}x_{m_{1},\cdots,m_{k-1}}y_{j}\\ &\times e^{2\pi i(m_{1}\theta_{1}+\cdots+m_{k-1}\theta_{k-1}+j\theta_{k})}e^{2\pi i(m_{1}n_{1}+\cdots+m_{k-1}n_{k-1}+jn_{k})\theta}\big{)};\end{split}$$

$$\begin{split}\mathbb{P}_{t}f_{k,(n)}(\theta)=\sum_{m_{i}}\sum_{j}\big{(}&x_{m_{1},\cdots,m_{k-1}}y_{j}e^{2\pi i(m_{1}\theta_{1}+\cdots+m_{k-1}\theta_{k-1}+j\theta_{k})}\times\\ &e^{-|m_{1}n_{1}+\cdots+m_{k-1}n_{k-1}+jn_{k}|t}e^{2\pi i(m_{1}n_{1}+\cdots+m_{k-1}n_{k-1}+jn_{k})\theta}\big{)}.\end{split}$$

$$\Vert {\mathbb{P}}_t{f}_{k,(n)}(\theta )\Vert <\frac{ϵ}{2^k},\forall t\ge l_{k-1},$$

$$\sum _{j=1}^k\Vert {\mathbb{P}}_{l_k}{f}_{j,(n)}(\theta )-{\mathbb{P}}_t{f}_{j,(n)}(\theta )\Vert <ϵ.$$

$$\Vert V_q({\mathbb{P}}_{t}f:t>0){\Vert }_{L^p}\le C_{p,q}\Vert f{\Vert }_{L^p},\forall f\in L^p([0,1];X).$$

$$\sum _{k=1}^m\mathbb{E}\Vert M_k-M_{k-1}{\Vert }^q\le C_q\underset{k}{\sup }\mathbb{E}\Vert M_k{\Vert }^q,$$

$$M_k-M_{k-1}={\varphi }_k({\epsilon }_1,\cdots ,{\epsilon }_{k-1}){\epsilon }_k,1\le k\le m,$$

$$\begin{array}{rl}& a_k(e^{2\pi i{\theta }_1},\cdots ,e^{2\pi i{\theta }_{k-1}})={\varphi }_k(sgn(cos2\pi {\theta }_1),\cdots ,sgn(cos2\pi {\theta }_{k-1})),\\ & b_k(e^{2\pi i{\theta }_k})=sgn(cos2\pi {\theta }_k).\end{array}$$

$$\Vert \sum _{k=1}^m({\mathbb{P}}_{l_1}{f}_{k,(n)}-{\mathbb{P}}_{l_0}{f}_{k,(n)})-f_{1,(n)}\Vert \le \Vert {\mathbb{P}}_{l_1}{f}_{1,(n)}-f_{1,(n)}\Vert +\Vert \sum _{k=2}^m{\mathbb{P}}_{l_1}{f}_{k,(n)}\Vert ,$$

$$\begin{array}{r}\Vert {\mathbb{P}}_{l_1}{f}_{1,(n)}-f_{1,(n)}\Vert <ϵ,\Vert \sum _{k=2}^m{\mathbb{P}}_{l_1}{f}_{k,(n)}\Vert \le \sum _{k=2}^m\frac{ϵ}{2^k}<ϵ.\end{array}$$

$$\Vert \sum _{k=1}^m({\mathbb{P}}_{l_1}{f}_{k,(n)}-{\mathbb{P}}_{l_0}{f}_{k,(n)})-f_{1,(n)}\Vert \le ϵ.$$

$$\Vert \sum _{k=1}^m({\mathbb{P}}_{l_i}{f}_{k,(n)}-{\mathbb{P}}_{l_{i-1}}{f}_{k,(n)})-f_{i,(n)}\Vert$$

$$\le \Vert \sum _{k=1}^{i-1}({\mathbb{P}}_{l_i}{f}_{k,(n)}-{\mathbb{P}}_{l_{i-1}}{f}_{k,(n)})\Vert +\Vert \sum _{k=i}^m({\mathbb{P}}_{l_i}{f}_{k,(n)}-{\mathbb{P}}_{l_{i-1}}{f}_{k,(n)})-f_{i,(n)}\Vert$$

$$=I+II.$$

$$I<\sum _{k=1}^{i-1}\Vert {\mathbb{P}}_{l_i}{f}_{k,(n)}-{\mathbb{P}}_{l_{i-1}}{f}_{k,(n)}\Vert <ϵ.$$

Full text

2000 Mathematics Subject Classification: Primary: 42B20, 42B25. Secondary: 46E30

Key words: variational inequalities, averaging operators, Hilbert transform, martingale cotype $q$, UMD

1. Introduction

It is well known that in the setting of vector-valued harmonic analysis, many results are related to the properties of geometry of Banach spaces. In particular, the so-called UMD, and convexity properties of Banach spaces have become the most fundamental tools in the study of vector-valued harmonic analysis. The boundedness of vector-valued singular integral operators is an important problem in this area of research. Burkholder [6] first proved that whenever a Banach space $X$ has the UMD property, the Hilbert transform is bounded on $L^{p}({\mathbb{R}}^{d};X)$ for any $1<p<\infty$. Later, Bourgain [2] established that the UMD property is indeed necessary for the boundedness of the Hilbert transform. Hytönen [16] proved that two-sided Littlewood-Paley-Stein $g$-function estimate is equivalent to the UMD property of Banach space $X$. Moreover, martingale cotype $q$ plays an important role in vector-valued analysis. The martingale cotype $q$ was introduced by Pisier [35] to characterise the properties of uniformly convex Banach spaces. In [37], Xu provided a characterization of the martingale cotype $q$ property via the vector-valued Littlewood-Paley theory associated to the Poisson kernel on the unit circle. Later on, Martínez et al [31] further characterized this property through general Littlewood-Paley-Stein theory. We refer [19] for an extensive study on vector-valued harmonic analysis.

On the other hand, variational inequalities have gained a lot of attention in past decades for many reasons, one of which is that we can measure the speed of convergence for the family of operators under consideration. In 1976, Lépingle [28] applied the regularity of Brownian motion to obtain the first scalar-valued variational inequality for martingales. Ten years later, motivated by Banach space geometry, Pisier and Xu [36] provided another proof of Lépingle’s result based on stopping time argument. Almost in the same time Bourgain [3] used Lépingle’s results to establish variational inequalities for the ergodic averages of a dynamical system. Bourgain’s work and Pisier-Xu’s method have inspired a new research direction in ergodic theory and harmonic analysis. In [20, 21, 22], Jones et al. proved variational inequalities for ergodic averages. Campbell et al. in [7] established the variational inequalities for Hilbert transform and averaging operators, and truncated singular integrals with the homogeneous type kernel [8]. Recently, Ding et al. [10] obtained the variational inequalities for truncated singular integrals with the rough type kernel. We recommend [4, 5, 23, 9, 8, 11, 12, 17, 24, 25, 29, 34, 32, 26, 33, 14, 30, 27, 18] and references therein for related work.

Among these works, some weighted norm and Banach lattice-valued variational inequality in harmonic analysis have also been established. It is worthy to mention that the first and third author [15] obtained the Banach space-valued variational inequalities for ergodic averages or averaging operators whenever the underlying Banach spaces are of martingale cotype $q$, while Hytönen et al [18] proved variational inequality for vector valued Walsh-Fourier series when the Banach spaces belong to a subclass of UMD spaces verifying tile-type $q$. In these two papers, finite cotype (or type) is also shown necessary for variational inequalities. However, regarding the sharp necessary conditions, they are still known to the best of the authors’ knowledge.

To further motivate the purpose of the present paper, let us recall Pisier-Xu’s result [36]. Let us start with some notations. Let $X$ be a Banach space and $\mathfrak{a}_{t}(x):(0,\infty)\times{\mathbb{R}}^{d}\rightarrow X$ be a vector-valued function. For $q\in[1,\infty)$, the seminorm $V_{q}$ is defined by

[TABLE]

where $Z$ is a subset of $(0,\infty)$ and the supremum runs over all finite increasing sequences in $Z$. Let $(\Sigma_{n})_{n\geq 0}$ be an increasing sequence of $\sigma$-algebras on a probability space $(\Omega,\Sigma,\mathbb{P})$. The associated conditional expectations are denoted by $(\mathbb{E}_{n})_{n\geq 0}$. We say that the Banach space $X$ is of martingale cotype $q$ with $2\leq q<\infty$, if there exists an absolute constant $C_{q}$ such that for all $f\in L^{q}(\Omega;X)$

[TABLE]

with convention $\mathbb{E}_{-1}f=0$.

In [36], Pisier and Xu proved the following vector-valued variational inequalities for martingales.

Theorem A**.**

Let $X$ be a Banach space.

1. (i)

If $X$ is of martingale cotype $q_{0}$ with $2\leq q_{0}<\infty$, then for every $1<p<\infty$ and $q>q_{0}$, there exists a positive constant $C_{p,q,X}$ such that

[TABLE] 2. (ii)

Suppose for some $p\in(1,\infty)$ and $q\in[2,\infty)$ the inequality (1.1) holds for a Banach space $X$. Then $X$ is of martingale cotype $q$.

Note that the above conclusion (ii) is quite easy in the martingale setting. Indeed, using standard extrapolation argument, we can assume that the inequality (1.1) holds for $p=q$. Then the conclusion follows trivially from the definitions. The situation changes dramatically in harmonic analysis.

Let $f:{\mathbb{R}}\rightarrow X$ be locally integrable. Let $B_{t}$ be an open ball centered at the origin and its radius equal to $t>0$. Then the central averaging operators are defined as follow

[TABLE]

where $|\cdot|$ denotes the Lebesgue measure.

As mentioned before, the authors in [15, Theorem 2.1] obtained the $L^{p}$-boundedness of vector-valued variational inequalities for averaging operators when the underlying Banach spaces satisfy martingale cotype property. But they could not obtain results similar to (ii) of Theorem A. In this paper, we do it.

Theorem 1.1**.**

Let $X$ be a Banach space.

1. (i)

If $X$ is of martingale cotype $q_{0}$ with $2\leq q_{0}<\infty$. Then for any $1<p<\infty$ and $q_{0}<q<\infty$, there exists a constant $C_{p,q}$ such that

[TABLE] 2. (ii)

Let $2\leq q<\infty$, and for some $p\in(1,\infty)$, there exists a positive constant $C_{p,q}$ such that

[TABLE]

then $X$ is of martingale cotype $q$.

Conclusion (i) has been obtained in [15]. To show conclusion (ii), we use several times the transference techniques and follows essentially Xu’s argument in his remarkable paper [37] based on Bourgain’s transference method. Many modifications are surely necessary in the present setting, see Section 2 for details.

Before stating our second result, we first introduce some notations. Recall that a Banach space $X$ is said to have the UMD property if for all scalars $r_{n}=\pm 1,~{}n=1,\cdots,N$, there exists a positive constant independent of $N$ such for all $1<p<\infty$ and all $X$-valued $L^{p}$-martingale differences $(df_{n})_{n\geq 1}$,

[TABLE]

Let $\mathcal{S}({\mathbb{R}})$ be the set of Schwartz functions on ${\mathbb{R}}$. Define $f\in\mathcal{S}({\mathbb{R}})\bigotimes X\subseteq L^{p}({\mathbb{R}};X)$ by setting $f(x)=\sum^{n}_{k=1}\varphi_{k}(x)b_{k}$, where $\varphi_{1},...,\varphi_{n}\in\mathcal{S}({\mathbb{R}})$ and $b_{1},...,b_{n}\in X$. For $f\in\mathcal{S}({\mathbb{R}})\otimes X$, the truncated Hilbert transform is defined by

[TABLE]

The second result of this paper is on vector-valued variational inequalities for Hilbert transform.

Theorem 1.2**.**

Let $X$ be a Banach space.

1. (i)

Let $2\leq q_{0}<\infty$, and $X$ be of martingale cotype $q_{0}$ satisfying UMD property. Then for any $1<p<\infty$ and $q_{0}<q<\infty$, there exist a constant $C_{p,q}$ such that

[TABLE] 2. (ii)

Let $2\leq q<\infty$. If for some $p\in(1,\infty)$, there exists a positive constant $C_{p,q}$ such that

[TABLE]

then $X$ satisfies UMD property and is of martingale cotype $q$.

With Theorem 1.1 at hand, the proof of Theorem 1.2 is relatively easy but some of the argument seems completely new. We use again several times the transference techniques for both (i) and (ii), see Section 3 for details.

Throughout this paper, by $C$ we always denote a positive constant that may vary from line to line.

2. the proof of Theorem 1.1

In this section, we follow closely Xu’s argument [37] to prove Theorem 1.1. The main ingredient is Bourgain’s transferenece method. We first introduce some notations.

Let $\mathbb{P}_{t}(\theta)=\sum_{n}e^{-t|n|}e^{2\pi in\theta}$ be the Poisson kernel of the unit disc. For $f\in L^{p}([0,1];X)$, denote the Poisson integral

[TABLE]

Let $m$ be a positive integer and $1\leq k\leq m$. Let $a_{k}(e^{2\pi i\theta_{1}},\cdots,e^{2\pi i\theta_{k-1}})$ be a trigonometric polynomial defined on $[0,1]^{k-1}$ with value in Banach space $X$,

[TABLE]

with convention $a_{k}=x_{0}$, when $k=1$. Let $b_{k}$ be a trigonometric polynomial with complex coefficients such that $\hat{b}(0)=0$,

[TABLE]

Let $\{n_{k}\}_{1\leq k\leq m}$ be a sequence of positive integers. Set

[TABLE]

In the following we fix $\theta_{1},\cdots,\theta_{m}$. All these functions are considered as trigonometric polynomials in $e^{i\theta}$. By the definition of $a_{k,(n)}(\theta)$ and $b_{k,(n)}(\theta)$, we have

[TABLE]

and

[TABLE]

The following lemma is a discrete version of Xu’s [37] Lemma 3.5.

Lemma 2.1**.**

Let $a_{k,(n)}$, $b_{k,(n)}$ and $f_{k,(n)}$ are defined as above. For any $\epsilon>0$, we can choose an increasing sequence $\{n_{k}\}_{1\leq k\leq m}$ of positive integers and a decreasing sequence $\{{l_{k}}\}_{0\leq k\leq{m}}\subset[0,+\infty]$ such that for all $\theta\in[0,1]$

[TABLE]

and for any $0\leq t\leq l_{k}$

[TABLE]

Proof.

First we set $l_{0}=+\infty$ and $n_{1}=1$. Suppose we have already defined $+\infty=l_{0}>l_{1}>\cdots>l_{k-1}$ and $1=n_{1}<\cdots<n_{k}$ with the required properties. Now, we define $l_{k}$ and $n_{k+1}$ as the following.

Since $f$ is a trigonometric polynomial, then $\mathbb{P}_{t}\ast f$ uniformly convergence to $f$ as $t\rightarrow 0^{+}$ on $\theta\in[0,1]$. Hence we can choose an $l_{k}$ sufficiently close to [math] and for any $0<t<l_{k}$ (2.3) holds. Next, we choose $n_{k+1}$ such that (2.2) at the step $k+1$ holds.

According to the definition $f_{k,n}$, we know that there is a finite integer $N$ such that $|m_{1}n_{1}+\cdots+m_{k}n_{k}|<N$ for all $|m_{1}|\leq N_{1},\cdots,|m_{k}|\leq N_{k}$. On the other hand, since $\hat{b}_{k+1}(0)=0$, if $|n_{k+1}|$ is sufficiently big (precisely, $n_{k+1}\in\mathbb{Z}\setminus[-N,N]$), then $|m_{1}n_{1}+\cdots+m_{k}n_{k}+jn_{k+1}|>0$ for all $|m_{1}|\leq N_{1},\cdots,|m_{k}|\leq N_{k}$ and $1\leq|j|\leq M$. Then by (2.1), we can choose $n_{k+1}$ such that (2.2) holds at step $k+1$.

Finally, let $l_{m}=0$, then we have set up the required sequences $(n_{k})_{1\leq k\leq m}$ and $(l_{k})_{0\leq k\leq{m}}$. ∎

The following lemma is cited from [15].

Lemma 2.2**.**

Let $X$ be a Banach space. If inequality (1.4) holds for some $p$, then for every $1<p<\infty$ we have

[TABLE]

By the method of the proof of Theorem 3.1 in [37], we now prove Theorem 1.1.

Proof of Theorem 1.1.

The first and third author proved the part (i) in [15, Theorem 2.1]. Hence, we only need to prove the part (ii).

Let $\Omega=\{-1,1\}^{\mathbb{N}}$ be the dyadic group equipped with its normalized Haar measure $P$. Let $m$ be a positive integer. According to [35], we know that to prove the Banach space $X$ is of martingale cotype $q$, it suffices to prove that there exists a constant $C$ such that for all finite Walsh-Paley martingale $M=(M_{k})_{1\leq k\leq m}$

[TABLE]

where denote $M_{-1}=0$.

Fix such a martingale $M=(M_{k})_{1\leq k\leq m}$. Let $(\varepsilon_{k})_{1\leq k\leq m}$ be the coordinate sequence of $\Omega$. By the representation of Walsh-Paley martingales (cf. [19]), we know that there exist functions $\phi_{k}:\{-1,1\}^{k-1}\rightarrow X$ such that

[TABLE]

where $\varepsilon_{k}\in\{-1,1\}^{k}$ and $\phi_{1}$ is a constant belong to $X$.

Following [1], we use transference from $\Omega$ to the group $[0,1]^{\mathbb{N}}$. For $1\leq k\leq m$, set

[TABLE]

Since $a_{k}(e^{2\pi i\theta_{1}},\cdots,e^{2\pi i\theta_{k-1}})\in L^{q}([0,1]^{k-1};X)$ and $b_{k}(e^{2\pi i\theta_{k}})\in L^{1}([0,1])$, by Cesàro summability, we know that for any $\epsilon>0$, there are two trigonometric polynomials $a^{\prime}_{k}$ and $b^{\prime}_{k}$ with values respectively in $X$ and $\mathbb{C}$ such that $\|a^{\prime}_{k}-a_{k}\|_{q}<\epsilon$ and $\|b^{\prime}_{k}-b_{k}\|_{q}<\epsilon$. Note that $\int_{0}^{1}{sgn(cos2\pi\theta)}d\theta=0$. Then $\hat{b^{\prime}_{k}}(0)=0$. By this consideration, we assume that $a_{k}$ and $b_{k}$ are trigonometric polynomials with values respectively in $X$ and $\mathbb{C}$ and $\hat{b_{k}}(0)=0$, such that $\|f_{k,(n)}-(M_{k}-M_{k-1})\|_{q}<\epsilon$.

In the following, we keep all notations introduced in Lemma 2.1. By Lemma 2.1 we have a sequence $\{n_{i}\}_{1\leq i\leq m}$ and $\{l_{i}\}_{0\leq i\leq m}$ such that (2.2) and (2.3) hold.

Firstly, for $i=1$, we have

[TABLE]

combining (2.3) with (2.2), we obtain the following estimate

[TABLE]

Hence for $i=1$, we get

[TABLE]

For $1<i\leq m$, we first have the following inequality

[TABLE]

Now, we estimate $I$ and $II$. By (2.3), the first term $I$ is controlled by

[TABLE]

To handle part $II$, one can observe the following inequality

[TABLE]

For $III$ and $V$, by (2.2) we obtain

[TABLE]

For $IV$, using (2.3) again, we have

[TABLE]

Therefore, we have $\|\sum_{k=1}^{m}(\mathbb{P}_{l_{i}}f_{k,(n)}-\mathbb{P}_{l_{i-1}}f_{k,(n)})-f_{i,(n)}\|\leq 3\epsilon$. By this estimate we see that

[TABLE]

Hence we can observe that

[TABLE]

On the other hand, by our assumption that inequality (1.4) holds, and Lemma 2.2, we have

[TABLE]

Combining (2.6) with (2.7), we have

[TABLE]

By the Fubini theorem and the change of variables

[TABLE]

we obtain

[TABLE]

Letting $\epsilon\rightarrow 0$, we deduce the desired inequality (2.5). Thus, we proved that $X$ is of martingale cotype $q$. This completes the proof of the part (ii). ∎

3. the proof of Theorem 1.2

In this section, we prove Theorem 1.2. Before proving the theorem, we need some notations and lemmas.

Let $P(x)=\frac{1}{\pi}\frac{1}{1+x^{2}}$ be the Poisson function in ${\mathbb{R}}$. For $\epsilon>0$, set $P_{\epsilon}(x)=\frac{1}{\epsilon}P(\epsilon^{-1}x)$, and define

[TABLE]

Let $Q_{\epsilon}$ denote the conjugate Poisson kernel, that is $Q_{\epsilon}(x)=\frac{1}{\pi}\frac{x}{\epsilon^{2}+x^{2}}$. Define

[TABLE]

Then we have the following equality

[TABLE]

The following lemma can be obtained by the vector-valued variational inequalities for averaging operators (see [15, Lemma 3.2] or [21, Corollary 2.6]).

Lemma 3.1**.**

Let $X$ be a Banach space. If $X$ is of martingale cotype $q_{0}$ with $q_{0}\geq 2$, then for every $1<p<\infty$ and $q>q_{0}$,

[TABLE]

We also need the following lemma.

Lemma 3.2**.**

If the Banach space $X$ satisfies UMD property, and $1<p<\infty$, then the following two inequalities are equivalent:

1. (1)

[TABLE] 2. (2)

[TABLE]

The following lemma extends the results of Lemma 3.1 for more general functions, the proof just follows the scalar case as in [7]. We omit the details.

Lemma 3.3**.**

Let $X$ be a Banach space. Let $\Phi$ be a scalar-valued function defined on ${\mathbb{R}}$ and $f\in L^{p}({\mathbb{R}};X)$. Define $\Phi_{t}(x)=\frac{1}{t}\Phi(\frac{x}{t})$, and set $\Phi_{t}f(x)=\Phi_{t}\ast f(x)=\int_{{\mathbb{R}}}\Phi_{t}(y)f(x-y)dy$. If the Banach space $X$ is of martingale cotype $q_{0}$ with $q_{0}\geq 2$, then for every $q>q_{0}$ and $1<p<\infty$, we have the following:

1. (1)

Let $\Phi$ be supported on $[0,1]$. Assume $\Phi$ is differentiable on $(0,1)$, and that $\int_{0}^{1}x|\Phi^{\prime}(x)|dx<\infty$, then $V_{q}(\Phi_{t}f:t>0)$ is $L^{p}$ bounded. 2. (2)

Let $\Phi$ be supported on $[1,\infty)$. Assume $\Phi$ is differentiable on $(0,1)$, and that $\lim_{x\rightarrow\infty}\Phi(x)=0$. Moreover, $\Phi(x)$ satisfies $\int_{1}^{\infty}x|\Phi^{\prime}(x)|dx<\infty$, then $V_{q}(\Phi_{t}f:t>0)$ is $L^{p}$ bounded.

Proof of Theorem 1.2 (i).

For $f\in\mathcal{S}({\mathbb{R}})\bigotimes X$, by the triangle inequality, we have

[TABLE]

By our assumption that the Banach space $X$ is of martingale cotype $q_{0}$ and satisfies UMD property, by Theorem 1.1 (i), Lemma 3.1 and Lemma 3.2, it will be enough to estimate $V_{q}([H_{\epsilon}-Q_{\epsilon}]f(x):\epsilon>0)$.

For $f\in\mathcal{S}({\mathbb{R}})\bigotimes X$, we observe

[TABLE]

Denote $\phi^{+}(y)=\frac{1}{\pi}\frac{1}{y(y^{2}+1)}\mathds{1}_{[1,\infty)}(y)$, $\phi^{-}(y)=\frac{1}{\pi}\frac{1}{y(y^{2}+1)}\mathds{1}_{(-\infty,-1]}(y)$, $A^{+}_{\epsilon}f(x)=f\ast\phi^{+}_{\epsilon}(x)$ and $A^{-}_{\epsilon}f(x)=f\ast\phi^{-}_{\epsilon}(x)$. Also let $\rho^{+}(y)=\frac{1}{\pi}\frac{y}{y^{2}+1}\mathds{1}_{[0,1]}(y)$, $\rho^{-}(y)=\frac{1}{\pi}\frac{y}{y^{2}+1}\mathds{1}_{[-1,0]}(y)$, $G^{+}_{\epsilon}f(x)=f\ast\rho^{+}_{\epsilon}(x)$ and $G^{-}_{\epsilon}f(x)=f\ast\rho^{-}_{\epsilon}(x)$. By Lemma 3.3 $\|V_{q}(A^{+}_{\epsilon}f:\epsilon>0)\|_{L^{p}}\leq C_{q}\|f\|_{L^{p}}$ and $\|V_{q}(A^{-}_{\epsilon}f:\epsilon>0)\|_{L^{p}}\leq C_{q}\|f\|_{L^{p}}$. Similar estimates also hold for operators $V_{q}(G^{+}_{\epsilon}f:\epsilon>0)$ and $V_{q}(G^{-}_{\epsilon}f:\epsilon>0)$. This completes to the proof of the part (i) of Theorem  1.2. ∎

Now let us prove the part (ii) of Theorem  1.2. Before proving this part, we first introduce some lemmas.

Let $R>\epsilon>0$. Clearly, the function $y^{-1}\mathds{1}_{\{\epsilon<|y|<R\}}$ belong to $L^{q}({\mathbb{R}})$, $1<q\leq\infty$. Hence by Hölder’s inequality,

[TABLE]

is well defined for all $f\in L^{p}({\mathbb{R}};X)$, $p\geq 1$. The same is true for $H_{\epsilon}f$.

In the following, we will prove the main lemma in this section.

Lemma 3.4**.**

Let $X$ be a Banach space. For $2\leq q<\infty$, if there exists some $1<p<\infty$ and a positive constant $C_{p,q}$ such that

[TABLE]

then $X$ has UMD property and we have the following inequality

[TABLE]

Proof.

Since variational inequalities imply the pointwise convergence for the family of operators under consideration. Then by condition (3.3), we have for every $f\in L^{p}({\mathbb{R}};X)$

[TABLE]

Furthermore,

[TABLE]

Hence, by the $L^{p}$-boundedness of vector-valued Hilbert transform, we know that the Banach space $X$ is of UMD (cf. [19, Theorem 5.1.1]).

Next we will prove inequality (3.4). Since

[TABLE]

Therefore,

[TABLE]

Note that

[TABLE]

By the pointwise estimate, we have

[TABLE]

Using the above inequality and equality (3.1), we obtain

[TABLE]

where in the last inequality we have used that the Banach space $X$ satisfies the UMD property. Thus we obtain inequality (3.4). ∎

Associating with Poisson kernel $P_{\epsilon}$, the first and third author [15, Lemma 3.4] proved the following lemma.

Lemma 3.5**.**

Let $\mathbb{P}_{\epsilon}$ be the Poisson kernel of the unit disc, which was defined in section 2. Let $X$ be a Banach space. For $1<p<\infty$, if inequality (3.4) holds, then for every $f\in L^{p}([0,1];X)$ we have

[TABLE]

With the above lemmas, we are now in a position to prove the part (ii) of Theorem 1.2.

Proof of Theorem 1.2 (ii).

Using Lemma 3.4 we obtained that the Banach space $X$ satisfies the UMD property. Moreover, combining inequalities (3.4) with Lemma 3.5, we can establish inequalities (3.5). In section 2, we proved that inequalities (3.5) implying the Banach space $X$ is of martingale cotype $q$. This completes the proof of the part (ii) of Theorem 1.2. ∎