# Vector-valued Hilbert transforms along curves

Guixiang Hong1 and Honghai Liu2∗ 1School of Mathematics and Statistics, Wuhan
University, Wuhan 430072, China and Instituto de Ciencias Matemáticas, CSIC-
UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás
Cabrera 13-15. 28049, Madrid, Spain<EMAIL_ADDRESS>2 School of
Mathematics and Information Science, Henan Polytechnic University, Jiaozuo,
Henan 454003, China<EMAIL_ADDRESS>
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.
∗ Corresponding author)

###### Abstract.

In this paper, we show that Hilbert transforms along some curves are bounded
on $L^{p}({\mathbb{R}}^{n};X)$ for some $1<p<\infty$ and some UMD spaces $X$.
In particular, we prove that Hilbert transforms along some curves are
completely $L^{p}$-bounded in the terminology from operator space theory.
Moreover, we obtain the $L^{p}(\mathbb{R}^{n};X)$-boundedness of anisotropic
singular integrals by using the ”method of rotations” of Calderón-Zygmund. All
these results extend already existing related ones.

###### Key words and phrases:

Hilbert transforms along curves, Weighted Hörmander condition, UMD spaces,
Completely bounded, Analytic interpolation.

###### 2010 Mathematics Subject Classification:

Primary 43A32; Secondary 46B99.

##  1\. Introduction

The question of whether the mapping properties of singular integral operators
could be extended to the Lebesgue-Bôhner spaces $L^{p}(\mathbb{R}^{n};X)$
($1<p<\infty$) of vector-valued functions was taken up by several authors in
the 60’s. In [1], Benedek, Calderón and Panzone observed that the boundedness
on $L^{p_{0}}(\mathbb{R}^{n};X)$ for one $1<p_{0}<\infty$ of a singular
integral operator, together with Hörmander’s condition, implies its
boundedness on $L^{p}(\mathbb{R}^{n};X)$ for all $1<p<\infty$. However, to
actually get the $L^{p_{0}}(\mathbb{R}^{n};X)$-boundedness (something that was
immediate for $p_{0}=2$ in the scalar-valued), turned out to be a
significantly difficult task except in the case $X=L^{p_{0}}(\Omega)$ for some
measure space $\Omega$.

The first progress made in this direction is Burkholder’s extension [3] of
Riesz’s classical theorem on the $L^{p}$-boundedness of the Hilbert transform,
where it was shown that if the underlying Banach space $X$ satisfies the so
called UMD-property, then the Hilbert transform is bounded on
$L^{p}(\mathbb{R};X)$ for any $1<p<\infty$. Moreover, the UMD-property was
shown by Bourgain [2] to be necessary for the boundedness of the Hilbert
transform. It is well-known that the Hilbert transform is a prototype of
singular integral operators and Fourier multipliers, its boundedness motivates
McConnell’s [17] and Zimmermann’s [28] results on vector-valued Marcinkiewicz-
Mihlin multipliers, and Hytönen and Weis’s [12] results on vector-valued
singular convolution integrals.

Particularly, if $X$ equals $S_{p}$–the Schatten class, the
$L^{p}(\mathbb{R}^{n};S_{p})$-boundedness is called complete
$L^{p}$-boundedness in the light of noncommutative harmonic analysis. In this
setting, the complete $L^{2}$-boundedness is immediately available because
$S_{2}$ is a Hilbert space, and the Fourier transform (or almost orthogonality
principle) can be adapted. In order to obtain the complete
$L^{p}$-boundedness, so far as we know in the noncommutative harmonic
analysis, there are only two ways. One way is to establish firstly the weak
type $(1,1)$ estimate, and then to use interpolation and the duality argument.
In this way, the convolution kernel need to satisfy the Lipschitz regularity
in order to conduct the pseudo-localization principle as done in [21] (see
also [10] for related results). The other way is to get $(L^{\infty},BMO)$
(the noncommutative BMO space) estimate, then to use interpolation and the
duality argument. In this case, the kernel is required to satisfy the
Hörmander’s condition as done in [18] and [15]. However, to get the complete
$L^{p}$-boundedness is not a trivial work when the kernel does not satisfy the
Lipschitz regularity and the Hörmander condition, see e.g. [9] for more
information.

The purpose of our project is to extend the vector-valued singular integrals
theory to more general setting. We consider vector-valued singular Radon
transforms, which are given by the following principal-valued integral

$\mathscr{T}f(x)={\rm p.v.}\int_{\mathbb{R}^{k}}f(x-\Gamma(t))K(t)dt,\ \ f\in
C_{0}^{\infty}({\mathbb{R}}^{n})\otimes X,$

where $X$ is a Banach space, $K$ is a Calderón-Zygmund kernel in
$\mathbb{R}^{k}$ and $\Gamma:\mathbb{R}^{k}\rightarrow{\mathbb{R}}^{n}$ is a
surface in ${\mathbb{R}}^{n}$ with $\Gamma(0)=0$, $n\geq 2$. Precisely, we are
interested in the boundedness of $\mathscr{T}$ on $L^{p}(\mathbb{R}^{n};X)$,
where $p\in(1,\infty)$ and $X$ is some Banach space. Obviously, $\mathscr{T}$
are classical vector-valued singular convolution integrals if $k=n$ and
$\Gamma(t)=(t_{1},t_{2},\cdots,t_{n})$, and related results have been
introduced in the previous paragraphs. On the other hand, if $X=\mathbb{R}$,
$\mathscr{T}$ are classical singular integrals associated to surfaces, which
have been well-studied by Stein, Nagel, Wainger, Christ and so on, see [27]
for a survey of results through 1978 and [6] through 1999.

In the present paper, we start with the investigation of Hilbert transforms
along curves in the hope of providing the insight and inspiration for
subsequent development of this subject, as the role played by the classical
Hilbert transform in the classical vector-valued Calderón-Zygmund theory.
Vector-valued Hilbert transforms along curves are defined by

$\mathscr{H}f(x)={\rm
p.v.}\int_{\mathbb{R}}f\big{(}x-\Gamma(t)\big{)}\frac{dt}{t},\ \ f\in
C_{0}^{\infty}({\mathbb{R}}^{n})\otimes X.$

In the scalar-valued case, the $L^{2}$-boundedness goes back the work [7] of
Fabes who proved it with $\Gamma(t)=(t^{\alpha},t^{\beta})$ using complex
integration. Then Stein and Wainger [26] obtained the $L^{2}$-boundedness for
all homogeneous curves by using Van der Corput’s estimates for trigonometric
integrals. The first breakthrough was the proof of the $L^{p}$-boundedness in
the papers of Nagel, Rivière and Wainger [19] as well as the paper of Nagel
and Wainger [20] using Stein’s complex interpolation. Since then, many related
results have been obtained, see Stein and Wainger’s survey paper [27] for the
curves having some curvature at the origin, the paper of Carlsson et al [5]
and the references therein for the flat curves in $\mathbb{R}^{2}$. However,
all results about vector-valued singular integrals mentioned previously can
not be directly applied to Hilbert transforms along curves on
$L^{p}(\mathbb{R}^{n};X)$, because they are no longer Calderón-Zygmund
operators. Therefore this study is a move beyond the vector-valued Calderón-
Zygmund theory.

In the present paper, we extend Nagel, Rivière and Wainger as well as Nagel
and Wainger’s results mentioned above to the vector-valued setting by
combining their original arguments and some idea developed recently by Hytönen
and Weis [14] in the vector-valued Calderón-Zygmund theory. To state our
results, we need to recall and introduce some notations. Denote by
$\epsilon_{j}$, $j\in\mathbb{Z}$, the Rademacher system of independent random
variables on a probability space $(\Omega,\Sigma,\mathbf{P})$ verifying
$\mathbf{P}(\epsilon_{j}=1)=\mathbf{P}(\epsilon_{j}=-1)=1/2$. Let
$\mathbb{E}=\int(\cdot)d\mathbf{P}$ be the corresponding expectation. The main
Banach space geometry property of $X$ we are concerned in this paper is the
UMD property (see e.g. [3]), i.e. the following inequality holds:

$\big{(}\mathbb{E}\big{\|}\sum_{k=1}^{N}\epsilon_{k}d_{k}\big{\|}_{X}^{2}\big{)}^{1/2}\leq
C\big{(}\mathbb{E}\big{\|}\sum_{k=1}^{N}d_{k}\big{\|}_{X}^{2}\big{)}^{1/2}$

for all $N\in\mathbb{N}$, all fixed signs $\epsilon_{k}\in\\{-1,1\\}$, all
$X$-valued martingale differences $(d_{k})_{k\geq 0}$. The following notation
is very useful for formulating the main results in this paper.

######  Definition 1.1.

Let $(a,b)\subseteq(0,1)$. We define $\mathcal{I}_{(a,b)}$ to be the set
consisting of UMD spaces with its element $X$ having the form
$X=[H,Y]_{\theta}$ such that $\theta\in(a,b)$, $H$ is a Hilbert space and $Y$
is another UMD space. $\mathcal{I}_{(0,1)}$ is denoted by $\mathcal{I}$ for
simplicity.

######  Remark 1.2.

(i). It is easy to check that all the noncommutative $L_{p}$ spaces
(containing commutative $L^{p}$ spaces) with $1<p<\infty$ belong to the class
$\mathcal{I}_{(|1-\frac{2}{p}|,1)}$. From the reflexivity of UMD space, in
general we have $X\in\mathcal{I}_{(a,b)}$ if and only if
$X^{\ast}\in\mathcal{I}_{(a,b)}$. Furthermore, if
$(a,b)\subseteq(c,d)\subseteq(0,1)$, then
$\mathcal{I}_{(a,b)}\subseteq\mathcal{I}_{(c,d)}$.

(ii). In [23], Rubio de Francia proved that for any UMD lattice $X$ there
exist $\theta\in(0,1)$, a Hilbert space $H$ and another UMD lattice $Y$ such
that $X=[H,Y]_{\theta}$. That means every UMD lattice $X$ belongs to
$\mathcal{I}$. In the same paper, the author also ask the open question “Is
every $B\in UMD$ intermediate between a ’worse’ $B_{0}$ and a Hilbert spaces
?” which in our language means “If $\mathcal{I}$ contains all UMD spaces?”.

The first result is on the Hilbert transform along the homogeneous curves
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
with each $\alpha_{i}>0$.

######  Theorem 1.3.

Let $X\in\mathcal{I}$ and $1<p<\infty$. Then there exists an absolute constant
$C_{p}$ such that

$\|\mathscr{H}f\|_{L^{p}(X)}\leq C_{p}\|f\|_{L^{p}(X)},\ \
f\in{L^{p}(\mathbb{R}^{n};X)}.$

This is a vector-valued version of Theorem 1 of Nagel, Rivière and Wainger in
[19]. Following the previous remark, Theorem 1.3 implies the complete
boundedness of Hilbert transforms along this kind of curves which is of
independent interest in the operator space theory. This result also partially
generalize the previous result by Rubio de Francia, Ruiz and Torra [22] where
they obtained Theorem 1.3 in the case $X=\ell^{q}$ with $1<q<\infty$. In [22],
the authors used indirectly Benedek, Calderón and Panzone’s strategy mentioned
previously. While the proof of Theorem 1.3 is motivated by the recent
development in the vector-valued Calderón-Zygmund theory [12], see Section 2
for related details.

Let $\delta_{t}$ be a one parameter group of dilations and
$\mathbf{e},\mathbf{f}$ be vectors in $\mathbb{R}^{n}$. A curve $\Gamma(t)$ is
called two-sided homogeneous if the following two conditions hold:

$\Gamma(t)=\left\\{\begin{array}[]{ccc}\delta_{t}\ \mathbf{e},&t>0,\\\
\delta_{-t}\ \mathbf{f},&t<0,\\\ 0,&t=0;\end{array}\right.$ (1.1)
$\\{\xi|\xi\cdot\Gamma(t)\equiv 0,t>0\\}=\\{\xi|\xi\cdot\Gamma(t)\equiv
0,t<0\\}.$

The curve
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
is a model with
$\delta_{t}x=(t^{\alpha_{1}}x_{1},t^{\alpha_{2}}x_{2},\cdots,t^{\alpha_{n}}x_{n})$,
$\mathbf{e}=\mathbf{1}$ and $\mathbf{f}=-\mathbf{1}$. We will see that the
same argument for this particular curve works for all the curves with the same
dilation but $\mathbf{e}=-\mathbf{f}$. Generalization of Theorem 1.3 to all
two-sided homogeneous curves in turn motivates us to consider the vector-
valued Calderón-Zygmund theory associated to one parameter group of dilations,
which is a project under progress.

As an application, Theorem 1.3 is used to deal with vector-valued anisotropic
singular integrals with homogeneous kernel by Calderón-Zygmund’s rotation
method. This work improves Hytönen’s Theorem 5.2 in [11] in some sense, see
Section 3 for more details.

In the next result, we deal with certain convex curves in $\mathbb{R}^{2}$
with the form $\Gamma(t)=\big{(}t,\gamma(t)\big{)}$, $\gamma(t)$ is some
convex function for $t\geq 0$.

######  Theorem 1.4.

Let $X$ be an UMD lattice belonging to the class $I_{(0,\frac{1}{5})}$,
$\gamma(t)$ be a continuous odd function, twice continuously differentiable,
increasing and convex for $t\geq 0$. Suppose also that $\gamma^{\prime\prime}$
is monotone for $t>0$ and there exists $C>0$ so that $\gamma^{\prime}(t)\leq
Ct\gamma^{\prime\prime}(t)$ for $t>0$. Then for $\frac{5}{3}<p<\frac{5}{2}$,
there exists an absolute constant $C_{p}$ such that

$\|\mathscr{H}f\|_{L^{p}(X)}\leq C_{p}\|f\|_{L^{p}(X)},\ \
f\in{L^{p}(\mathbb{R}^{n};X)}.$

A large class of functions $\gamma(t)$ satisfy the conditions in Theorem 1.4,
such as

$\gamma(t)=sgn(t)|t|^{\alpha},\ (\alpha\geq
2)\quad\textrm{and}\quad\gamma(t)=te^{-1/|t|}.$

The first one is homogeneous, while another one does not have any homogeneity.
This result is a vector-valued extension of Theorem 3.1 of Nagel and Wainger
in [20]. Theorem 1.4 also generalizes the second author’s result [16] in the
case $X=\ell^{q}$ with $5/3<q<5/2$. The proof of Theorem 1.4 is again
motivated by the recent development of the vector-valued Calderón-Zygmund
theory [14]. In fact, in Section 4, we prove a more general version, i.e.
Theorem 1.4 is also true if $X$ satisfies the following weaker condition:
there exist $\theta\in(0,\frac{1}{5})$, Hilbert space $H$ and UMD space $Y$
with property $(\alpha)$ (recalled in Section 4) such that $X=[H,Y]_{\theta}$.

##  2\. Proof of Theorem 1.3

The main arguments in this section are from [27], we will repeat some results
for completeness. Before the proof, we need some notations. Let matrix
$A=diag(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})$, then
$\Gamma^{\prime}(t)=A\Gamma(t)/t$ for $t>0$. We also define a norm function
$\rho(x)$ by the unique positive solution of

$\sum^{n}_{i=1}x^{2}_{i}\rho^{-2\alpha_{i}}=1$

and $\rho(0)=0$. This definition was introduced in the pioneering work on
anisotropic singular integrals of Fabes [7]. Obviously,
$\rho(\delta_{t}x)=t\rho(x)$ for $t>0$, $\rho(x)=1$ if and only if the
Euclidean norm $|x|=1$ which means $x$ is on the unit sphere
${\mathbf{S}}^{n-1}$. See also Proposition 1-9 in [27] for more properties of
$\rho$. By a change of variables, we assume $\alpha_{1}=1$ and $\alpha_{i}\geq
1$ for $2\leq i\leq n$, and set
$\Delta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$. Without lost of generality,
we assume that $\alpha_{i}\neq\alpha_{j}$ when $i\neq j$, then $\Gamma(t)$
does not lie in a proper subspace of $\mathbb{R}^{n}$. If not, $\Gamma$ lies
in some proper subspace, then the argument of Stein and Wainger in [27,
pp.1262] implies our desired result.

For $z\in\mathbb{C}$, we define an analytic family of operators
${\mathscr{H}}_{z}$ by

$\widehat{\mathscr{H}_{z}f}(\xi)=\\{\rho(\xi)\\}^{z}m_{z}(\xi)\hat{f}(\xi),$

where $m_{z}$ are given by

$m_{z}(\xi)={\rm p.v.}\int_{\mathbb{R}}e^{-2\pi
i\xi\cdot\Gamma(t)}|t|^{z}\frac{dt}{t}.$

Obviously, ${\mathscr{H}}_{0}$ is our original operator $\mathscr{H}$.

As in [27], the desired result will be concluded by analytic interpolation
once we show the following two estimates: For Hilbert space $H$

$\big{\|}{\mathscr{H}}_{z}f\big{\|}_{L^{2}({\mathbb{R}}^{n};H)}\leq
C(z)\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)},$ (2.1)

where $-1<Re(z)\leq\sigma$ for some $\sigma>0$ and $C(z)$ grows at most
polynomially in $|z|$, and for UMD space $Y$

$\|\mathscr{H}_{z}f\|_{L^{p}(\mathbb{R}^{n};Y)}\leq
C(z,p)\|f\|_{L^{p}(\mathbb{R}^{n};Y)},\ \ 1<p<\infty,$ (2.2)

where $-\beta\leq Re(z)\leq-\eta$ for arbitrarily positive $\eta$ and some
positive $\beta$ as well as $C(z,p)$ grows at most as fast as a polynomial in
$|z|$ for fixed $\eta$.

Indeed, we obtain Theorem 1.3 by performing twice the analytic interpolation
argument in [25] as follows. Let $T_{z}f(x)=e^{z^{2}}{\mathscr{H}}_{z}f(x)$.
Note that $|e^{z^{2}}|=e^{Re(z)^{2}-Im(z)^{2}}$, then by (2.1) there exists a
constant $M_{0}$ which is independent of $Im(z)$ such that

$\big{\|}T_{z}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}\leq
C(z)e^{-Im(z)^{2}}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}\leq
M_{0}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{n};H)}$ (2.3)

when $-1<{\rm Re}(z)<\sigma$. Also, for any UMD space $Y$ and
$q\in(1,\infty)$, by (2.2) there exists a constant $M_{1}$ which is
independent of $Im(z)$ such that

$\big{\|}T_{z}f\big{\|}_{L^{q}(\mathbb{R}^{n};{Y})}\leq
M_{1}\big{\|}f\big{\|}_{L^{q}(\mathbb{R}^{n};{Y})}\quad when\ \ -\beta<{\rm
Re}(z)<0.$ (2.4)

Obviously, this inequality holds in particular with $Y=H$.

For $1<p<\infty$, we choose $\theta_{1}\in(0,1)$, $\sigma_{1}<0$,
$0<\sigma_{0}<\sigma$ and $q_{1}\in(1,\infty)$ such that

$\sigma_{0}(1-\theta_{1})+\sigma_{1}\theta_{1}=:\sigma_{2}>0,\
\frac{1}{p}=\frac{1-\theta_{1}}{2}+\frac{\theta_{1}}{q_{1}}.$

Interpolating between (2.3) and (2.4) with $Y=H$, we have

$\big{\|}T_{z}f\big{\|}_{L^{p}(\mathbb{R}^{n};H)}\leq
C(p,z)\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{n};H)}\quad when\ \ {\rm
Re}(z)=\sigma_{2}>0.$ (2.5)

Note that $X=[H,Y]_{\theta}$ for some Hilbert space $H$, UMD space $Y$ and
$\theta\in(0,1)$. For fixed $\theta$, we choose $\sigma_{3}<0$ such that

$0=(1-\theta){\sigma_{2}}+\theta\sigma_{3}.$

In the same way, interpolating between (2.5) and (2.4) with $q=p$, we obtain

$\|\mathscr{H}f\|_{L^{p}(\mathbb{R}^{n};X)}=\|T_{0}f\|_{L^{p}(\mathbb{R}^{n};X)}\leq
C\|f\|_{L^{p}(\mathbb{R}^{n};X)}.$

The estimate (2.1) is trivial since Plancherel’s theorem remains true for
Hilbert space valued functions and the original arguments for Lemma 4.2 in
[27] work here. The novelty of the proof lies in the proof of (2.2). In the
case $Y=\ell^{q}$ with $1<q<\infty$, it has been proved in (2.2) in [22] by
Benedek, Calderón and Panzone’s argument since $L^{q}(\ell^{q})$-boundedness
is trivial. For general UMD space, we shall follow Hytönen and Weis’s idea
[14] established recently to prove the $L^{p}(Y)$ estimates simultaneously for
all $1<p<\infty$. The following subsection is devoted to the proof of estimate
(2.2).

###  2.1. The proof of (2.2)

The following proof is essentially the same as [11], we include it here for
the sake of completeness. From Lemma 4.4 of [27], we can write that

$\mathscr{H}_{z}f(x)=K_{z}\ast f(x),$

where

$K_{z}(x)=\int_{\mathbb{R}}h_{z}(x-\Gamma(t))|t|^{z}\frac{dt}{t}\ \text{and}\
\ \hat{h}_{z}(\xi)=\\{\rho(\xi)\\}^{z}.$

It is known that $h_{z}$ is a locally integrable function, $C^{\infty}$ away
from the origin satisfying

$h_{z}(\delta_{\lambda}x)=\lambda^{-\Delta-z}h_{z}(x),\;\lambda>0,\;x\neq 0.$

Moreover, each derivative of $h_{z}(x)$ is bounded by a polynomial in $|z|$,
if $\rho(x)=1$. In particular, $K_{z}$ has the homogeneity property
$\lambda^{\Delta}K_{z}\big{(}\delta_{\lambda}x\big{)}=K_{z}(x)$.

Let
$\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})=\\{\psi\in\mathscr{S}(\mathbb{R}^{n})|\
\hat{\psi}\in\mathscr{D}(\mathbb{R}^{n}),0\notin supp\ \hat{\psi}\\}$. Let
$\eta\in\mathcal{D}(\mathbb{R}^{n})$ have range $[0,1]$, vanish for
$\rho(\xi)\geq 2$ and equal $1$ for $\rho(\xi)\leq 1$. For $j\in\mathbb{Z}$,
we define $\hat{\phi}_{0}(\xi)=\eta(\xi)-\eta(\delta_{2}\xi)$,
$\hat{\varphi}_{j}(\xi)=\hat{\phi}_{0}(\delta_{2^{-j}}\xi)$ and
$\hat{\chi}_{j}(\xi)=\hat{\phi}_{j-1}(\xi)+\hat{\phi}_{j}(\xi)+\hat{\phi}_{j+1}(\xi)$.
Then $\hat{\varphi}_{j}(\xi)$ is supported in the annulus
$\\{2^{j-1}\leq\rho(\xi)\leq 2^{j+1}\\}$, and

$\displaystyle\sum_{j}\hat{\varphi_{j}}(\xi)=1\ \ \text{for}\ \ \xi\neq 0.$
(2.6)

Moreover, since $\hat{\chi}_{j}$ equals 1 on the support of $\hat{\phi}_{j}$,
we have

$\displaystyle\phi_{j}=\phi_{j}\ast\chi_{j}\ast\chi_{j}.$ (2.7)

The estimate (2.2) will be deduced from the following key estimate which will
be shown in the next subsection.

######  Proposition 2.1.

Let $\phi_{0}$ and $K_{z}$ be defined as above. We have

$\displaystyle\int_{\mathbb{R}^{n}}|\phi_{0}\ast
K_{z}(x)|\log^{n}(e+\rho(x))dx\leq C(z).$

With above preparations at hand, we finish the proof of the estimate (2.2).

###### Proof.

For fixed $z$, we denote $K_{z}$ by $K$ for simplicity. Given
$f\in\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})\otimes Y$,
$g\in\hat{\mathcal{D}}_{0}(\mathbb{R}^{n})\otimes Y^{*}$, by (2.6) and (2.7),
we have

$\displaystyle\langle g,K\ast f\rangle$ $\displaystyle=\langle\tilde{K}\ast
g,f\rangle=\sum_{j}\langle\phi_{j}\ast\tilde{K}\ast(\chi_{j}\ast
g),\chi_{j}\ast f\rangle,$

where the summation is finite and $\tilde{K}(x)=K(-x)$. Changing variable and
using the fact $\lambda^{\Delta}K_{z}(\delta_{\lambda}x)=K_{z}(x)$,

$(\phi_{j}\ast\tilde{K})\ast(\chi_{j}\ast
g)(x)=\int_{\mathbb{R}^{n}}\phi_{0}\ast\tilde{K}(y)(\chi_{j}\ast
g)(x-\delta_{2^{-j}}y)dy.$

Hence, by Hölder’s inequality and the Khintchine-Kahane inequality

$\displaystyle\big{|}\langle g,K\ast
f\rangle\big{|}=\big{|}\int_{\mathbb{R}^{n}}\mathbb{E}\langle\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y),\sum_{i}\epsilon_{i}\phi_{0}\ast K(y)\chi_{i}\ast
f\rangle dy\big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y)\|_{L^{p^{\prime}}(Y^{*})}\mathbb{E}\|\sum_{i}\epsilon_{i}\chi_{i}\ast
f\|_{L^{p}(Y)}|\phi_{0}\ast K(y)|dy.$

It is easy to check that $m=\sum_{j}\epsilon_{j}\hat{\chi}_{j}$ is an
anisotropic multiplier. Hence, by Theorem 3 in [11], we have

$\displaystyle\|\sum_{j}\epsilon_{j}\chi_{j}\ast
f\|_{L^{p}(\mathbb{R}^{n};Y)}\leq C_{p,X}\|f\|_{L^{p}(\mathbb{R}^{n};Y)}.$
(2.8)

By Proposition 2.1 and (2.8), we shall finish the proof by showing

$\displaystyle\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y)\|_{L^{p^{\prime}}(Y^{*})}\leq
C\log^{n}(e+\rho(y))\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g\|_{L^{p^{\prime}}(Y^{*})}.$

Let $e_{i}$ be the $i$-th standard unit vector. Above estimate is just a
$n$-fold application of

$\displaystyle\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g(\cdot-\delta_{2^{-j}}y_{i}e_{i})\|_{L^{p^{\prime}}(Y^{*})}\leq
C\log(e+\rho(y))\mathbb{E}\|\sum_{j}\epsilon_{j}\chi_{j}\ast
g\|_{L^{p^{\prime}}(Y^{*})},$

which follows from Lemma 10 of Bourgain [2]. ∎

###  2.2. The proof of Proposition 2.1

The proof of Proposition 2.1 is based on the following two lemmas. The first
one states that the kernel $K_{z}$ satisfies a weighted Hörmander condition,
which will be verified at the end of this subsection.

######  Lemma 2.2.

If $-\beta\leq Re(z)\leq-\eta$, then for sufficiently large constants $C_{0}$
and $C_{1}(z)$, we have

$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}(e+\rho(x))dx\leq
C_{1}(z)\log^{n}(e+\rho(y))$ (2.9)

for any $y\in\mathbb{R}^{n}\setminus\\{0\\}$. Moreover, $C_{1}(z)$ grows at
most as fast as a polynomial in $|z|$ for a fixed $\eta$.

The second lemma is a kind of decomposition lemma which has been established
in Lemma 4.10 of [14]. We reformulate it in our anisotropic case.

######  Lemma 2.3.

Let $\varphi\in\mathscr{S}(\mathbb{R}^{n})$ with vanishing integral. Then
there exists a decomposition $\varphi=\sum_{m\geq 0}\psi_{m}$ with the
following properties:

$\displaystyle\psi_{m}\in\mathcal{D}(\mathbb{R}^{n}),\;\mathrm{supp}\psi_{m}\subseteq\\{x|\
\rho(x)\leq C2^{\alpha m}\\},\;\int_{\mathbb{R}^{n}}\psi_{m}(y)dy=0,$

where $C$ and $\alpha$ are two universal constants only depending on the norm
$\rho$ and the dimension $n$, and for every $p\in[1,\infty]$ and every $M>0$,
the sequence of Lebesgue norms $\|\psi_{m}\|_{L^{p}}$, as well as
$\|\hat{\psi}_{m}\|_{L^{p}}$, is $\mathcal{O}(2^{-mM})$ as
$m\rightarrow\infty$.

###### Proof.

Let us give a quick explanation of this lemma. From Lemma 4.10 of [12],
$\psi_{m}$ is supported in $\\{x|\ |x|\leq 2^{m}\\}$. Fix $x\in\\{x|\ |x|\leq
2^{m}\\}$, by Proposition 1-9 of [27], if $\rho(x)\geq 1$, then

$\rho(x)\leq c_{1}|x|^{\alpha_{1}}\leq c_{1}2^{a_{1}m}$

and if $\rho(x)\leq 1$, then

$\rho(x)\leq c_{2}|x|^{a_{2}}\leq c_{2}2^{a_{2}m}$

with $c_{1},c_{2},a_{1},a_{2}$ positive constants. We obtain the desired
result by choosing $C=\max\\{c_{1},c_{2}\\}$ and
$\alpha=\max\\{a_{1},a_{2}\\}$. ∎

###### Proof of Proposition 2.1.

The main idea comes from [12], we include most details here for completeness.
By Lemma 2.3, we write $\phi_{0}=\sum_{m\geq 0}\psi_{m}$ with $\psi_{m}$’s
satisfying the properties stated in that lemma. Then we decompose $K_{z}$ into
pieces

$K_{z,m}(x)=K_{z}\ast\psi_{m}(x)$

and estimate each of them respectively.

We first estimate the integral outside the larger ellipsoid
$\mathcal{B}_{1}=\\{x|\ \rho(x)\leq CC_{1}2^{\alpha m}\\}$ with $C_{1}$ fixed
later depending on $C_{0}$. Recall that $\psi_{m}$ is supported in the
ellipsoid $\mathcal{B}_{0}=\\{x|\ \rho(x)\leq C2^{\alpha m}\\}$ and the
integral of $\psi_{m}$ vanishes, by Fubini’s theorem and Lemma 2.2, we obtain

$\displaystyle\int_{\mathcal{B}_{1}^{c}}|K_{z,m}(x)|\log^{n}(e+\rho(x))dx$
$\displaystyle=\int_{\mathcal{B}_{1}^{c}}|\int_{\mathcal{B}_{0}}K_{z}(x-y)\psi_{m}(y)dy|\log^{n}(e+\rho(x))dx$
$\displaystyle\leq\int_{\mathcal{B}_{0}}\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}(e+\rho(x))dx\psi_{m}(y)dy$
$\displaystyle\leq
C_{1}(z)\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))\psi_{m}(y)dy\leq
C_{1}(z)\|\psi_{m}\|_{L^{\infty}}\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))dy.$

By Lemma 2.3, the last quantity is of order $\mathcal{O}(2^{-m})$ as
$m\rightarrow\infty$ since $\|\psi_{m}\|_{L^{\infty}}\leq C_{M}2^{-mM}$ for
$M>0$ while

$\int_{\mathcal{B}_{0}}\log^{n}(e+\rho(y))dy\leq C2^{mN}$

for a fixed $N$.

Inside the ellipsoid $\mathcal{B}_{1}$, the computation is easier because of
the fact $\|\hat{K}_{z}\|_{L^{\infty}}\leq C(z)$, then

$\displaystyle\int_{\mathcal{B}_{1}}|K_{z,m}(x)|$
$\displaystyle\log^{n}(e+\rho(x))dx\leq\|K_{z,m}\|_{L^{\infty}}\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx$
$\displaystyle\leq\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\|\hat{K}_{z,m}\|_{L^{1}}$
$\displaystyle=\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\int_{\mathbb{R}^{n}}|\hat{K}_{z}(\xi)\hat{\psi}_{m}(\xi)|d\xi$
$\displaystyle\leq\|\hat{K}_{z}\|_{L^{\infty}}\|\hat{\psi}_{m}\|_{L^{1}}\int_{\mathcal{B}_{1}}\log^{n}(e+\rho(x))dx\leq
C(z)2^{-m}.$

The last inequality holds due to the same reason that for the case outside the
ellipsoid. Finally, we obtain Proposition 2.1 by summing over $m$. ∎

To complete the proof of Proposition 2.1, we still need to show Lemma 2.2.

###### Proof of Lemma 2.2.

We follow the main sketch provided in [27], but improve related estimates. To
verify $K_{z}$ satisfying (2.9), we may assume that $\rho(y)=1$, it suffices
to prove that

$\int_{\rho(x)\geq
C_{0}}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx\leq C(z).$ (2.10)

In fact, we set $\lambda=\rho(y)$ and $y^{\prime}=y/\lambda$. Obviously,
$\rho(y^{\prime})=1$. By a linear transformation
$x=\delta_{\lambda}x^{\prime}$ and the homogeneity of $K_{z}$, we have

$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle=$ $\displaystyle\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\lambda\rho(x^{\prime})\big{)}dx^{\prime}.$

If $\lambda=\rho(y)\geq 6$, it is trivial that

$\log\big{(}e+\lambda\rho(x^{\prime})\big{)}\leq\log\big{(}e+\lambda\big{)}+\log\big{(}e+\rho(x^{\prime})\big{)}\leq\log\big{(}e+\lambda\big{)}\log\big{(}e+\rho(x^{\prime})\big{)},$

where we use the assumption that $C_{0}\geq 6$. Then,

$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\rho(x^{\prime})\big{)}dx^{\prime}\log^{n}\big{(}e+\rho(y)\big{)}$
$\displaystyle\leq C(z)\log^{n}\big{(}e+\rho(y)\big{)}.$

When $\lambda=\rho(y)<6$, by (2.10), we get

$\displaystyle\int_{\rho(x)\geq
C_{0}\rho(y)}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq 2^{n}\int_{\rho(x^{\prime})\geq
C_{0}}|K_{z}(x^{\prime}-y^{\prime})-K_{z}(x^{\prime})|\log^{n}\big{(}e+\rho(x^{\prime})\big{)}dx^{\prime}$
$\displaystyle\leq C(z)\leq C(z)\log^{n}\big{(}e+\rho(y)\big{)}.$

To prove (2.10), we define $K_{z}^{1}$ and $K_{z}^{2}$ by

$K_{z}^{1}(x)=\int_{|t|\leq 1}h_{z}(x-\Gamma(t))|t|^{z}\frac{dt}{t}\
\text{and}\ K_{z}^{2}(x)=K_{z}(x)-K_{z}^{1}(x),$

respectively. We split the integral as

$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}(x-y)-K_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle+\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x-y)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle+\int_{\rho(x)\geq
C_{0}}|K_{z}^{2}(x-y)-K_{z}^{2}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx.$

To estimate first two summands, we need a estimate related to $h_{z}$, which
can be found in [27, pp.1273]. The homogeneity and smoothness of $h_{z}$ away
from origin imply that

$|h_{z}(x-y)-h_{z}(x)|\leq C(z)\frac{|y|}{\\{\rho(x)\\}^{\Delta+Re(z)+\mu}}$
(2.11)

for some $\mu>0$, provide $|y|/|x|$ is sufficiently small.

We set $\beta=\min\\{\mu,1\\}$. For the first integral, by using Fubini’s
theorem and (2.11), we have

$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq C_{0}}\int_{|t|\leq
1}|h_{z}(x-\Gamma(t))-h_{z}(x)||t|^{Re(z)-1}dt\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{|t|\leq 1}|t|^{Re(z)-1}\int_{\rho(x)\geq
C_{0}}|h_{z}(x-\Gamma(t))-h_{z}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dxdt$
$\displaystyle\leq\int_{|t|\leq 1}|t|^{Re(z)-1}|\Gamma(t)|\int_{\rho(x)\geq
C_{0}}\rho(x)^{-[\Delta+Re(z)+\mu]}\log^{n}\big{(}e+\rho(x)\big{)}dxdt$
$\displaystyle\leq C(z),$

where we use the fact that $-\beta<Re(z)<0$.

The norm function $\rho(x)$ have the property of $\rho(x+y)\leq
c\big{(}\rho(x)+\rho(y)\big{)}$ for some $c>0$(see Proposition 1-9 in [27]).
Specially, we set $C_{0}\geq\max\\{6,3c\\}$. Note that
$\rho(x-y)\geq\frac{1}{c}\rho(x)-\rho(y)\geq\frac{C_{0}}{c}-1\geq 2$ and
$\rho(x)\leq c[\rho(x-y)+\rho(y)]\leq c\rho(x-y)+c$. Using a linear
transformation, we treat the second summand as the first one,

$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{1}(x-y)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq\int_{\rho(x)\geq
2}|K_{z}^{1}(x)|\log^{n}\big{(}e+c+c\rho(x)\big{)}dx\leq C(z).$

Finally, using Fubini’s theorem, we have

$\displaystyle\int_{\rho(x)\geq
C_{0}}|K_{z}^{2}(x-y)-K_{z}^{2}(x)|\log^{n}\big{(}e+\rho(x)\big{)}dx$
$\displaystyle\leq$ $\displaystyle\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{0}}\big{|}h_{z}\big{(}x-y-\Gamma(t)\big{)}-h_{z}\big{(}x-\Gamma(t)\big{)}\big{|}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}.$

We divide the inner integral above according to the distance between $x$ and
$\Gamma(t)$. Note that $\rho(y)=1$, if $|y|/|x-\Gamma(t)|$ is sufficient
small, that is $|x-\Gamma(t)|$ is away from the origin, we can get that
$\rho(x-\Gamma(t))\geq C_{2}$, where $C_{2}$ is an appropriate constant. In
this case, by (2.11) and a linear transformation, we obtain the following
estimate

$\displaystyle\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\geq
C_{2}\end{subarray}}\big{|}h_{z}\big{(}x-y-\Gamma(t)\big{)}-h_{z}\big{(}x-\Gamma(t)\big{)}\big{|}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq
C_{0}\\\ \rho(x-\Gamma(t))\geq
C_{2}\end{subarray}}\frac{|y|}{\\{\rho\big{(}x-\Gamma(t)\big{)}\\}^{\Delta+\mu+Re(z)}}\log^{n}\big{(}e+\rho(x)\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{2}}\frac{1}{\\{\rho(x)\\}^{\Delta+\mu+Re(z)}}\log^{n}\big{(}e+c\rho(x)+ct\big{)}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\rho(x)\geq
C_{2}}\frac{1}{\\{\rho(x)\\}^{\Delta+\mu+Re(z)}}\big{\\{}\log^{n}\big{(}e+\rho(x)\big{)}+\log^{n}\big{(}e+t\big{)}\big{\\}}\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq C,$

where we use the fact that for fixed $|t|\geq 1$, $\rho(x)\leq
c[\rho(x-\Gamma(t))+\rho(\Gamma(t))]=c[\rho(x-\Gamma(t))+t]$.

It is trivial that $\rho\big{(}x+y+\Gamma(t)\big{)}\leq
c^{2}[\rho(x)+\rho(y)+\rho(\Gamma(t))]=c^{2}[1+\rho(x)+t]$. Then, the
remainder can be controlled by

$\displaystyle\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}[|h_{z}(x-y-\Gamma(t))|+|h_{z}(x-\Gamma(t))|]\log^{n}(e+\rho(x))\frac{dxdt}{|t|^{1-Re(z)}}$
$\displaystyle\leq\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq
C_{0}\\\ \rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}|h_{z}\big{(}x-y-\Gamma(t)\big{)}|\log^{n}\big{(}e+\rho(x)\big{)}dx|t|^{Re(z)-1}dt$
$\displaystyle+\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\geq C_{0}\\\
\rho(x-\Gamma(t))\leq
C_{2}\end{subarray}}|h_{z}\big{(}x-\Gamma(t)\big{)}|\log^{n}\big{(}e+\rho(x)\big{)}dx|t|^{Re(z)-1}dt$
$\displaystyle\leq C\int_{|t|\geq 1}\int_{\begin{subarray}{c}\rho(x)\leq
c(C_{2}+1)\end{subarray}}|h_{z}(x)|dx|t|^{Re(z)-1}\log^{n}(e+t)dt$
$\displaystyle\leq C(z),$

where we use the fact that $h_{z}$ is locally integrable. ∎

##  3\. Anisotropic singular integrals

It was shown by Calderón and Zygmund [4] that the $L^{p}$-boundedness of
singular integrals with rough kernels can be deduced from the
$L^{p}$-boundedness of the (directional) Hilbert transform using the method of
rotations. In this section, we show a similar phenomenon happens, that is, the
$L^{p}(X)$-boundedness of Hilbert transforms along curve
$\Gamma(t)=(|t|^{\alpha_{1}}sgnt,|t|^{\alpha_{2}}sgnt,\cdots,|t|^{\alpha_{n}}sgnt)$
considered in the previous section implies the $L^{p}(X)$ boundedness of
singular integrals $T_{\Omega}$ with kernels of the form
$K(x)=\Omega(x)\rho(x)^{-\Delta}$, where $\Omega$ is a function on
$\mathbb{R}^{n}\setminus\\{0\\}$ satisfying the homogeneity
$\Omega(\delta_{t}x)=\Omega(x)\ \text{for all}\ t>0,$ size condition

$\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}|\Omega(\omega)|d\omega<\infty,$
(3.1)

and the cancelation condition

$\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega=0,$

which can be understood from the following change-of-variable formula

$dx=t^{\Delta-1}\sum^{n}_{i=1}\alpha_{i}{\omega}^{2}_{i}dtd\omega.$

######  Theorem 3.1.

Let $X\in\mathcal{I}$. If $\Omega$ is odd, then the operators $T_{\Omega}$
described previously are bounded on $L^{p}(\mathbb{R}^{n};X)$ for
$1<p<\infty$.

Guliev [8] has obtained the boundedness of anisotropic singular integrals with
scalar valued-kernels on UMD lattices. Recently, Hytönen[11] generalized some
work of Guliev to the anisotropic singular integrals with operator-valued
kernels acting on UMD space. While their arguments require that $\Omega(x)$
should satisfy a kind of $L^{\infty}$-Dini condition, which is a much more
restricted condition than ours. So, Theorem 3.1 is a generalization of Hytönen
and Guliev’s result in this sense.

###### Proof.

Changing the variables, we find

$\displaystyle T_{\Omega}f(x)$ $\displaystyle={\rm
p.v.}\int_{\mathbb{R}^{n}}f\big{(}x-\delta_{\rho(y)}\delta_{\rho(y)}^{-1}y\big{)}\Omega(\delta_{\rho(y)}^{-1}y)\\{\rho(y)\\}^{-\Delta}dy$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbf{S}^{n-1}}f\big{(}x-\delta_{t}\omega\big{)}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega\frac{dt}{t}.$
(3.2)

Note that $\Omega$ is odd, by a linear transformation, we also have

$T_{\Omega}f(x)=\int^{0}_{-\infty}\int_{\mathbf{S}^{n-1}}f\big{(}x+\delta_{(-t)}\omega\big{)}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)d\omega\frac{dt}{t}.$
(3.3)

Using Fubini theorem, and adding (3.2) and (3.3) together, we get

$T_{\Omega}f(x)=\frac{1}{2}\int_{\mathbf{S}^{n-1}}\sum^{n}_{i=1}\alpha_{i}\omega^{2}_{i}\Omega(\omega)\big{[}\int_{-\infty}^{0}f\big{(}x+\delta_{(-t)}\omega\big{)}\frac{dt}{t}+\int_{0}^{\infty}f\big{(}x-\delta_{t}\omega\big{)}\frac{dt}{t}\big{]}d\omega.$

Then, it suffices to prove that

$\|\int_{-\infty}^{0}f\big{(}x+\delta_{(-t)}\omega\big{)}\frac{dt}{t}+\int_{0}^{\infty}f\big{(}x-\delta_{t}\omega\big{)}\frac{dt}{t}\|_{L^{p}(\mathbb{R}^{n};\mathbf{X})}\leq
C_{p}\|f\|_{L^{p}(\mathbb{R}^{n};X)},$

where the constant $C_{p}$ is independent of $\omega$.

For fixed $\omega\in\mathbf{S}^{n-1}$, define $\Gamma_{\omega}(t)$ as the
curve in the form of (1.1) associated to the dilation $\delta_{t}$ with
$\mathbf{e}=\omega$ and $\mathbf{f}=-\omega$, then the quantity inside the
norm of the previous inequality is the Hilbert transform along the curve
$\Gamma_{\omega}(t)$. The same arguments for the proof of Theorem 1.3 work
also for the curve $\Gamma_{\omega}(t)$, and we obtain the desired result. ∎

In the classical case (dilation given by $\delta_{t}x=tx$), it is known that
the boundedness of $T_{\Omega}$ is also obtained for the even function
$\Omega$ under a stronger size condition $\Omega\in
L\log^{+}L(\mathbf{S}^{n-1})$. The main ingredient is the existence of Riesz
transforms $R_{j},\;j=1,2,\cdots,n$, such that

  1. (i)

$-\sum^{n}_{j=1}R_{j}\circ R_{j}=I$,

  2. (ii)

the kernel of $T_{\Omega}\circ R_{j}$ is still homogeneous, and the associated
$\Omega_{j}$ is an odd function satisfying size condition (3.1).

In the anisotropic setting, it seems very difficult to find some replacements
for Riesz transforms such that similar properties as (i) and (ii) hold. Hence
we leave it as an open problem that whether Theorem 3.1 is still true for the
even function $\Omega$ under a stronger size condition.

##  4\. The proof of Theorem 1.4

The main argument for the proof is similar to that for Theorem 1.3. We first
introduce a family of analytic operators. For $z\in\mathbb{C}$, we define an
analytic family of operators ${\mathscr{H}}_{z}$ by

$\widehat{\mathscr{H}_{z}f}(\xi,\eta)=m_{z}(\xi,\eta)\hat{f}(\xi,\eta),$

where $m_{z}$ are given by

$m_{z}(\xi,\eta)={\rm p.v.}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\frac{dt}{t}.$

Obviously, ${\mathscr{H}}_{0}$ is our original operator $\mathscr{H}$.

Following the idea in [20], it suffices to prove the following two estimates:

$\big{\|}{\mathscr{H}}_{z}f\big{\|}_{L^{2}({\mathbb{R}}^{2};H)}\leq
C_{\delta}\big{[}1+|Im(z)|\big{]}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)},$
(4.1)

where $Re(z)=\frac{1}{4}-\delta$ for some $\delta>0$, and

$\big{\|}\mathscr{H}_{z}f\big{\|}_{L^{q}({\mathbb{R}}^{2};{Y})}\leq
C\big{[}1+|Im(z)|\big{]}^{2}\big{\|}f\big{\|}_{L^{q}({\mathbb{R}}^{2};{Y})},$
(4.2)

where $Y$ is an UMD lattice, $Re(z)<-1$, $1<q<\infty$, the constant $C$
depends on $Re(z)$ and is independent of $Im(z)$.

Indeed, we finish the proof by analytic interpolation argument [25]. Let
$T_{z}f(x)=e^{z^{2}}{\mathscr{H}}_{z}f(x)$. Note that
$|e^{z^{2}}|=e^{Re(z)^{2}-Im(z)^{2}}$, by (4.1) there exists a constant
$M_{0}$ which is independent of $Im(z)$ such that

$\big{\|}T_{z}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}\leq
C_{\delta}e^{-Im(z)^{2}}\big{[}1+|Im(z)|\big{]}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}\leq
M_{0}\big{\|}f\big{\|}_{L^{2}(\mathbb{R}^{2};H)}$

when ${\rm Re}(z)=\frac{1}{4}-\delta$. Also, for UMD lattice $Y$ and
$q\in(1,\infty)$, by (4.2) there exists a constant $M_{1}$ which is
independent of $Im(z)$ such that

$\big{\|}T_{z}f\big{\|}_{L^{q}(\mathbb{R}^{2};{Y})}\leq
M_{1}\big{\|}f\big{\|}_{L^{q}(\mathbb{R}^{2};{Y})}\quad when\ \ {\rm
Re}(z)<-1.$

This inequality also holds in particular with $Y=H$.

For $\frac{5}{3}<p\leq 2$, there exist $1<q<\infty$ and
$\theta_{0}\in(0,\frac{1}{5})$ so that

$\frac{1}{p}=\frac{1-\theta_{0}}{2}+\frac{\theta_{0}}{q}\ \ \text{and}\ \
(\frac{1}{4}-\delta)(1-\theta_{0})+(-1-\varepsilon_{0})\theta_{0}=:\sigma_{1}\in(0,\frac{1}{4})$

for some $\varepsilon_{0}>0$ and $0<\delta<\frac{1}{4}$. By interpolation of
analytic operators, we have

$\big{\|}T_{z}f\big{\|}_{L^{p}(\mathbb{R}^{2};H)}\leq
C(z)\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{2};H)}\quad for\ \ {\rm
Re}(z)=\sigma_{1}\in(0,1/4).$

Given an UMD lattice $X\in\mathcal{I}_{(0,1/5)}$, there exist a
$\theta\in(0,\frac{1}{5})$, a Hilbert space $H$ and another UMD lattice $Y$,
such that
$L^{p}(\mathbb{R}^{2};X)=[L^{p}(\mathbb{R}^{2};H),L^{p}(\mathbb{R}^{2};{Y})]_{\theta}$.
For such a $\theta$ and appropriate $\sigma_{1}$, we choose
$\varepsilon_{1}>0$ such that
$(1-\theta)\sigma_{1}+\theta(-1-\varepsilon_{1})=0$. Using interpolation of
analytic operators once more, we obtain

$\big{\|}\mathscr{H}f\big{\|}_{L^{p}(\mathbb{R}^{2};X)}\leq
C\big{\|}f\big{\|}_{L^{p}(\mathbb{R}^{2};X)}$

for $\frac{5}{3}<p\leq 2$. The duality argument implies the result for $2\leq
p<\frac{5}{2}$. This completes the proof of Theorem 1.4.

The estimate (4.1) holds since Plancherel’s theorem works also for Hilbert
space valued functions and the original argument in [20] can be repeated in
the present situation. The novelty of the proof lies in the estimate (4.2),
for which we need the vector-valued Fourier multiplier theorem established
recently.

Let us firstly recall some notations. A Banach space $X$ satisfies property
$(\alpha)$ if there is a positive constant $C$ such that

$\mathbb{E}\mathbb{E}^{\prime}\bigg{|}\sum^{N}_{k,l=1}\epsilon_{k}\epsilon_{l}^{\prime}\alpha_{kl}x_{kl}\bigg{|}_{X}\leq
C\mathbb{E}\mathbb{E}^{\prime}\bigg{|}\sum^{N}_{k,l=1}\epsilon_{k}\epsilon_{l}^{\prime}x_{kl}\bigg{|}_{X}$

for all $N\in\mathbb{N}$, all vectors $x_{kl}\in X$ and scalars
$|\alpha_{kl}|\leq 1$ $(1\leq k,l\leq N)$, where $\epsilon_{k}$,
$k\in\mathbb{Z}$ and $\epsilon_{l}^{\prime}$, $l\in\mathbb{Z}$ are two
identical independent sequences.

######  Remark 4.1.

The commutative $L^{p}$ spaces satisfy property $(\alpha)$ for all $1\leq
p<\infty$. Also, this property is inherited from $X$ by $L^{p}(\mu,X)$ for
$p\in[1,\infty)$. Every Banach space with a local unconditional structure and
finite cotype, in particular every Banach lattice, has property $(\alpha)$.

Let $m:{\mathbb{R}}^{n}\rightarrow\mathbb{C}$ be a bounded function, the
associated operator $T_{m}$ is defined on the test functions
$f\in{\mathscr{S}}({\mathbb{R}}^{n})\otimes X$ by

$T_{m}f(x)=(m\hat{f})^{\vee}(x).$

The sufficiency part of the following vector-valued Fourier multiplier theorem
was proved by Štrkalj and Weis [24], while the necessity of those conditions
was obtained by Hytönen and Weis [14].

######  Lemma 4.2.

The Marcinkiewicz-Lizorkin condition $|\xi^{\beta}||D^{\beta}m(\xi)|\leq C$
for all $\beta\in\\{0,1\\}^{n}$ is sufficient for the
$L^{p}({\mathbb{R}}^{n};X)$-boundedness of $T_{m}$, $n>1$, if and only if $X$
is an UMD space with property $(\alpha)$.

In view of Lemma 4.2 and Remark 4.1, to prove the estimate (4.2), it suffices
to show that the following functions

$m_{z}(\xi,\eta),\ \xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta),\
\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta),\
\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$

are uniformly bounded on $\mathbb{R}^{2}$ for $Re(z)<-1$.

The uniform boundedness of $m_{z}(\xi,\eta)$ is trivial, it can be showed by
minor modification of the proof of (4.1). Without repetition, we omit the
proof. The following estimates are essentially proved in [20], we include them
here for the sake of completeness.

The boundedness of $\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)$.
Integration by part implies that

$\displaystyle\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-2\pi i\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\xi\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\frac{d}{dt}(e^{-2\pi i\xi
t})e^{-2\pi i\eta\gamma(t)}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\bigg{|}^{\infty}_{-\infty}$
$\displaystyle+$ $\displaystyle 2\pi i\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 2z\eta^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\gamma(t)\gamma^{\prime}(t)dt.$

Note that $Re(z)<-1$, for $t\in{\mathbb{R}}$, we have
$\big{|}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\big{|}=\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\leq
1$. The boundary terms are bounded by $1$.

For $Re(z)<-1$, making the change of variables $u=|\eta|\gamma(t)$, we obtain

$\displaystyle\bigg{|}\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}\gamma^{\prime}(t)|\eta|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}dt$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}\big{(}1+u^{2}\big{)}^{Re(z)}du\leq\pi.$

In a similar way, the second integrated term can be dominated by

$\displaystyle\bigg{|}z\eta^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\gamma(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{\infty}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)-1}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$ $\displaystyle|z|\int_{0}^{\infty}(1+u)^{Re(z)-1}du\leq
1+|Im(z)|.$

Therefore, for $Re(z)<-1$,

$\big{|}\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)\big{|}\leq
C\big{[}1+|Im(z)|\big{]}.$

The boundedness of $\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta)$.
Integrating by parts, we obtain

$\displaystyle\eta\frac{\partial m_{z}}{\partial\eta}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-2\pi i\ {\rm p.v.}\int_{\mathbb{R}}e^{-2\pi
i[\xi
t+\eta\gamma(t)]}\eta\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\frac{dt}{t}$
$\displaystyle+$ $\displaystyle 2z\ {\rm p.v.}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\frac{dt}{t}.$

To estimate above two integrals, we follow the argument used in the proof of
(4.1). For the first integral, for any $\varepsilon>0$, it suffices to bound
the following two parts

$\int_{\varepsilon<|t|<t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\
\ \text{and}\ \ \int_{|t|\geq
t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}.$

Recall that $t_{0}>0$ was chosen so that $|\eta|\gamma(t_{0})=1$, and
$\gamma(t)\leq t\gamma^{\prime}(t)$ because of the convexity. Thus,

$\displaystyle\int_{\varepsilon<|t|<t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\leq
2|\eta|\int_{0}^{t_{0}}\frac{\gamma(t)}{t}dt\leq
2|\eta|\int_{0}^{t_{0}}\gamma^{\prime}(t)dt\leq 2.$

For $Re(z)<-1$, an elementary calculation implies that

$\int_{|t|\geq
t_{0}}|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\frac{dt}{|t|}\leq
2|\eta|^{2Re(z)+1}\int_{t_{0}}^{\infty}\gamma^{2Re(z)}(t)\frac{\gamma(t)}{t}dt\leq
2.$

Similarly, the second integral can be controlled by

$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\frac{dt}{t}\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{t_{0}}\eta^{2}\gamma^{2}(t)\frac{dt}{t}+2|z|\int_{t_{0}}^{\infty}\eta^{2}\gamma^{2}(t)\big{[}\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)-1}\frac{dt}{t}$
$\displaystyle\leq$ $\displaystyle
2|z|\eta^{2}\int_{0}^{t_{0}}\gamma(t)\gamma^{\prime}(t)dt+2|z|\eta^{2Re(z)}\int_{t_{0}}^{\infty}\gamma^{2Re(z)-1}(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$ $\displaystyle|z|+\frac{|z|}{|Re(z)|}\leq
2|Re(z)|\big{[}1+|Im(z)|\big{]}.$

Therefore, for $Re(z)<-1$,

$\big{|}\xi\frac{\partial m_{z}}{\partial\xi}(\xi,\eta)\big{|}\leq
C\big{[}1+|Im(z)|\big{]}.$

The boundedness of
$\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$. To deal
with $\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$, we
rewrite it as

$\displaystyle\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)$
$\displaystyle=$ $\displaystyle-4\pi^{2}\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta\gamma^{2}(t)dt.$

For the first term, integrating by parts, we obtain

$\displaystyle 4\pi^{2}\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\gamma(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle 2\pi
i\int_{\mathbb{R}}\frac{d}{dt}\big{(}e^{-2\pi i\xi t}\big{)}e^{-2\pi
i\eta\gamma(t)}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle=$ $\displaystyle 2\pi ie^{-2\pi i[\xi
t+\eta\gamma(t)]}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\bigg{|}_{-\infty}^{\infty}$
$\displaystyle-$ $\displaystyle 4\pi^{2}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 2\pi i\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta^{3}\gamma^{2}(t)\gamma^{\prime}(t)dt.$

Obviously, for $Re(z)<-1$, $t\in\mathbb{R}$, $\big{|}2\pi ie^{-2\pi i[\xi
t+\eta\gamma(t)]}[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}\big{|}\leq
2\pi|\eta||\gamma(t)|\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\leq 2\pi$.
So, the boundary terms are bounded by $2\pi$.

For the first integrated term, making the change of variables
$u=\eta^{2}\gamma^{2}(t)$, we have

$\displaystyle\bigg{|}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)[\eta\gamma(t)]\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{\infty}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{Re(z)}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\big{(}1+u\big{)}^{Re(z)}du\leq\frac{1}{|Re(z)+1|}.$

The second integrated terms can be treated in the same way, let
$u=\eta\gamma(t)$,

$\bigg{|}\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z}dt\bigg{|}\leq\int_{\mathbb{R}}(1+u^{2})^{Re(z)}du\leq\pi.$

Similarly, a trivial calculation shows that

$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta^{3}\gamma^{2}(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$ $\displaystyle
2|z|\int_{0}^{\infty}u^{2}\big{(}1+u^{2}\big{)}^{Re(z)-1}du\leq\pi|z|.$

The second term can be handled similarly. Integrating by parts, we decompose
it as

$\displaystyle 4\pi iz\xi\eta\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\eta\gamma^{2}(t)dt$
$\displaystyle=$ $\displaystyle 2\pi
iz\int_{\mathbb{R}}\frac{d}{dt}\big{(}e^{-2\pi i\xi t}\big{)}e^{-2\pi
i\eta\gamma(t)}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle=$ $\displaystyle 2\pi ize^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\bigg{|}_{-\infty}^{\infty}$
$\displaystyle-$ $\displaystyle 4\pi^{2}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle-$ $\displaystyle 4\pi iz\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma(t)\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt$
$\displaystyle-$ $\displaystyle 4\pi iz(z-1)\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-2}\eta^{2}\gamma(t)\gamma^{\prime}(t)dt.$

Obviously, for $Re(z)<-1$, $t\in\mathbb{R}$, $\big{|}ze^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}\big{|}\leq|z|$.
The boundary terms are dominated by $4\pi|z|$.

For the first integrated term, by making the change of variables
$u=\eta\gamma(t)$, we have the estimate

$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta\gamma^{\prime}(t)\eta^{2}\gamma^{2}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt\bigg{|}$
$\displaystyle\leq|z|\int_{\mathbb{R}}u^{2}(1+u^{2})^{Re(z)-1}dt$
$\displaystyle\leq\pi|z|.$

To estimate the second integrated terms, we make the transformation
$u=\eta^{2}\gamma^{2}(t)$ and get

$\displaystyle\bigg{|}z\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\eta^{2}\gamma(t)\gamma^{\prime}(t)\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-1}dt\bigg{|}$
$\displaystyle\leq|z|\int_{0}^{\infty}(1+u)^{Re(z)-1}du$
$\displaystyle\leq\frac{|z|}{|Re(z)|}.$

Similarly, the third integrated terms can be treated as

$\displaystyle\bigg{|}z(z-1)\int_{\mathbb{R}}e^{-2\pi i[\xi
t+\eta\gamma(t)]}\big{[}1+\eta^{2}\gamma^{2}(t)\big{]}^{z-2}\eta^{4}\gamma^{3}(t)\gamma^{\prime}(t)dt\bigg{|}$
$\displaystyle\leq$
$\displaystyle|z(z-1)|\int_{0}^{\infty}(1+u)^{Re(z)-1}du\leq\frac{|z(z-1)|}{|Re(z)|}.$

Note that for $Re(z)<-1$, we have the following elementary estimates

$|z|\leq|Re(z)|\big{[}1+|Im(z)|\big{]}\ \ \text{and}\ \
|z-1|\leq|Re(z)-1|\big{[}1+|Im(z)|\big{]}.$

Finally, combining the above eight estimates, we obtain

$\big{|}\xi\eta\frac{\partial^{2}m_{z}}{\partial\xi\partial\eta}(\xi,\eta)\big{|}\leq
C\big{[}1+Im(z)\big{]}^{2}.$

This completes the proof of Theorem 1.4.

Acknowledgement. The first author is supported in part by MINECO: ICMAT Severo
Ochoa project SEV-2011-0087 and ERC Grant StG-256997-CZOSQP (EU); The second
author is supported in part by NSFC 11371057 and 11471033. The authors would
like to thank the referee for many valuable and useful comments and
suggestions which have improved this paper.

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