# Maximal operators associated with Fourier multipliers and applications

Jin Bong Lee  Research Institute of Mathematics, Seoul National University,
Seoul 08826, Republic of Korea<EMAIL_ADDRESS>and  Jinsol Seo  Department
of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce a criterion for maximal operators associated with
Fourier multipliers to be bounded on $L^{p}(\mathbb{R}^{d})$ for each
$p\in(1,\infty)$. Noteworthy examples satisfying the criterion are multipliers
of the Mikhlin type or limited decay which are not necessarily radial. To do
so, we make use of modified square function estimates and bilinear
interpolation. For the bilinear interpolation we introduce a function space
$\Sigma^{2}(\mathcal{B})$ where $\mathcal{B}$ denotes a Banach space of
functions on $\mathbb{R}^{d}$, which is a variant of weighted Sobolev spaces.
In result, we obtain convergence results for fractional half-wave equations
and surface averages as well as the $L^{p}$ boundedness for the maximal
operators.

###### Key words and phrases:

Fourier muitipliers, Maximal operators, Bilinear interpolation

###### 2020 Mathematics Subject Classification:

42B15, 42B25, 42B35, 42B37

J. B. Lee is supported by the National Research Foundation of Korea(NRF) grant
2021R1C1C2008252 and the National Research Foundation of Korea(NRF) grant
2022R1A4A1018904

##  1\. Introduction

Maximal operators and Fourier multipliers are crucial objects in harmonic
analysis due to their importance in harmonic analysis itself and vast
applications in other areas of mathematics such as theory of partial
differential equations. Maximal operators of our particular interest are given
in terms of Fourier multipliers. One of the best examples of this kind could
be the spherical maximal function

$\displaystyle\mathcal{M}_{sph}(f)(x)$
$\displaystyle:=\sup_{t>0}\bigg{|}\int_{\mathbb{S}^{d-1}}f(x-ty)d\sigma_{sph}(y)\bigg{|},$

where $d\sigma_{sph}$ denotes the spherical measure on $\mathbb{S}^{d-1}$
induced by the Lebesgue measure. For $f\in\mathscr{S}(\mathbb{R}^{d})$, we
make use of the Fourier inversion formula to write

$\displaystyle\mathcal{M}_{sph}(f)(x)$
$\displaystyle:=\sup_{t>0}\bigg{|}\int_{\mathbb{R}^{d}}e^{2\pi i\langle
x,\xi\rangle}\widehat{d\sigma_{sph}}(t\xi)\widehat{f}(\xi)d\xi\bigg{|},$

which is the form of
$\sup_{t>0}\big{|}\big{(}m(t\cdot)\widehat{f}(\cdot)\big{)}\check{\,\,}\big{|}$.
Note that $\widehat{d\sigma_{sph}}$ is the Fourier transform of the surface-
carried measure of $\mathbb{S}^{d-1}$ and equals to
$\frac{J_{(d-2)/2}(|\xi|)}{|\xi|^{(d-2)/2}}$ up to constant, where
$J_{\nu}(r)$ is the Bessel function of order $\nu$. $\mathcal{M}_{sph}$ is
bounded on $L^{p}(\mathbb{R}^{d})$ for $p>\frac{d}{d-1}$ when $d\geq 3$ due to
E. M. Stein [19, 21] and for $d=2$ by J. Bourgain [3].

In studies of $\mathcal{M}_{sph}$, what has been mainly concerned is the
relation between the $L^{p}$ boundedness and geometric aspects of
$\mathcal{M}_{sph}$. Generally speaking, geometric properties of
$\widehat{d\sigma_{sph}}$ are one of many examples which allow maximal
operators associated with the Fourier multipliers to be bounded on
$L^{p}(\mathbb{R}^{d})$. Thus, one can naturally question that for which
conditions on the Fourier multipliers $m$
$\mathcal{M}_{m}(f):=\sup_{t>0}\big{|}(m(t\cdot)\widehat{f}(\cdot))\check{\,}\big{|}$
is bounded on $L^{p}$.

In [17], Rubio de Francia showed mapping properties of $\mathcal{M}_{m}$ when
$m$ satisfies the limited decay condition, $i.e$,

$|\partial^{\gamma}m(\xi)|\leq C|\xi|^{-a}$ for a given $a>0$,

which is a generalisation of geometric properties of
$\widehat{d\sigma_{sph}}$. Results in [17] are based on the following square
function estimates:

######  Theorem A ([17], Lemma 4).

Let $m$ be a function of class $C^{s}(\mathbb{R}^{d})$ with $s>\frac{d}{2}$
and supported in $\\{1/2<|\xi|<2\\}$, and let the $g$-function be given by

$G_{m}(f)(x)=\biggl{(}\int_{0}^{\infty}\big{|}\big{(}m(t\xi)\widehat{f}(\xi)\big{)}\check{\,}(x)\big{|}^{2}\frac{dt}{t}\biggr{)}^{1/2}.$

Then for $\beta>d/2$ we have

$\big{\|}G_{m}(f)\|_{L^{1}(\mathbb{R}^{d})}\leq
C_{\beta}\|m\|_{L_{\beta}^{2}(\mathbb{R}^{d})}\|f\|_{H^{1}(\mathbb{R}^{d})}.$

We also introduce results of Dappa and Trebels [6], which could be understood
as a radial analogue of Theorem A.

######  Theorem B ([6],Theorem 1).

Let $\rho\in C^{[d/2+1]}(\mathbb{R}^{d}\setminus\\{0\\})$ be a positive,
$A_{t}$-homogeneous distance function. Let $m$ be a measurable function on
$(0,\infty)$ which vanishes at infinity.

  1. (1)

If $m$ satisfies

(1.1) $\displaystyle\int_{0}^{\infty}t^{\lambda-1}|m^{(\lambda)}(t)|dt\leq B,$
$\displaystyle\bigg{(}\int_{0}^{\infty}|t^{\lambda}m^{(\lambda)}(t)|^{2}\frac{dt}{t}\bigg{)}^{1/2}\leq
B,$

for $\lambda>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}$, then
$\mathcal{M}_{m\circ\rho}$ is of strong type $(p,p)$, $p\in(1,\infty)$, also
$\mathcal{M}_{m\circ\rho}$ is of weak type $(1,1)$.

  2. (2)

If $m$ satisfies (1.1) for
$\lambda>(d-1)\bigl{(}\frac{1}{2}-\frac{1}{p}\bigr{)}+1$, then
$\mathcal{M}_{m\circ\rho}$ is of strong type $(p,p)$, $2\leq p\leq\infty$.

By $A_{t}$-homogeneous distance, we mean a continuous function on
$\mathbb{R}^{d}$ with

$\rho>0,\quad\rho(A_{t}\xi)=t\rho(\xi),\quad\text{for all $t>0$,
$\xi\in\mathbb{R}^{d}$},$

where $A_{t}$ is a dilation matrix given by
$A_{t}:=e^{P\log{t}}=\sum_{n=0}^{\infty}\frac{(\log{t})^{n}}{n!}P^{n}$ for
$P=(p_{ij})$ introduced in [20]. Since one of examples of $m\circ\rho$ is a
radial Fourier multiplier, our main result (Theorem 1.1) can be understood as
a non-radial generalisation for $m\circ\rho$. When $m$ is radial, Theorem B is
extended to spectral multipliers and we recommend [7, 13] and references
therein.

In [5], Christ, Grafakos, Hozík and Seeger studied $\mathcal{M}_{m}$ when $m$
is the Hörmander-Mikhlin multiplier.

######  Theorem C ([5], Corollary 1.3).

Suppose $1<p<\infty,r=\min\\{p,2\\}$, and $\beta>d/r$. Suppose that

$\displaystyle\|m(2^{k}\cdot)\widehat{\psi}(\cdot)\|_{L_{\beta}^{r}(\mathbb{R}^{d})}$
$\displaystyle\leq\omega(k),\quad k\in\mathbb{Z},$ (1.2)
$\displaystyle\omega^{*}(0)+\sum_{l=1}^{\infty}\frac{\omega^{*}(l)}{l}$
$\displaystyle<\infty.$

Then $\mathcal{M}_{m}$ is bounded on $L^{p}(\mathbb{R}^{d})$. Note that
$\omega^{*}(t):=\sup\\{\lambda>0:\text{card}\big{(}\\{k:|\omega(k)|>\lambda\\}\big{)}>t\\}$.
Moreover, when $p=\infty$ $\mathcal{M}_{m}$ maps $L^{\infty}$ to $BMO$.

######  Remark 1.

In [5] the authors also present a counter example which yields that being the
Mikhlin multiplier is not sufficient for $m$ to guarantee the $L^{p}$ mapping
property of $\mathcal{M}_{m}$. In [10], Grafakos, Hozík and Seeger proved that
the same result holds when the denominator $l$ in the summation of (1.2) is
replaced into $l\sqrt{\log(l)}$ and $\alpha>d/p+1/p^{\prime}$ for $p\in(1,2]$,
$\alpha>d/2+1/p$ for $p>2$. For more details in this direction, we recommend
[5, 10] and references therein.

Considering historical results, one of novelties of this paper is that we
obtain a low regularity result in terms of Theorem C with an operator norm
derived from Theorems A and B. We should note that our result, Theorem 1.1,
doesn’t yield an improvement for Theorem C. However, since Theorem 1.1
requires reduced regularity of $m$, so the regularity difference allows us to
consider not only Mikhlin type multipliers but also certain singular Fourier
multipliers introduced by Miyachi in [15] and Fourier multipliers of limited
decay in [17]. To do so, we first modify square function estimates for
$\mathcal{M}_{m}$ by making use of fractional calculus and bilinear complex
interpolation. For the bilinear interpolation, we consider the square function
as a bilinear operator for $m$ and $f$. Also we introduce an appropriate
function space for $m$ to apply the interpolation.

To state main theorem, we recall notations for maximal Fourier multiplier
operators.

$\displaystyle\mathcal{M}_{m}(f)(x)$
$\displaystyle:=\sup_{t>0}\big{|}T_{m(t\cdot)}(f)(x)\big{|},$ $\displaystyle
T_{m}(f)(x)$ $\displaystyle:=\int_{\mathbb{R}^{d}}e^{2\pi
ix\cdot\xi}m(\xi)\widehat{f}(\xi)d\xi.$

We also make use of a function space $\Sigma^{2}(\mathcal{B})$ whose norm is
given by

$\|f\|_{\Sigma^{2}(\mathcal{B})}:=\biggl{(}\sum_{j\in\mathbb{Z}}\big{\|}f(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{\mathcal{B}}^{2}\biggr{)}^{1/2},$

where $\mbox{supp}(\widehat{\psi})$ is in $\\{\frac{1}{2}<|\xi|<2\\}$ as
defined in (2.1) and $\mathcal{B}$ denotes a Banach space of functions on
$\mathbb{R}^{d}$. Properties of $\Sigma^{2}(\mathcal{B})$ are introduced in
Section 2. In most cases, we choose $\mathcal{B}=L_{s}^{p}$ or $B_{p,p}^{s}$.
We note that $B_{p,q}^{s}(\mathbb{R}^{d})$ is a space of tempered
distributions $f$ whose norm is given by

$\|f\|_{B_{p,q}^{s}(\mathbb{R}^{d})}:=\|S_{0}f\|_{L^{p}(\mathbb{R}^{d})}+\bigg{(}\sum_{j=1}^{\infty}\big{(}2^{js}\|\psi_{j}\ast
f\|_{L^{p}(\mathbb{R}^{d})}\big{)}^{q}\bigg{)}^{1/q},\quad S_{0}f:=\sum_{j\leq
0}\psi_{j}\ast f,$

where $\psi_{j}(\cdot)=2^{jd}\psi(2^{j}\cdot)$. Then for $p=q$ we let
$B_{p,p}^{s}(\mathbb{R}^{d})=B_{p}^{s}(\mathbb{R}^{d})$, and
$B_{p}^{s}(\mathbb{R}^{d})$ satisfies the embedding property
$L_{s}^{p}(\mathbb{R}^{d})\subseteq B_{p}^{s}(\mathbb{R}^{d})$ for $p\geq 2$
and $s\in\mathbb{R}$. Now we introduce our main theorem.

######  Theorem 1.1.

Let $p\in(1,\infty)$. If $m\in\Sigma^{2}(B_{p_{0}}^{s})$,
$\frac{1}{p_{0}}=|\frac{1}{p}-\frac{1}{2}|$ and
$s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}$, then we have

$\displaystyle\big{\|}\mathcal{M}_{m}(f)\big{\|}_{L^{p}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

Note that $B_{p_{0}}^{s}=B_{p_{0},p_{0}}^{s}(\mathbb{R}^{d})$ denotes the
Besov space on $\mathbb{R}^{d}$. Moreover, when $p=1$ or $p=\infty$,
$\mathcal{M}_{m}$ satisfies $(H^{1}\to L^{1})$, $(L^{\infty}\to BMO)$
estimates, respectively.

In proving Theorem 1.1, we first obtain a pointwise upper bound of
$\mathcal{M}_{m}$ by a Hilbert space valued operator $T_{\widetilde{m}}$,
where $\widetilde{m}$ is a variant of $m$ by using the fractional calculus. We
note that the vector-valued operator satisfies $(L^{1}\to H^{1})$, $(BMO\to
L^{\infty})$ estimates due to [16]. Then we consider the vector-valued
operator $T_{\widetilde{m}}(f)$ as a bilinear operator for $m$ and $f$, and
make use of multilinear interpolation introduced by Calderón [4, paragraph
10.1]. We also introduce an appropriate function space for $m$ and
$\widetilde{m}$. The function space and the multilinear perspective allows us
to apply the multilinear interpolation, which yields interpolation among
conditions on $m$ so that we obtain condition for $\mathcal{M}_{m}$ to be
bounded on each $L^{p}(\mathbb{R}^{d})$.

######  Remark 2.

For $m$ in Theorem 1.1, we can say that
$\|m\|_{\Sigma^{2}(C(\mathbb{R}^{d}))}<\infty$ since
$B_{p_{0}}^{s}(\mathbb{R}^{d})\subset C(\mathbb{R}^{d})$. Thus it follows that
$m(\xi)\to 0$ as $|\xi|\to 0$ or $|\xi|\to\infty$. If one considers $m$ such
that $m(\xi)\not\to 0$ as $|\xi|\to 0$, Theorem 1.1 still holds if $m$ is of
class $C^{\infty}$ near the origin. Hence we have

$T^{*}(f):=\sup_{t>0}\big{|}\big{(}m(t\cdot)\varphi_{0}(t\cdot)\widehat{f}(\cdot)\big{)}\check{\,}\big{|}\lesssim\mathcal{M}f,$

where $\varphi_{0}\in C^{\infty}(\mathbb{R}^{d})$, $\varphi_{0}\equiv 1$ in
$B(0,1)$ and vanishes outside of $B(0,2)$. Although $m$ is not smooth, if $m$
is of class $C^{[\frac{d}{2}]+2}(\mathbb{R}^{d})$, then one can use [16,
Theorem II.2.1] to obtain the $L^{p}$ boundedness of $T^{*}$ for
$1<p\leq\infty$.

The novelty of Theorem 1.1 is that we actually obtain maximal estimates for
$\mathcal{M}_{m}$ when $m$ is not differentiable at the origin; the condition
of Theorem 1.1 allows us to consider $m$ whose derivatives blow up at the
origin. For examples of these kinds, we present Propositions 1.5 and 1.7.

Even if $m$ is not smooth and is not of class
$C^{[\frac{d}{2}]+2}(\mathbb{R}^{d})$, $L^{p}$ estimates for $\mathcal{M}_{m}$
still follow from Theorem 1.1.

######  Corollary 1.2.

Let $p\in(1,\infty)$, and $m$ be a function of class
$B^{s}_{p_{0}}(\mathbb{R}^{d})$ where
$s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}$. Then, for any fixed
$\phi_{0}\in C_{c}^{\infty}(\mathbb{R}^{d})$ we have

$\|\mathcal{M}_{m\phi_{0}}(f)\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|m\|_{B^{s}_{p_{0}}(\mathbb{R}^{d})}\|f\|_{L_{p}(\mathbb{R}^{d})}\,.$

Moreover, when $p=1$ or $p=\infty$, $\mathcal{M}_{m\phi_{0}}$ satisfies
$(H^{1}\to L^{1})$, $(L^{\infty}\to BMO)$ estimates, respectively. In
particular, if $m$ is a function of class $L^{2}_{s}$ with $s>\frac{d+1}{2}$
and has compact support, then $\|\mathcal{M}_{m}\|_{L^{p}\rightarrow L^{p}}$
is bounded for any $p\in(1,\infty)$.

Due to $\frac{d+1}{2}<[\frac{d}{2}]+2$, Corollary 1.2 improves the condition
in Remark 2 that $m$ is of class $C^{[\frac{d}{2}]+2}(\mathbb{R}^{d})$. To our
best knowledge, Corollary 1.2 is the first improvement for $m$ supported near
the origin since Rubio de Francia suggested the condition $m\in
C^{[\frac{d}{2}]+2}(\mathbb{R}^{d})$ in [17].

Now we introduce distinguishing features of Theorem 1.1 in terms of Theorem B
and Theorem C. In [6], the authors actually suggested a condition for non-
radial $m$ under which $\mathcal{M}_{m}$ is of strong type $(p,p)$ for
$1\in(1,\infty)$. We note that Theorem 1.1 requires $m$ to be regular in terms
of $s>\frac{d+1}{2}$ for $L^{p}$ boundedness for all $p\in(1,\infty)$, while
conditions in [6] require $s>d$ for the same result. For results of radial $m$
in [6], Theorem 1.1 yields the following counter part:

######  Corollary 1.3.

Let $m$ be a radial symmetric function given by $m(\xi)=h(|\xi|)$ and
$s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}$. Then we have

(1.3)
$\displaystyle\|\mathcal{M}_{m}(f)\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|h\|_{{\Sigma^{2}(L_{s}^{2}(\mathbb{R}))}}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

Corollary 1.3 is similar with the result of [6] in the radial symmetric case.
We note that (1.3) maybe useful in some situations due to the norm equivalence
such as Lemma 4.1 or Proposition 4.2.

We now investigate results of Theorem 1.1 and Theorem C, which is [5,
Corollary 1.3]. In [5, Corollary 1.3] for $p,q\in(1,\infty)$ the condition

(1.4)
$\displaystyle\sum_{j\in\mathbb{Z}}\|m(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{r}}^{q}<\infty,\quad
r=\min\\{p,2\\},\,\,s>\frac{d}{r}$

yields the $L^{p}$ boundedness of $\mathcal{M}_{m}$. Recall that in Theorem
1.1, $\mathcal{M}_{m}$ is bounded on $L^{p}(\mathbb{R}^{d})$ whenever

(1.5)
$\displaystyle\sum_{j\in\mathbb{Z}}\|m(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{B_{p_{0}}^{s}}^{2}<\infty,\quad\frac{1}{p_{0}}=\Big{|}\frac{1}{p}-\frac{1}{2}\Big{|},\,\,s>d\Big{|}\frac{1}{p}-\frac{1}{2}\Big{|}+\frac{1}{2}.$

It should be noted that the quantities (1.4) and (1.5) cannot be directly
compared. When $p\leq 2$, however, comparing the ranges of $s$ in (1.4) and
(1.5), one can check a gain of regularity by $\frac{d-1}{2}$. Such gain of
regularity allows us to consider non-Mikhlin type Fourier multipliers. For
examples of non-Mikhlin type, we suggest singular Fourier multipliers in [15]
or multipliers of limited decay in [17], which cannot be handled by the
condition (1.4). We also present a convergence type result for those Fourier
multipliers, which are Propositions 1.5 and 1.7.

On the other hand, for $p>2d$ (1.5) is dominated by (1.4) with $q=2$, $r=2$,
$\alpha>d/2$, so one can actually improve Theorem 1.1 by interpolating $L^{2}$
result of this paper and $L^{\infty}\to BMO$ result of [5]. We should remark
that the interpolation yields

(1.6) $\displaystyle\|\mathcal{M}_{m}\|_{L^{p}\to
L^{p}}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})},\quad\text{for}\quad
s>d\Big{(}\frac{1}{2}-\frac{1}{p}\Big{)}+\frac{1}{p},\quad\text{and $p\geq
2$},$

where we adapt interpolation for vector-valued Hardy spaces and BMO in [2].
Note that (1.6) yields a gain of regularity by $\frac{1}{2}-\frac{1}{p}$
compared to (1.5) whenever $p\geq 2$. Due to (1.6), we remark that every
results derived from Theorem 1.1 can be improved when $p\geq 2$.

Now we introduce applications of Theorem 1.1 whose proofs are given in Section
6. First of all, we define the notion of Fourier multipliers with $slow$
$decay$. A function $m$ is said to be a Fourier multiplier with slow decay if
$m$ satisfies

(1.7) $\displaystyle|\partial^{\gamma}m(\xi)|\leq
C|\xi|^{-\delta|\gamma|},\quad\text{for some $\delta\in(0,1)$}.$

It can be directly checked that the slow decay condition (1.7) cannot be
implied by the Mikhlin condition. We also give a non-trivial example of
multipliers with slow decay. Let $m_{\beta}$ be a Mikhlin multiplier such that
$|\partial^{\gamma}m_{\beta}(\xi)|\lesssim|\xi|^{-\beta-|\gamma|}$ and
$m_{\beta}$ vanishes near the origin. Then for $\alpha\in(0,1)$ we define
$m_{\alpha,\beta}$, $\mathfrak{M}_{\alpha,\beta}$ as following:

(1.8) $\displaystyle m_{\alpha,\beta}(\xi)$
$\displaystyle:=e^{i|\xi|^{\alpha}}m_{\beta}(\xi),$
$\displaystyle\mathfrak{M}_{\alpha,\beta}(f)(x)$
$\displaystyle:=\sup_{t>0}\big{|}T_{m_{\alpha,\beta}(t\cdot)}(f)(x)\big{|}.$

Then $m_{\alpha,\beta}$ is slowly decaying with $\delta=1-\alpha$ in (1.7). In
terms of Theorem C, one can see that $\mathfrak{M}_{\alpha,\beta}$ is bounded
on $L^{p}(\mathbb{R}^{d})$ for $\frac{1}{p}<\frac{\beta/\alpha}{d}$. Applying
Theorem 1.1 to $\mathfrak{M}_{\alpha,\beta}$, however, we can make an
improvement on the range of $p$.

######  Corollary 1.4.

$\mathfrak{M}_{\alpha,\beta}$ is bounded on $L^{p}(\mathbb{R}^{d})$ for
$|\frac{1}{p}-\frac{1}{2}|<\frac{\beta/\alpha}{d}-\frac{1}{2d}$.

The proof of Corollary 1.4 is given in Section 6. Here we present a breif
reasoning for the range $\frac{1}{p}<\frac{\beta/\alpha}{d}+\frac{d-1}{2d}$.
It is straight forward to see that
$|\partial^{\gamma}m_{\alpha,\beta}(\xi)|\lesssim|\xi|^{-(\beta+(1-\alpha)k)}$
for $|\gamma|=k$. To apply Theorem 1.1 we check $k=\beta+(1-\alpha)k$, so
$k=\beta/\alpha$. Then we must have
$k=\beta/\alpha>d(\frac{1}{p}-\frac{1}{2})+\frac{1}{2}$, which gives
$\frac{1}{p}<\frac{\beta/\alpha}{d}+\frac{d-1}{2d}$. In [15], one can find
that for radial $m_{\alpha,\beta}$ the maximal operator
$\mathfrak{M}_{\alpha,\beta}$ is bounded on $L^{p}(\mathbb{R}^{d})$ for
$p\in(1,\infty)$ when $\beta>\frac{d\alpha}{2}$ since the Fourier transform of
$m_{\alpha,\beta}$ is in $L^{1}$ for those $\beta$, $\alpha$. In our case, for
$m_{\alpha,\beta}$ which is not necessarilly radial
$\mathfrak{M}_{\alpha,\beta}$ is bounded on $L^{p}(\mathbb{R}^{d})$ for
$p\in(1,\infty)$ when $\beta>\frac{d\alpha}{2}+\frac{\alpha}{2}$. One can
understand the difference $\frac{\alpha}{2}$ is a radial counter part.

As an application of Corollary 1.4, we can study
$e^{-it(-\Delta)^{\frac{\alpha}{2}}}$, which is a solution of a half wave
equation or fractional Schrödinger equations. Particularly, for $\alpha=1$ one
can relate $e^{-it(-\Delta)^{\frac{1}{2}}}$ to a Fourier intergral operator of
degree 1, which is a central object in harmonic analysis. In studies on
dispersive equations, the Fourier integral operators, and the related topics,
$e^{-it(-\Delta)^{\frac{\alpha}{2}}}(f)$ is considered as an extension
operator $\mathcal{F}f(x,t)$, and $\big{(}L^{2}\to L_{t}^{q}L_{x}^{r}\big{)}$
estimates of $\mathcal{F}f$ is one of main goals of studying Fourier integral
operators and dispersive equations. In this paper, we do not discuss the mixed
norm estimates for $e^{-it(-\Delta)^{\frac{\alpha}{2}}}$, and for $\alpha=1$
we recommend [11, 14] and references therein. Instead of the mixed norm
estimates, we are interested in $L^{p}$ behavior of

$U_{\alpha,\beta}(f)(x,t)=\frac{e^{-it(-\Delta)^{\frac{\alpha}{2}}}-I}{t^{\beta}}(f)(x),\quad\text{as
$t\to 0$.}$

######  Proposition 1.5.

For $\alpha\in(0,1)$ and $\beta\in(1/2,1)$, $U_{\alpha,\beta}(f)(x,t)\to 0$ in
$L^{p}(\mathbb{R}^{d})$ as $t\to 0$, whenever
$\frac{d-2\beta+1}{2d}<\frac{1}{p}<\frac{d+2\beta-1}{2d}$, and
$f\in\dot{L}_{\alpha\beta}^{p}(\mathbb{R}^{d})$. In particular, under the same
condition on $\alpha,\beta$ and $f$, we have for $a.e.$ $x$

$\Big{|}e^{-it(-\Delta)^{\frac{\alpha}{2}}}f(x)-f(x)\Big{|}=O(t^{\beta}),\quad
t\to 0.$

Note that $f\in\dot{L}_{s}^{p}$ is equivalent to
$(|\cdot|^{s}\widehat{f})\check{\,}\in L^{p}$.

Theorem 1.1 can be also applied to $\mathcal{M}_{m}$ when $m$ enjoys both
Mikhlin’s condition and the limited decay. Let $a>0$, $b>\frac{1}{2}$. We
assume that $m_{a,b}$ is of class $L^{2}_{s,loc}$ for $s>\frac{d+1}{2}$ and
satisfies that for given $p\in(1,\infty)$ and $j>0$,

(1.9)
$\displaystyle\|m_{a,b}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\lesssim
2^{-j\min(a,b-s)},\quad\frac{1}{p_{0}}=\big{|}\frac{1}{p}-\frac{1}{2}\big{|},\quad
s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}.$

Under the condition (1.9) we have

######  Corollary 1.6.

Let $m_{a,b}$ be a Fourier multiplier satisfying (1.9). Then
$\mathcal{M}_{a,b}$ is bounded on $L^{p}(\mathbb{R}^{d})$ for
$\frac{d+1-2b}{2d}<\frac{1}{p}<\frac{d-1+2b}{2d}$.

The condition (1.9) can be understood in a two-fold manner. If $m$ satisfies

(1.10)
$\displaystyle\|m(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\lesssim
2^{-ja},\quad j>0,$

then $m$ can be regarded as a Mikhlin multiplier such that
$|\partial^{\gamma}m(\xi)|\lesssim(1+|\xi|)^{-a}|\xi|^{-|\gamma|}$ and it
satisfies conditions of Theorem C, which guarantees that $\mathcal{M}_{m}$ to
be bounded on $L^{p}$ for certain $p$. On the other hand, if we have

(1.11)
$\displaystyle\|m(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\lesssim
2^{-j(b-s)},\quad j>0,$

then $m$ is a multiplier of the limited decay whose maximal estimates are
introduced in [17]. It can be checked that (1.10) behaves worse than (1.11)
for $s<b-a$ and (1.11) enjoys worse decay than that of (1.10) for $s>b-a$.
Thus, (1.9) considers the worst case when the cases (1.10) and (1.11) are
combined.

Using Corollary 1.6, we obtain a pointwise convergence type result. For $m\in
C_{loc}^{[\frac{d+1}{2}]+1}(\mathbb{R}^{d})$ such that
$|\partial^{\gamma}m(\xi)|\leq(1+|\xi|)^{-\beta}$ for any multi-index
$\gamma$, and $m(0)=1$, we have

######  Proposition 1.7.

For $\alpha\in(0,1)$ and $f\in\dot{L}_{\alpha}^{p}(\mathbb{R}^{d})$ with
$\frac{d+1-2(\alpha+\beta)}{2d}<\frac{1}{p}<\frac{d-1+2(\alpha+\beta)}{2d}$,
we have

$\lim_{t\to 0}\frac{f(x)-T_{m(t\cdot)}f(x)}{t^{\alpha}}=0,\quad\text{for
almost every $x\in\mathbb{R}^{d}$}.$

A noteworthy example for $m$ in Proposition 1.7 is the Fourier transform of a
surface-carried measure $\sigma$ on $S\subset\mathbb{R}^{d}$, where $S$ is a
hypersurface of nonvanishing $d-1$ principal curvatures.

######  Notation.

Let $A\lesssim B$ denote $A\leq CB$, where $C$ is independent of $A$ and $B$.
$B(x,R)$ denotes a ball in $\mathbb{R}^{d}$ of radius $R$ centered at $x$.

##  2\. Preliminaries

###  2.1. Function spaces

Choose $\phi\in\mathscr{S}(\mathbb{R}^{d})$ such that $\widehat{\phi}\equiv 1$
on $B(0,1)$ and $\widehat{\phi}\equiv 0$ on $B(0,2)^{c}$. Define

(2.1)
$\displaystyle\widehat{\psi}_{j}=\widehat{\psi}(\cdot/2^{j})=\widehat{\phi}(\cdot/2^{j})-\widehat{\phi}(\cdot/2^{j-1})\quad\text{for}\quad
j\in\mathbb{Z}.$

Then we introduce function spaces appeared in our arguments. Let $\mathcal{B}$
be a Banach space continuously embedded in the space of distributions
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$. For $q\in(1,\infty)$ and
$\theta\in\mathbb{R}$, we define normed spaces

$\displaystyle\Sigma_{\theta}^{q}(\mathcal{B})$
$\displaystyle:=\\{f\in\mathscr{D}^{\prime}(\mathbb{R}^{d}\setminus\\{0\\}):\|f\|_{\Sigma_{\theta}^{q}(\mathcal{B})}^{q}:=\sum_{j\in\mathbb{Z}}2^{q\theta}\big{\|}f(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{\mathcal{B}}^{q}<\infty\\},$
$\displaystyle l_{\theta}^{q}(\mathcal{B})$
$\displaystyle:=\\{\\{f_{j}\\}_{j\in\mathbb{Z}}:f_{j}\in\mathcal{B},\,\,\text{and}\,\,\sum_{j\in\mathbb{Z}}2^{q\theta}\|f_{j}\|_{\mathcal{B}}<\infty\\}.$

Note that $\Sigma_{\theta}^{q}(\mathcal{B})$ is a Banach space where we
additionally assume that

(2.2)
$\displaystyle\|\widehat{\psi}f\|_{\mathcal{B}}\lesssim\|f\|_{\mathcal{B}}\quad\text{and}\quad\|f(2\,\cdot\,)\|_{\mathcal{B}}\simeq\|f\|_{\mathcal{B}}.$

Let $S$ and $R$ be mappings given by

$\displaystyle S(f)$
$\displaystyle=\\{f(2^{j}\cdot)\widehat{\psi}(\cdot)\\}_{j\in\mathbb{Z}}=:\\{f_{j}\\}_{j\in\mathbb{Z}},$
$\displaystyle R\bigl{(}\\{f_{j}\\}\bigr{)}$
$\displaystyle=\sum_{j}\big{(}\widehat{\psi}_{j-1}+\widehat{\psi}_{j}+\widehat{\psi}_{j+1}\big{)}(\cdot)f_{j}(2^{-j}\cdot).$

Then by (2.2) $S\in
L\big{(}\Sigma_{\theta}^{q}(\mathcal{B}),l_{\theta}^{q}(\mathcal{B})\big{)}$
and $R\in
L\big{(}l_{\theta}^{q}(\mathcal{B}),\Sigma_{\theta}^{q}(\mathcal{B})\big{)}$,
respectively, where $L(A,B)$ denotes a space of bounded linear maps from $A$
to $B$. Moreover, one can check that $R\circ S$ is an identity map of
$Id_{\Sigma_{\theta}^{q}(\mathcal{B})}$. Hence by [22, Theorem 1.2.4]
$\Sigma_{\theta}^{q}(\mathcal{B})$ is isomorphic to a closed subspace of
$l_{\theta}^{q}(\mathcal{B})$ which is a reflexive Banach space when
$\mathcal{B}$ is reflexive. If $\theta=0$, then we simply write
$\Sigma^{q}(\mathcal{B})$.

Let $(\mathcal{A},\mathcal{B})$ be an interpolation couple of Banach spaces,
$q=(1-\delta)q_{0}+\delta q_{1}$, and
$\theta=(1-\delta)\theta_{0}+\delta\theta_{1}$. Then we have from [22, Theorem
1.2.4 and 1.18.1]

(2.3)
$\displaystyle[\Sigma_{\theta_{0}}^{q_{0}}(\mathcal{A}),\Sigma_{\theta_{1}}^{q_{1}}(\mathcal{B})]_{\delta}=\Sigma_{\theta}^{q}\bigl{(}[\mathcal{A},\mathcal{B}]_{\delta}\bigr{)}.$

For instance, if we choose $\mathcal{A}=L_{s_{1}}^{2}$,
$\mathcal{B}=L_{s_{2}}^{2}$, $s_{1},s_{2}\geq 0$ with $\theta=0$, $q=2$, then
we have by (2.3)

(2.4)
$\displaystyle[\Sigma^{2}(L_{s_{1}}^{2}),\Sigma^{2}(L_{s_{2}}^{2})]_{\delta}=\Sigma^{2}(L_{s}^{2}),\quad
s=(1-\delta)s_{1}+\delta s_{2},\,\delta\in(0,1).$

Another example for $\Sigma^{q}(\mathcal{B})$ is the weighted Sobolev space
$H_{\theta}^{p,\gamma}$ introduced in [12], which equals
$\Sigma^{p}_{\theta/p}(L^{p}_{\gamma})$. When $\gamma\in\mathbb{N}\cup\\{0\\}$
we have

(2.5) $\displaystyle
H_{\theta}^{p,\gamma}(\mathbb{R}^{d}\setminus\\{0\\})=\\{f\in\mathscr{S}^{\prime}(\mathbb{R}^{d}):\|f\|_{H_{\theta}^{p,\gamma}(\mathbb{R}^{d}\setminus\\{0\\})}^{p}:=\sum_{l=0}^{\gamma}\int_{\mathbb{R}^{d}}\big{|}D_{x}^{l}(f)(x)\big{|}^{p}|x|^{pl+\theta-d}dx<\infty\\}.$

Indeed, one can check that by the change of variables,

$\displaystyle\sum_{l=0}^{\gamma}\int_{\mathbb{R}^{d}}\big{|}D_{x}^{l}(f)(x)\big{|}^{p}|x|^{pl+\theta-d}dx$
$\displaystyle\simeq\sum_{l=0}^{\gamma}\sum_{j\in\mathbb{Z}}2^{j(pl+\theta-d)}\int_{\mathbb{R}^{d}}\big{|}D_{x}^{l}(f)(x)\big{|}^{p}|\widehat{\psi}_{j}(x)|^{p}dx$
$\displaystyle=\sum_{l=0}^{\gamma}\sum_{j\in\mathbb{Z}}2^{j\theta}\int_{\mathbb{R}^{d}}\big{|}D_{x}^{l}(f(2^{j}\cdot))\big{|}^{p}|\widehat{\psi}(x)|^{p}dx$
$\displaystyle\simeq\sum_{l=0}^{\gamma}\sum_{j\in\mathbb{Z}}2^{j\theta}\int_{\mathbb{R}^{d}}\big{|}D_{x}^{l}(f(2^{j}\cdot)\widehat{\psi}(\cdot))(x)\big{|}^{p}dx.$

Note that $|\cdot|^{-d}$ is just a scaling factor and for $\theta=0$ it
follows that $H_{0}^{p,\gamma}$-norm is scaling invariant. For more
information of $H_{\theta}^{p,\gamma}$, we recommend [12] and references
therein.

###  2.2. Fractional calculus

To study a maximal operator, classical square function argument makes use of
the fundamental theorem of calculus with respect to $t>0$. In doing so, one
must take derivative with respect to $t$, which in turn yields loss of
regularity of a symbol. Such loss may be optimized if we use a fractional
analogue of the fundamental theorem.

We define the Riemann-Liouville integrals and fractional derivatives of order
$\alpha\in(0,1)$. For all $t\in(0,\infty)$

$\displaystyle I_{0+}^{\alpha}f(t)$
$\displaystyle:=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s)ds,$
$\displaystyle D_{0+}^{\alpha}F(t)$
$\displaystyle:=\frac{d}{dt}\big{(}I_{0+}^{1-\alpha}F\big{)}(t)$

where $f$ is a locally integrable function on $[0,\infty)$ and $F$ satisfies
$I_{0+}^{1-\alpha}F$ is absolutely continuous. For further use of
$I_{0+}^{\alpha}$ and $D_{0+}^{\alpha}$, we introduce the following lemma
which is crucial in the proof of Theorem 1.1:

######  Lemma 2.1.

Let $F\in C_{loc}\big{(}[0,\infty)\big{)}\cap
C^{0,\beta}_{loc}\big{(}(0,\infty)\big{)}$ for a fixed $\beta\in(0,1]$. Then
for any $\alpha\in(0,\beta)$, $D^{\alpha}_{0+}F$ satisfies the following
identity:

(2.6) $\displaystyle
D^{\alpha}_{0+}F(t)=\frac{1}{\Gamma(1-\alpha)}\left(\frac{F(t)}{t^{\alpha}}+\int_{0}^{t}\frac{F(t)-F(s)}{(t-s)^{1+\alpha}}ds\right)\,.$

Moreover, we have

(2.7) $\displaystyle
F(t)=I_{0+}^{\alpha}D_{0+}^{\alpha}F(t)+\frac{t^{\alpha}}{\Gamma(\alpha)}F(0)\,.$

###### Proof.

From the definition of $I^{1-\alpha}F(t)$, for $h>0$ we have

$\displaystyle\frac{I^{1-\alpha}F(t+h)-I^{1-\alpha}F(t)}{h}$
$\displaystyle=\,$
$\displaystyle\frac{1}{h}\int_{t}^{t+h}\big{(}(t+h-s)^{-\alpha})\big{(}F(s)-F(t)\big{)}ds$
$\displaystyle+\frac{1}{h}\int_{0}^{t}\big{(}(t+h-s)^{-\alpha}-(t-s)^{-\alpha})\big{(}F(s)-F(t)\big{)}ds$
$\displaystyle+\frac{1}{h}F(t)\Big{(}\int_{0}^{t+h}(t+h-s)^{-\alpha}-\int_{0}^{t}(t-s)^{-\alpha}ds\Big{)}$
$\displaystyle=:$ $\displaystyle I_{1}+I_{2}+I_{3}\,.$

Define

$A=\sup_{0\leq s\leq 2t}|F(s)|\quad\text{and}\quad B=\sup_{t/2\leq s\leq
2t}\frac{|F(t)-F(s)|}{|t-s|^{\beta}}.$

For $0<h<t/2$, by change of variables, we have

$\displaystyle|I_{1}|\leq\int_{0}^{1}(hr)^{-\alpha}\big{|}F(t+h(1-r))-F(t)\big{|}ds\leq
h^{\beta-\alpha}B\int_{0}^{1}r^{-\alpha}(1-r)^{\beta}dr.$

Therefore, $I_{1}\rightarrow 0$ as $h\rightarrow 0$. For $I_{2}$, we make use
of the fundamental theorem of calculus to obtain

$\displaystyle
I_{2}=-\alpha\int_{0}^{t}\int_{0}^{1}(t-s+hr)^{-1-\alpha}dr\big{(}F(s)-F(t)\big{)}ds\,.$

Note that

$\displaystyle\int_{0}^{t}\int_{0}^{1}(t-s+hr)^{-1-\alpha}\big{|}F(s)-F(t)\big{|}drds$
$\displaystyle\leq\,$
$\displaystyle\int_{0}^{t}(t-s)^{-1-\alpha}|F(s)-F(t)|ds$
$\displaystyle\lesssim\,$ $\displaystyle
At^{-\alpha}+Bt^{\beta-\alpha}<\infty\,.$

Thus by the dominated convergence theorem, it follows that

$I_{2}\rightarrow-\alpha\int_{0}^{t}(t-s)^{-1-\alpha}\big{(}F(s)-F(t)\big{)}ds\quad\text{as}\quad
h\to 0.$

For $I_{3}$, by direct calculation we obtain

$I_{3}=\frac{F(t)}{1-\alpha}\times\frac{(t+h)^{1-\alpha}-t^{1-\alpha}}{h}\rightarrow\frac{F(t)}{t^{\alpha}}\,.$

For $\frac{I^{1-\alpha}F(t)-I^{1-\alpha}F(t-h)}{h}$ and $h>0$, we have

$\displaystyle\frac{I^{1-\alpha}F(t)-I^{1-\alpha}F(t-h)}{h}=\,$
$\displaystyle\frac{1}{h}\int_{t-h}^{t}\big{(}(t-s)^{-\alpha})\big{(}F(s)-F(t-h)\big{)}ds$
$\displaystyle+\frac{1}{h}\int_{0}^{t-h}\big{(}(t-s)^{-\alpha}-(t-h-s)^{-\alpha})\big{(}F(s)-F(t-h)\big{)}ds$
$\displaystyle+\frac{1}{h}F(t-h)\Big{(}\int_{0}^{t}(t-s)^{-\alpha}-\int_{0}^{t-h}(t-h-s)^{-\alpha}ds\Big{)}.$

Then by the same argument it follows that for $h>0$

$\frac{I^{1-\alpha}F(t)-I^{1-\alpha}F(t-h)}{h}\rightarrow\frac{F(t)}{t^{\alpha}}+\int_{0}^{t}\frac{F(t)-F(s)}{(t-s)^{1+\alpha}}ds\quad\text{as}\quad
h\to 0\,.$

Therefore (2.6) is proved.

From the assumption of $F$ we obtain $D^{\alpha}_{0+}F(t)\in
L^{1}_{loc}([0,\infty))$, and the equality (2.7) holds by [18, Theorem 2.4].
The lemma is proved. ∎

##  3\. Proof of Theorem 1.1

Let $\varepsilon\in(0,\frac{1}{6})$ be sufficiently small and
$s>d|\frac{1}{p}-\frac{1}{2}|+\frac{1}{2}+3\varepsilon$. Since
$\Sigma^{2}(B^{s}_{p_{0}})\subset\Sigma^{2}(C^{0,\frac{1}{2}+3\varepsilon})$,
we consider $m$ as an element of
$C_{loc}^{0,\frac{1}{2}+3\varepsilon}(\mathbb{R}^{d}\setminus\\{0\\})$ and

$\|m(2^{j}\cdot)\widehat{\psi}\|_{C^{0,1/2+3\varepsilon}}\rightarrow
0\quad\text{as}\quad j\rightarrow-\infty.$

This implies that $m(\xi)\rightarrow 0$ as $\xi\rightarrow 0$. Thus we want to
show that for such $m$,

$\displaystyle\big{\|}\mathcal{M}_{m}(f)\big{\|}_{L^{p}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

To do this we introduce a square function type estimate using fractional
calculus. From Lemma 2.1 we obtain that

$m(t\xi)=\frac{1}{\Gamma(\frac{1}{2}-\varepsilon)}I^{\frac{1}{2}+\varepsilon}\big{(}(\cdot)^{-\frac{1}{2}-\varepsilon}\widetilde{m}(\cdot\xi)\big{)}(t),$

where

$\widetilde{m}(t\xi):=m(t\xi)+(1/2+\varepsilon)\int_{0}^{1}\frac{m(t\xi)-m(st\xi)}{(1-s)^{\frac{3}{2}+\varepsilon}}ds\,.$

Note that for fixed $\xi\in\mathbb{R}^{d}$ with $|\xi|\in(2^{j-1},2^{j+1})$
for some $j\in\mathbb{Z}$, we have

$\displaystyle|\widetilde{m}(\xi)|\lesssim\|m\|_{L^{\infty}}+\Big{|}\int_{0}^{1}\frac{m(\xi)-m(s\xi)}{(1-s)^{\frac{3}{2}+\varepsilon}}ds\Big{|}\lesssim\|m\|_{L^{\infty}}+\Big{|}\int_{1/2}^{1}\frac{m(\xi)-m(s\xi)}{(1-s)^{\frac{3}{2}+\varepsilon}}ds\Big{|}.$

Note that
$\|m\|_{\infty}\lesssim\|m\|_{\Sigma(L^{\infty})}\lesssim\|m\|_{\Sigma(C^{0,1/2+3\varepsilon})}$.
We also have

$\displaystyle\Big{|}\int_{1/2}^{1}\frac{m(\xi)-m(s\xi)}{(1-s)^{\frac{3}{2}+\varepsilon}}ds\Big{|}$
$\displaystyle\lesssim|\xi|^{\frac{1}{2}+3\varepsilon}\sup_{1/2<s<1}\frac{|m(\xi)-m(s\xi)|}{|\xi-s\xi|^{\frac{1}{2}+3\varepsilon}}$
$\displaystyle\lesssim
2^{j(\frac{1}{2}+3\varepsilon)}\sum_{k=j-1}^{j+1}\big{[}m\widehat{\psi}_{k}\big{]}_{C^{0,1/2+3\varepsilon}}$
$\displaystyle\lesssim\sum_{k=j-1}^{j+1}\|m(2^{k}\cdot)\widehat{\psi}(\cdot)\|_{C^{0,1/2+3\varepsilon}}\lesssim\|m\|_{\Sigma^{2}(C^{0,1/2+3\varepsilon})}<\infty.$

Since we choose arbitrary $\xi\in\mathbb{R}^{d}$, $\widetilde{m}$ is bounded.
By the boundedness of $\widetilde{m}$, we make use of Fubini’s theorem to
obtain

$T_{m(t\cdot)}f=\frac{1}{\Gamma(\frac{1}{2}-\varepsilon)}I^{\frac{1}{2}+\varepsilon}\big{(}t^{-\frac{1}{2}-\varepsilon}T_{\widetilde{m}(t\cdot)}f\big{)}.$

Thus it follows that

(3.1) $\displaystyle\big{|}T_{m(t\cdot)}f\big{|}^{2}$
$\displaystyle=\big{|}\frac{1}{\Gamma(\frac{1}{2}-\varepsilon)}I^{\frac{1}{2}+\varepsilon}\big{(}t^{-\frac{1}{2}-\varepsilon}T_{\widetilde{m}(t\cdot)}f\big{)}\big{|}^{2}$
$\displaystyle=\bigg{|}\frac{1}{\Gamma(\frac{1}{2}-\varepsilon)}\int_{0}^{t}(t-s)^{-\frac{1}{2}+\varepsilon}s^{-\frac{1}{2}-\varepsilon}T_{\widetilde{m}(s\cdot)}fds\bigg{|}^{2}$
$\displaystyle\lesssim\int_{0}^{t}(t-s)^{-1+2\varepsilon}s^{-2\varepsilon}ds\times\int_{0}^{t}\big{|}T_{\widetilde{m}(s\cdot)}f\big{|}^{2}\frac{ds}{s}.$

By taking supremum over $t>0$ on (3.1), we obtain

(3.2)
$\displaystyle\big{|}\mathcal{M}_{m}f|^{2}\lesssim\int_{0}^{\infty}\big{|}T_{\widetilde{m}(t\cdot)}f\big{|}^{2}\frac{dt}{t}=:G_{\widetilde{m}}(f)^{2}.$

Thanks to (3.2), we obtain $L^{2}(\frac{dt}{t})$-valued operator instead of
supremum over $t>0$ in price of $\widetilde{m}$. Since we desire a condition
of $m$ for the $L^{p}$ boundedness, the following mapping property of
$m\mapsto\widetilde{m}$ is required:

######  Lemma 3.1.

For $\beta\geq 0$

(3.3) $\displaystyle\sup_{\xi\neq
0}\Big{(}\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}\,\Big{)}^{1/2}$
$\displaystyle\lesssim\|m\|_{\Sigma^{2}(C^{0,1/2+3\varepsilon})},$ (3.4)
$\displaystyle\|\widetilde{m}\|_{\Sigma^{2}(L_{\beta}^{2}(\mathbb{R}^{d}))}$
$\displaystyle\lesssim\|m\|_{\Sigma^{2}(L_{\beta+1/2+3\varepsilon}^{2}(\mathbb{R}^{d}))}.$

Additionally, to perform bilinear interpolation, we need initial estimates for
$G_{m}$ in terms of the left hand sides of (3.3) and (3.4), which is Lemma
3.2.

######  Lemma 3.2.

For the square function

$\displaystyle
G_{m}(f)=\bigg{(}\int_{0}^{\infty}\big{|}T_{m(t\cdot)}f\big{|}^{2}\frac{dt}{t}\bigg{)}^{1/2},$

we have

$\displaystyle\big{\|}G_{m}(f)\big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim\sup_{\xi\neq
0}\Big{(}\int_{0}^{\infty}|m(t\xi)|^{2}\frac{dt}{t}\,\Big{)}^{1/2}\|f\|_{L^{2}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}G_{m}(f)\big{\|}_{L^{1}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L_{s}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})},$
$\displaystyle\big{\|}G_{m}(f)\big{\|}_{BMO(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L_{s}^{2})}\|f\|_{L^{\infty}(\mathbb{R}^{d})}.$

For further reasoning, we temporarily assume Lemmas 3.1 and 3.2. We present
their proofs in Sections 4 and 5. Then by Lemmas 3.1, 3.2, we conclude that

(3.5)
$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{2}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|m\|_{\Sigma^{2}(C^{0,1/2+3\varepsilon})}\|f\|_{L^{2}(\mathbb{R}^{d})},$
(3.6)
$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{1}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|m\|_{\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})}.$

We interpolate (3.5), (3.6) in terms of Lemma 5.2 to obtain

$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{p}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})},$

for $p_{0}=|1/p-1/2|^{-1}$ and $s\geq
d|\frac{1}{p}-\frac{1}{2}|+\frac{1}{2}+3\varepsilon$. We present the details
for the interpolation in subsection 5.2 and note that the interpolation is
valid since we consider $G_{\widetilde{m}}$ as an $L^{2}(\frac{dt}{t})$-valued
bilinear operator $B$ in the sense that

$B(m,f)(x,t):=T_{\widetilde{m}(t\,\cdot\,)}f(x)\quad\text{and}\quad\big{\|}B(m,f)\big{\|}_{L^{2}(\frac{dt}{t})}=G_{\widetilde{m}}(f).$

Then by (3.2), we have

$\displaystyle\big{\|}\mathcal{M}_{m}(f)\big{\|}_{L^{p}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})}.$

This proves the theorem.

##  4\. Proof of Lemma 3.1

###  4.1. Proof of (3.3)

We first show that $\sup_{\xi\neq
0}\big{(}\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}\big{)}^{1/2}\lesssim\|m\|_{\Sigma^{2}(C^{0,1/2+3\varepsilon})}$.
Recall that

$\widetilde{m}(\xi)=m(\xi)+(1/2+\varepsilon)\int_{0}^{1}(1-s)^{-\frac{3}{2}-\varepsilon}\bigl{(}m(\xi)-m(s\xi)\bigr{)}ds,$

where $m$ vanishes near the origin. Then for $m$ we have

$\displaystyle\sup_{\xi\not=0}\int_{0}^{\infty}|m(t\xi)|^{2}\frac{dt}{t}\,$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\sup_{\xi\not=0}\int_{2^{j}/|\xi|}^{2^{j+1}/|\xi|}|m(t\xi)|^{2}\frac{dt}{t}$
$\displaystyle\lesssim\sum_{j\in\mathbb{Z}}\|m(2^{j}\,\cdot\,)\widehat{\psi}\|_{\infty}^{2}\leq\sum_{j\in\mathbb{Z}}\|m(2^{j}\,\cdot\,)\widehat{\psi}\|_{C^{0,1/2+3\varepsilon}(\mathbb{R}^{d})}^{2},$

Thus, without loss of generality we set

(4.1)
$\displaystyle\widetilde{m}(\xi)=\int_{0}^{1}(1-s)^{-\frac{3}{2}-\varepsilon}\bigl{(}m(\xi)-m(s\xi)\bigr{)}ds.$

Due to

$\displaystyle|\widetilde{m}(\xi)|\lesssim\int_{0}^{1/2}|m(\xi)|+|m(s\xi)|ds+\int_{1/2}^{1}(1-s)^{-\frac{3}{2}-\varepsilon}|m(\xi)-m(s\xi)|ds,$

we have

(4.2)
$\displaystyle\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}\lesssim\int_{0}^{\infty}\Big{(}|m(t\xi)|^{2}+I(t\xi)^{2}\Big{)}\frac{dt}{t}=\sum_{j\in\mathbb{Z}}\int_{|\xi|^{-1}}^{2|\xi|^{-1}}\Big{(}|m(2^{j}t\xi)|^{2}+I(2^{j}t\xi)^{2}\Big{)}\frac{dt}{t}$

where

$I(\xi)=\int_{1/2}^{1}(1-s)^{-\frac{3}{2}-\varepsilon}|m(\xi)-m(s\xi)|ds.$

Put $\widehat{\eta}=\widehat{\psi}_{-1}+\widehat{\psi}_{0}+\widehat{\psi}_{1}$
which equals to 1 on $\\{\xi\,:\,1/2\leq|\xi|\leq 2\\}$. For $|\xi|^{-1}\leq
t\leq 2|\xi|^{-1}$ and $1/2\leq s\leq 1$, we obtain

$\frac{1}{2}\leq|st\xi|\leq|t\xi|\leq 2.$

Therefore we have

$\displaystyle|m(2^{j}t\xi)|\leq\|m(2^{j}\cdot)\widehat{\eta}\|_{\infty}\,\,,\,\,\,\,\frac{|m(2^{j}t\xi)-m(2^{j}st\xi)|}{\big{(}(1-s)t|\xi|\big{)}^{1/2+3\varepsilon}}\leq\big{[}m(2^{j}\cdot)\widehat{\eta}\,\big{]}_{C^{0,1/2+3\varepsilon}},$

which implies

$\displaystyle|m(2^{j}t\xi)|+I(2^{j}t\xi)\,$
$\displaystyle=|m(2^{j}t\xi)|+(t|\xi|)^{1/2+3\varepsilon}\int_{1/2}^{1}(1-s)^{-1+2\varepsilon}\frac{|m(2^{j}t\xi)-m(2^{j}st\xi)|}{\big{(}(1-s)t|\xi|\big{)}^{1/2+3\varepsilon}}ds$
(4.3)
$\displaystyle\lesssim\|m(2^{j}\cdot)\widehat{\eta}\|_{C^{0,1/2+3\varepsilon}}\,.$

By (4.2), (4.1), and that $C^{0,\frac{1}{2}+3\varepsilon}$ satisfying the
property (2.2), we have

$\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}\lesssim\sum_{j}\|m(2^{j}\cdot)\widehat{\eta}\|^{2}_{C^{0,1/2+3\varepsilon}}\simeq\sum_{j}\|m(2^{j}\cdot)\widehat{\psi}\|_{C^{0,1/2+3\varepsilon}}^{2}.$

###  4.2. Proof of (3.4)

Recall that for $H_{\theta}^{p,\gamma}$ in (2.5) we have
$\Sigma^{2}(L_{\beta}^{2})=H_{0}^{2,\beta}$. Since
$C_{c}^{\infty}(\mathbb{R}^{d})$ is dense in every $H_{\theta}^{p,\gamma}$ due
to [12, Proposition 2.2.2], it suffices to show that the mapping
$m\mapsto\widetilde{m}$ initially defined for $m\in
C_{c}^{\infty}(\mathbb{R}^{d})$ can be bounded from
$\Sigma^{2}\big{(}L_{\beta+1/2+2\varepsilon}^{2}\big{)}$ to
$\Sigma^{2}\big{(}L_{\beta}^{2}\big{)}$. Since
$\|m\|_{\Sigma^{2}(L_{\beta}^{2})}\lesssim\|m\|_{\Sigma^{2}(L_{\beta+1/2+2\varepsilon}^{2})}$
we consider $\widetilde{m}$ as in (4.1). We actually prove the mapping
property of $m\to\widetilde{m}$ for $\beta=0$, which yields the case of
$\beta\in\mathbb{N}$. For real $\beta$, we interpolate the results of
$\beta\in\mathbb{N}$ cases in terms of (2.4). We make use of the following
equivalent quantities to $\|m\|_{\Sigma^{2}(L_{n}^{2})}$,
$\|m\|_{\Sigma^{2}(L_{n+\alpha}^{2})}$ for $n\in\mathbb{N}\cup\\{0\\}$ and
$\alpha\in(0,1)$.

(4.4) $\displaystyle\|m\|_{\Sigma^{2}(L_{n}^{2})}^{2}$
$\displaystyle\simeq\sum_{\gamma:|\gamma|\leq
n}\int_{\mathbb{R}^{d}}\big{|}|\xi|^{|\gamma|}\partial_{\xi}^{\gamma}m(\xi)\big{|}^{2}\frac{d\xi}{|\xi|^{d}},$
(4.5) $\displaystyle\|m\|_{\Sigma^{2}(L_{n+\alpha}^{2})}^{2}$
$\displaystyle\simeq\|m\|_{\Sigma^{2}(L^{2}_{n})}+\|D^{n}m\|_{\Sigma^{2}_{n}(L^{2}_{\alpha})},$
(4.6) $\displaystyle\|m\|_{\Sigma_{n}^{2}(L_{\alpha}^{2})}^{2}$
$\displaystyle\simeq\int_{\mathbb{R}^{d}}|m(\xi)|^{2}|\xi|^{2n-d}d\xi+\int_{\mathbb{R}^{d}}\biggl{(}\int_{|y-\xi|\leq\frac{|\xi|}{2}}\frac{\big{|}m(y)-m(\xi)\big{|}^{2}}{|y-\xi|^{d+2\alpha}}dy\biggr{)}\frac{d\xi}{|\xi|^{d-2\alpha-2n}}.$

(4.4) and (4.5) are given in Lemma 4.1, and (4.6) is a special case of
Proposition 4.2 in Subsection 4.4. Since
$m\in\Sigma^{2}(L_{n+\alpha}^{2})\subset\Sigma^{2}(L_{n}^{2})$, without loss
of any generality we put

$\widetilde{m}(\xi)=\int_{0}^{1}(1-s)^{-\frac{3}{2}-\varepsilon}\bigl{(}m(\xi)-m(s\xi)\bigr{)}ds$

and let

$\int_{0}^{1}(\cdots)=\int_{0}^{1-\delta}(\cdots)+\int_{1-\delta}^{1}(\cdots)=I+II$

for some $\delta>0$ chosen later. Then we consider $I$ and $II$ separately.
For $I$, we have

$\displaystyle\big{\|}I\big{\|}_{\Sigma^{2}(L^{2})}\leq\int_{0}^{1-\delta}(1-s)^{-\frac{3}{2}-\varepsilon}\big{(}\big{\|}m\big{\|}_{\Sigma^{2}(L^{2})}+\big{\|}m(s\cdot)\big{\|}_{\Sigma^{2}(L^{2})}\big{)}ds.$

Since the right hand side of (4.4) is scaling invariant, it follows that

$\displaystyle\big{\|}I\big{\|}_{\Sigma^{2}(L^{2})}\leq
2\delta^{-\frac{3}{2}}\big{\|}m\big{\|}_{\Sigma^{2}(L^{2})}.$

Thus it remains to handle $II$-term. For $II$, we claim that if
$s\in(1-\delta,1)$, then

(4.7)
$\displaystyle\big{\|}m(\cdot)-m(s\cdot)\big{\|}_{\Sigma^{2}(L^{2})}\lesssim(1-s)^{1/2+2\varepsilon}\big{\|}m\big{\|}_{\Sigma^{2}(L_{1/2+2\varepsilon}^{2})},$

which is given in the next subsection. By (4.7), we can show that

$\displaystyle\big{\|}II\big{\|}_{\Sigma^{2}(L^{2})}\lesssim\delta^{1-\varepsilon}\big{\|}m\big{\|}_{\Sigma^{2}(L_{1/2+2\varepsilon}^{2})}.$

Thus we have shown that

(4.8)
$\displaystyle\|\widetilde{m}\|_{\Sigma^{2}(L^{2})}\lesssim\|m\|_{\Sigma^{2}(L_{1/2+2\varepsilon}^{2})}.$

The inequality (4.8) means that $m\in\Sigma^{2}(L_{1/2+2\varepsilon}^{2})$
implies $\widetilde{m}\in\Sigma^{2}(L^{2})$. We want to generalize this
statement to the case of $m\in\Sigma^{2}(L_{n+1/2+2\varepsilon}^{2})$ and
$\widetilde{m}\in\Sigma^{2}(L_{n}^{2})$.

To do so, note that by (4.4) and (4.5) we have

$\displaystyle\|m\|_{\Sigma^{2}(L_{n+1/2+2\varepsilon}^{2})}$
$\displaystyle\simeq\sum_{|\gamma|\leq
n}\big{\|}|\xi|^{|\gamma|}\partial^{\gamma}m(\xi)\big{\|}_{\Sigma^{2}(L_{1/2+2\varepsilon}^{2})},$
$\displaystyle\|\widetilde{m}\|_{\Sigma^{2}(L_{n}^{2})}$
$\displaystyle\simeq\sum_{|\gamma|\leq
n}\big{\|}|\xi|^{|\gamma|}\partial^{\gamma}\widetilde{m}(\xi)\big{\|}_{\Sigma^{2}(L^{2})}.$

Also, one can observe that
$\Big{(}|\xi|^{|\gamma|}\partial^{\gamma}m(\xi)\Big{)}^{\thicksim}=|\xi|^{|\gamma|}\partial^{\gamma}\widetilde{m}(\xi)$.
Then from (4.8) we have

$\displaystyle\|\widetilde{m}\|_{\Sigma^{2}(L_{n}^{2})}\,$
$\displaystyle\simeq\sum_{|\gamma|\leq
n}\big{\|}|\xi|^{|\gamma|}\partial^{\gamma}\widetilde{m}(\xi)\big{\|}_{\Sigma^{2}(L^{2})}$
$\displaystyle=\sum_{|\gamma|\leq
n}\big{\|}\Big{(}|\xi|^{|\gamma|}\partial^{\gamma}m(\xi)\Big{)}^{\thicksim}\big{\|}_{\Sigma^{2}(L^{2})}$
$\displaystyle\lesssim\sum_{|\gamma|\leq
n}\big{\|}|\xi|^{|\gamma|}\partial^{\gamma}m(\xi)\big{\|}_{\Sigma^{2}(L^{2}_{1/2+\varepsilon})}$
$\displaystyle\simeq\|m\|_{\Sigma^{2}(L_{n+1/2+\varepsilon}^{2})}.$

Thus we have shown that $m\in\Sigma^{2}(L_{n+1/2+\varepsilon}^{2})$ implies
$\widetilde{m}\in\Sigma^{2}(L_{n}^{2})$ for $n=0,1,2,\cdots$.

###  4.3. Proof of the claim (4.7)

Recall that $\|m(\cdot)-m(s\cdot)\|_{\Sigma^{2}(L^{2}(\mathbb{R}^{d}))}^{2}$
is

$\displaystyle\int_{\mathbb{R}^{d}}\big{|}m(\xi)-m(s\xi)\big{|}^{2}\frac{d\xi}{|\xi|^{d}}.$

Put $\delta=\frac{1}{8}$. For a fixed $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$,
put

$\displaystyle B_{1,\xi}$
$\displaystyle=\\{y\in\mathbb{R}^{d}:|y-\xi|<\delta(1-s)|\xi|\\},$
$\displaystyle B_{2,\xi}$
$\displaystyle=\\{z\in\mathbb{R}^{d}:|z-s\xi|<\delta(1-s)|s\xi|\\}\,,$

and make use of the triangle inequality

$|m(\xi)-m(s\xi)|\leq|m(\xi)-m(y)|+|m(y)-m(z)|+|m(z)-m(s\xi)|$

to obtain

$\displaystyle\|m(\cdot)-m(s\cdot)\|_{\Sigma^{2}(L^{2})}^{2}$
$\displaystyle\lesssim\int_{\mathbb{R}^{d}}\int_{B_{2,\xi}}\int_{B_{1,\xi}}\Big{(}|m(\xi)-m(y)|^{2}+|m(y)-m(z)|^{2}+|m(z)-m(s\xi)|^{2}\Big{)}\frac{dy}{|B_{1,\xi}|}\frac{dz}{|B_{2,\xi}|}\frac{d\xi}{|\xi|^{d}}$
$\displaystyle=:II_{1}+II_{2}+II_{3}.$

We deal with $II_{1}$, $II_{2}$, and $II_{3}$, separately.

For $II_{1}$, since $y\in B_{1,\xi}$ satisfies $|y-\xi|<\delta(1-s)|\xi|$, we
have

$\displaystyle II_{1}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{B_{1,\xi}}|m(\xi)-m(y)|^{2}\frac{dy}{|B_{1,\xi}|}\frac{d\xi}{|\xi|^{d}}$
$\displaystyle\simeq\int_{\mathbb{R}^{d}}\int_{B_{1,\xi}}\frac{|m(\xi)-m(y)|^{2}}{\big{(}\delta(1-s)|\xi|\big{)}^{d}}dy\frac{d\xi}{|\xi|^{d}}$
$\displaystyle=\delta^{-d}\int_{\mathbb{R}^{d}}\bigg{(}\int_{B_{1,\xi}}\frac{|m(\xi)-m(y)|^{2}}{\big{(}(1-s)|\xi|\big{)}^{d+1+4\varepsilon}}dy\bigg{)}\big{(}(1-s)|\xi|\big{)}^{1+4\varepsilon}\frac{d\xi}{|\xi|^{d}}$
$\displaystyle\leq\delta^{1+4\varepsilon}(1-s)^{1+4\varepsilon}\int_{\mathbb{R}^{d}}\int_{|y-\xi|<\delta|\xi|}\frac{|m(\xi)-m(y)|^{2}}{|\xi-y|^{d+1+4\varepsilon}}dy\frac{d\xi}{|\xi|^{d-1-4\varepsilon}}.$

Since $\delta<1/2$, it follows that

(4.9) $\displaystyle
II_{1}\lesssim(1-s)^{1+4\varepsilon}\|m\|_{\Sigma^{2}(L_{1/2+2\varepsilon})}^{2}.$

For $II_{3}$, we perform the argument of the case $II_{1}$ to obtain

$\displaystyle II_{3}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{B_{2,\xi}}|m(z)-m(s\xi)|^{2}\frac{dz}{|B_{2,\xi}|}\frac{d\xi}{|\xi|^{d}}$
$\displaystyle\simeq\int_{\mathbb{R}^{d}}\int_{B_{2,\xi}}\frac{|m(z)-m(s\xi)|^{2}}{\bigl{(}\delta(1-s)|s\xi|\bigr{)}^{d}}dz\frac{d\xi}{|\xi|^{d}}$
$\displaystyle\leq\delta^{1+4\varepsilon}(1-s)^{1+4\varepsilon}\int_{\mathbb{R}^{d}}\int_{|z-\xi|<\delta|\xi|}\frac{|m(z)-m(\xi)|^{2}}{|z-\xi|^{d+1+4\varepsilon}}dz\frac{d\xi}{|\xi|^{d-1-4\varepsilon}}.$

Again, since $\delta<1/2$ we have

(4.10) $\displaystyle
II_{3}\lesssim(1-s)^{1+4\varepsilon}\|m\|_{\Sigma^{2}(L_{1/2+2\varepsilon})}^{2}.$

Lastly, we consider the term $II_{2}$. In this case, we need to handle both
$y$\- and $z$\- integrals. To do this, define a set $E$ for
$s\in(1-\delta,1)$.

$E:=\\{(y,z,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{d}:|y-\xi|\leq\delta(1-s)|\xi|,|z-s\xi|\leq\delta(1-s)|s\xi|\\}.$

Observe that on $E$ we have $|y-\xi|<\delta(1-s)|\xi|\leq\delta^{2}|\xi|$,
which yields

$\displaystyle|y|<2|\xi|,\quad|\xi|<2|y|.$

Moreover,

$\displaystyle|z-y|\leq|z-s\xi|+(1-s)|\xi|+|\xi-y|\leq 2(1-s)|\xi|\leq
2\delta|\xi|\leq 4\delta|y|,\quad s>1-\delta.$

Now we can estimate $II_{2}$.

$\displaystyle II_{2}$
$\displaystyle=\int_{\mathbb{R}^{d}}\int_{B_{2,\xi}}\int_{B_{1,\xi}}|m(y)-m(z)|^{2}\frac{dy}{|B_{1,\xi}|}\frac{dz}{|B_{2,\xi}|}\frac{d\xi}{|\xi|^{d}}$
$\displaystyle\simeq\int\int\int_{E}\,\,\frac{|m(y)-m(z)|^{2}}{(1-s)^{2d}s^{d}|\xi|^{3d}}\,\,dydzd\xi$
$\displaystyle\lesssim(1-s)^{-d+1+4\varepsilon}\int\int\int_{E}\,\,\frac{|m(y)-m(z)|^{2}}{|y-z|^{d+1+4\varepsilon}|y|^{2d-1-4\varepsilon}}\,\,dydzd\xi,$

where we use $|\xi|\simeq|y|$, $|y-z|<2(1-s)|\xi|$, and $s>1-\delta$ in the
last inequality. Since $\delta=\frac{1}{8}$, if we apply the change of
variables in terms of $E$, then $II_{2}$ is dominated by
$\delta^{-2d}(1-\delta)^{-d}$ times of the following quantity:

$\displaystyle(1-s)^{-d+1+4\varepsilon}\int_{y\in\mathbb{R}^{d}\setminus\\{0\\}}\biggl{(}\int_{|z-y|<4\delta|y|}\frac{|m(y)-m(z)|^{2}}{|y-z|^{d+1+4\varepsilon}}dz\biggr{)}\bigg{(}\int_{|\xi-y|<2\delta(1-s)|y|}d\xi\bigg{)}\frac{dy}{|y|^{2d-1-4\varepsilon}}$
$\displaystyle\leq(1-s)^{-d+1+4\varepsilon}\int_{y\in\mathbb{R}^{d}\setminus\\{0\\}}\biggl{(}\int_{|z-y|<4\delta|y|}\frac{|m(y)-m(z)|^{2}}{|y-z|^{d+1+4\varepsilon}}dz\biggr{)}\delta^{d}(1-s)^{d}|y|^{d}\frac{dy}{|y|^{2d-1-4\varepsilon}}$
$\displaystyle=\delta^{d}(1-s)^{1+4\varepsilon}\int_{y\in\mathbb{R}^{d}\setminus\\{0\\}}\biggl{(}\int_{|z-y|<4\delta|y|}\frac{|m(y)-m(z)|^{2}}{|y-z|^{d+1+4\varepsilon}}dz\biggr{)}\frac{dy}{|y|^{d-1-4\varepsilon}}.$

Since $4\delta=1/2$ and Proposition 4.2, we have

(4.11) $\displaystyle
II_{2}\lesssim(1-s)^{1+4\varepsilon}\|m\|_{\Sigma^{2}(L_{1/2+2\varepsilon})}^{2}.$

Together with (4.9), (4.10), and (4.11), we prove

$\displaystyle\big{\|}m(\cdot)-m(s\cdot)\big{\|}_{\Sigma^{2}(L^{2})}\lesssim(1-s)^{\frac{1}{2}+2\varepsilon}\big{\|}m\big{\|}_{\Sigma^{2}\big{(}L_{1/2+2\varepsilon}^{2}\big{)}},$

which is (4.7). This proves the lemma.

###  4.4. Equivalent norm for $\Sigma^{2}(L_{s}^{2})$

We show the equivalence of (4.5). Recall that
$\Sigma_{\theta}^{2}(\mathcal{B})$ is equipped with the norm given by

$\displaystyle\big{\|}f\big{\|}_{\Sigma_{\theta}^{2}(\mathcal{B})}:=\Big{(}\sum_{j\in\mathbb{Z}}2^{2j\theta}\big{\|}f(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{\mathcal{B}}^{2}\Big{)}^{1/2}.$

For $\mathcal{B}=L_{n}^{2}$ or $L_{n+\alpha}^{2}$ with $\alpha\in(0,1)$, the
following equivalences are known:

######  Lemma 4.1.

[12, Proposition 2.2.3, Theorem 4.1] Let $n\in\mathbb{N}_{0}$ and
$\alpha\in(0,1)$ .

  1. (1)

$\|m\|_{\Sigma^{2}_{\theta}(L^{2}_{n})}^{2}\simeq\sum_{k=0}^{n}\int_{\Omega}|D^{k}m|^{2}|x|^{2\theta+2k-d}dx$

  2. (2)

$\|m\|_{\Sigma^{2}_{\theta}(L^{2}_{n+\alpha})}\simeq\|m\|_{\Sigma^{2}_{\theta}(L^{2}_{n})}+\|D^{n}m\|_{\Sigma^{2}_{\theta+n}(L^{2}_{\alpha})}$

Then by making use of Lemma 4.1, we can obtain the equivalence (4.6).

######  Proposition 4.2.

For any $\alpha\in(0,1)$, $p\in(1,\infty)$ and $\theta\in\mathbb{R}$,

$\displaystyle\|m\|_{\Sigma^{2}_{2\theta}(L^{2}_{\alpha})}^{2}\simeq\int_{\Omega}|m|^{2}|x|^{2\theta-d}dx+\int_{\Omega}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\Big{)}|x|^{2\alpha+2\theta-d}dx$

###### Proof.

From the representation of Besov norm on $\mathbb{R}^{d}$ in [22, p.189, 190],

$\|m\|_{L^{2}_{\alpha}}^{2}\simeq\|m\|_{L_{2}}^{2}+\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dxdy,$

we obtain

$\displaystyle\|m\|_{\Sigma^{2}_{\theta}(L^{2}_{\alpha})}^{2}$
$\displaystyle\simeq\sum_{k\in\mathbb{Z}}2^{2k\theta}\|m(2^{k}\cdot)\widehat{\psi}\|_{2}^{2}+\sum_{k\in\mathbb{Z}}2^{2k\theta}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{|m(2^{k}x)\widehat{\psi}(x)-m(2^{k}y)\widehat{\psi}(y)|^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle=:I_{0}+I_{1}\,.$

By Lemma 4.1, we have

(4.12) $\displaystyle
I_{0}\simeq\int_{\mathbb{R}^{d}}|m|^{2}|x|^{2\theta-d}dx\,.$

For $I_{1}$ we perform the change of variables $2^{k}x\to x$, $2^{k}y\to y$ so
that

$\displaystyle I_{1}\,$
$\displaystyle:=\sum_{k\in\mathbb{Z}}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{|m(x)\widehat{\psi}(2^{-k}x)-m(y)\widehat{\psi}(2^{-k}y)|^{2}}{|x-y|^{d+2\alpha}}2^{k(2\theta-d+2\alpha)}dxdy$
$\displaystyle=\sum_{k\in\mathbb{Z}}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\frac{\big{|}|x|^{\theta-d/2+\alpha}m(x)\eta(2^{-k}x)-|y|^{\theta-d/2+\alpha}m(y)\eta(2^{-k}y)\big{|}^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\iint_{|x-y|\geq\frac{|x|}{2}}\frac{|\eta(2^{-k}x)|^{2}|x|^{\theta-d+2\alpha}|m(x)|^{2}}{|x-y|^{d+2\alpha}}+\frac{|\eta(2^{-k}y)|^{2}|y|^{\theta-d+2\alpha}|m(y)|^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle\quad+\sum_{k\in\mathbb{Z}}\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}\eta(2^{-k}x)-\eta(2^{-k}y)\big{|}^{2}}{|x-y|^{d+2\alpha}}|x|^{2\theta-d+2\alpha}|m(x)|^{2}dxdy$
$\displaystyle\quad+\sum_{k\in\mathbb{Z}}\iint_{|x-y|<\frac{|x|}{2}}|\eta(2^{-k}y)|^{2}\frac{\big{|}|x|^{\theta-d/2+\alpha}m(x)-|y|^{\theta-d/2+\alpha}m(y)\big{|}^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle=:I_{1,1}+I_{1,2}+I_{1,3}$

where $\eta(x)=|x|^{-\theta+d/2-\alpha}\widehat{\psi}(x)$. Note that we apply
$|x|^{\theta-d/2+\alpha}m(x)\,\eta(2^{-k}y)$ for the last inequality when
$|x-y|<\frac{|x|}{2}$.

To progress further, observe that for any $a\in\mathbb{R}$

(4.13) $\displaystyle\sum_{k\in\mathbb{Z}}2^{ak}|\eta(2^{-k}x)|^{2}$
$\displaystyle\simeq|x|^{a},$ (4.14)
$\displaystyle\sum_{k\in\mathbb{Z}}2^{ak}|\big{(}\nabla\eta\big{)}(2^{-k}x)|^{2}$
$\displaystyle\lesssim|x|^{a}.$

For $\alpha>0$ we have

$\displaystyle\int_{|x-y|\geq\frac{|x|}{3}}|x-y|^{-d-2\alpha}dy$
$\displaystyle\simeq|x|^{-2\alpha},$
$\displaystyle\int_{|x-y|\geq\frac{|x|}{2}}|x-y|^{-d-2\alpha}dx$
$\displaystyle\leq\int_{|x|\geq|y|}|x|^{-d-2\alpha}dx\simeq|y|^{-2\alpha}.$

Then by (4.13) we can estimate $I_{1,1}$.

(4.15) $\displaystyle
I_{1,1}\lesssim\iint_{|x-y|\geq\frac{|x|}{3}}\frac{|x|^{2\theta-d+2\alpha}|m(x)|^{2}}{|x-y|^{d+2\alpha}}dydx\lesssim\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx\simeq
I_{0}.$

For $I_{1,2}$ we make use of (4.14) to obtain

(4.16) $\displaystyle\sum_{k}|\eta(2^{-k}x)-\eta(2^{-k}y)|^{2}$
$\displaystyle\leq\sum_{k}|x-y|^{2}2^{-2k}\Big{(}\int_{0}^{1}\big{|}\big{(}\nabla\eta\big{)}\big{(}2^{-k}(1-r)x+2^{-k}ry\big{)}\big{|}dr\Big{)}^{2}$
$\displaystyle\leq|x-y|^{2}\int_{0}^{1}\sum_{k}2^{-2k}\big{|}\big{(}\nabla\eta\big{)}\big{(}2^{-k}(1-r)x+2^{-k}ry\big{)}\big{|}^{2}dr$
$\displaystyle\lesssim|x-y|^{2}\int_{0}^{1}|(1-r)x+ry|^{-2}dr$
$\displaystyle\lesssim|x-y|^{2}|x|^{-2}$

when $|x-y|<|x|/2$. By (4.16) we have

$\displaystyle
I_{1,2}\,\lesssim\iint_{|x-y|<\frac{|x|}{2}}|x-y|^{-d+2(1-\alpha)}|m(x)|^{2}|x|^{2\theta-d+2(\alpha-1)}dxdy.$

Since

$\displaystyle\int_{|x-y|<\frac{|x|}{2}}|x-y|^{-d+2(1-\alpha)}dy\lesssim|x|^{2(1-\alpha)},$

it follows that

(4.17) $\displaystyle
I_{1,2}\,\lesssim\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx\simeq
I_{0}.$

Therefore, by (4.12), (4.15), (4.17) we have

(4.18)
$\displaystyle\|m\|_{\Sigma_{\theta}^{2}(L^{2}_{\alpha})}^{2}\lesssim\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx+I_{1,3}\,.$

For $I_{1,3}$ we apply (4.13) so that

$I_{1,3}\simeq\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}m(x)-|y|^{\theta-d/2+\alpha}m(y)\big{|}^{2}}{|x-y|^{d+2\alpha}}dxdy.$

Then we have

(4.19)
$\displaystyle\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}m(x)-|y|^{\theta-d/2+\alpha}m(y)\big{|}^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle\lesssim$
$\displaystyle\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}m(x)-|x|^{\theta-d/2+\alpha}m(y)\big{|}^{2}}{|x-y|^{d+2\alpha}}dxdy$
$\displaystyle+\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}-|y|^{\theta-d/2+\alpha}\big{|}^{2}|m(y)|^{2}}{|x-y|^{d+2\alpha}}dxdy.$

Note that the first term in the RHS of (4.19) can be written as

(4.20)
$\displaystyle\int_{\mathbb{R}^{d}}\bigg{(}\int_{|x-y|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\bigg{)}|x|^{2\alpha+2\theta-d}dx,$

which is the desired quantity. For the second term, we make use of the
inequality

$\big{|}|x|^{a}-|y|^{a}\big{|}\lesssim_{a}|x-y||y|^{a-1}$

whenever $|x-y|\leq|x|/2$ so that

$\displaystyle\int_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}-|y|^{\theta-d/2+\alpha}\big{|}^{2}}{|x-y|^{d+2\alpha}}dx$
$\displaystyle\lesssim|y|^{2\theta-d+2\alpha-2}\int_{|x-y|<|y|}|x-y|^{-d-2\alpha+2}dx$
$\displaystyle\simeq|y|^{2\theta-d+2\alpha-2}\times|y|^{2-2\alpha}=|y|^{2\theta-d}.$

Thus we have

(4.21)
$\displaystyle\iint_{|x-y|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}-|y|^{\theta-d/2+\alpha}\big{|}^{2}|m(y)|^{2}}{|x-y|^{d+2\alpha}}dxdy\lesssim\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx.$

By (4.18), (4.20), (4.21), we have

(4.22)
$\displaystyle\|m\|_{\Sigma_{\theta}^{2}(L^{2}_{\alpha})}^{2}\,\lesssim\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx+\int_{\mathbb{R}^{d}}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\Big{)}|x|^{2\alpha+2\theta-d}dx.$

Finally, we show that the converse of (4.22) also holds. Observe that

$\displaystyle\big{|}|x|^{N}m(x)-|x|^{N}m(y)\big{|}^{2}\lesssim\big{|}|x|^{N}m(x)-|y|^{N}m(y)\big{|}^{2}+\big{|}|x|^{N}-|y|^{N}\big{|}^{2}|m(y)|^{2},$

which yields

$\displaystyle\int_{\mathbb{R}^{d}}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\Big{)}|x|^{2\alpha+2\theta-d}dx$
$\displaystyle\quad\lesssim
I_{1,3}+\int_{\mathbb{R}^{d}}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{\big{|}|x|^{\theta-d/2+\alpha}-|y|^{\theta-d/2+\alpha}\big{|}^{2}}{|x-y|^{d+2\alpha}}dx\Big{)}|m(y)|^{2}dy$
$\displaystyle\quad\lesssim I_{1,3}+I_{0},$

where the last inequality holds due to (4.21). Moreover, one can observe that

$\displaystyle\big{|}|x|^{N}m(x)\eta(2^{-k}y)-|y|^{N}m(y)\eta(2^{-k}y)\big{|}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\big{|}|x|^{N}m(x)\big{|}^{2}|\eta(2^{-k}y)-\eta(2^{-k}x)\big{|}^{2}+\big{|}|x|^{N}m(x)\eta(2^{-k}x)-|y|^{N}m(y)\eta(2^{-k}y)\big{|}^{2},$

which yields

$\displaystyle I_{1,3}\lesssim I_{1,2}+I_{1}.$

Since $I_{1,2}\lesssim I_{0}$ and
$I_{0}+I_{1}\simeq\|m\|_{\Sigma_{\theta}^{2}(L_{\alpha}^{2})}^{2}$, we have
shown that

(4.23)
$\displaystyle\int_{\mathbb{R}^{d}}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\Big{)}|x|^{2\alpha+2\theta-d}dx\lesssim\|m\|_{\Sigma_{\theta}^{2}(L_{\alpha}^{2})}^{2}.$

By (4.23), it follows that

$\displaystyle\int_{\mathbb{R}^{d}}|m(x)|^{2}|x|^{2\theta-d}dx+\int_{\mathbb{R}^{d}}\Big{(}\int_{|y-x|<\frac{|x|}{2}}\frac{|m(x)-m(y)|^{2}}{|x-y|^{d+2\alpha}}dy\Big{)}|x|^{2\alpha+2\theta-d}dx\lesssim\|m\|_{\Sigma_{\theta}^{2}(L^{2}_{\alpha})}^{2}.$

This proves the proposition. ∎

##  5\. Proof of Lemma 3.2 and bilinear interpolation

###  5.1. Proof of Lemma 3.2

We note that Lemma 3.2 is a modified version of the following result in [17]:

######  Lemma 5.1 ([17], Lemma 4).

Let $m$ be a function of class $C^{s}(\mathbb{R}^{d})$ with $s>\frac{d}{2}$
and supported in $\\{1/2<|\xi|<2\\}$, and let the $g$-function be given by

$G_{m}(f)(x)=\biggl{(}\int_{0}^{\infty}\big{|}\big{(}m(t\xi)\widehat{f}(\xi)\big{)}\check{\,}(x)\big{|}^{2}\frac{dt}{t}\biggr{)}^{1/2}.$

Then for $\beta>d/2$ we have

$\big{\|}G_{m}(f)\|_{L^{1}(\mathbb{R}^{d})}\leq
C_{\beta}\|m\|_{L_{\beta}^{2}(\mathbb{R}^{d})}\|f\|_{H^{1}(\mathbb{R}^{d})}.$

In [17], Lemma 5.1 is proved by showing the following inequality due to the
Calderón-Zygmund theory:

(5.1)
$\displaystyle\int_{|x|>2|y|}\biggl{(}\int_{0}^{\infty}\big{|}K_{t}(x-y)-K_{t}(x)\big{|}^{2}\frac{dt}{t}\biggr{)}^{1/2}dx\leq
C_{\beta}\|m\|_{L_{\beta}^{2}(\mathbb{R}^{d})}.$

To perform the Calderón-Zygmund theory, one needs to obtain initial
$L^{2}$-estimates. For the $L^{2}$ estimate, we make use of the Plancherel
theorem to obtain

$\displaystyle\|G_{m}(f)\|_{L^{2}}^{2}$
$\displaystyle=\int_{\mathbb{R}^{d}}|\widehat{f}(\xi)|^{2}\biggl{(}\int_{0}^{\infty}|m(t\xi)|^{2}\frac{dt}{t}\biggr{)}d\xi$
(5.2)
$\displaystyle\leq\sup_{\xi\in\mathbb{R}^{d}\setminus\\{\vec{0}\\}}\|m(t\xi)\|_{\mathcal{H}(\mathbb{R}_{+})}^{2}\|f\|_{L^{2}(\mathbb{R}^{d})}^{2},$

where $\mathcal{H}(\mathbb{R}_{+})$ denotes the Hilbert space
$L^{2}(\mathbb{R}_{+},\frac{dt}{t})$. Note that (5.2) yields the first
assertion of Lemma 3.2. Moreover, let $2^{k}\leq|\xi_{0}|\leq 2^{k+1}$ for
some $k\in\mathbb{Z}$. Then we have

$\displaystyle\|m(t\xi_{0})\|_{\mathcal{H}(\mathbb{R}_{+})}^{2}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\int_{2^{j}}^{2^{j+1}}|m(t\xi_{0})|^{2}\frac{dt}{t}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\sup_{t\in[2^{j},2^{j+1}]}|m(t2^{k}\xi_{0}^{\prime})|^{2}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\sup_{1\leq|\xi^{\prime}|\leq
4}|m(2^{j+k}\xi^{\prime})|^{2}\lesssim\sum_{j\in\mathbb{Z}}\big{\|}m(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L^{\infty}(\mathbb{R}^{d})}^{2},$

where $\xi_{0}=2^{k}\xi_{0}^{\prime}$ and we apply the change of variable for
the second inequality. Thus it follows that if $m\in\Sigma^{2}(L^{\infty})$,
then

(5.3)
$\displaystyle\big{\|}G_{m}(f)\big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L^{\infty})}\|f\|_{L^{2}(\mathbb{R}^{d})}.$

Back to (5.1), observe that by Hörmander’s multiplier theorem in vector-valued
setting, we have

$\displaystyle\int_{|x|>2|y|}\biggl{(}\int_{0}^{\infty}\big{|}K_{t}(x-y)-K_{t}(x)\big{|}^{2}\frac{dt}{t}\biggr{)}^{1/2}dx$
$\displaystyle\leq\sup_{j\in\mathbb{Z}}\big{\|}m(2^{j}t\xi)\widehat{\psi}(\xi)\big{\|}_{L_{\beta}^{2}(\mathbb{R}^{d};\mathcal{H}(\mathbb{R}_{+}))}.$

Then (5.1) can be accomplished by demonstrating the following inequality: For
$k\in\mathbb{N}_{0}$ and $\mu(t2^{j}\xi)=m(t2^{j}\xi)\widehat{\psi}(\xi)$

$\displaystyle\big{\|}m(t2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{k}^{2}(\mathbb{R}^{d};\mathcal{H}(\mathbb{R}_{+}))}^{2}$
$\displaystyle=\sum_{l:|l|\leq
k}\int_{\mathbb{R}^{d}}\|D_{\xi}^{l}(\mu(t2^{j}\xi))\|_{\mathcal{H}}^{2}d\xi$
$\displaystyle\leq N(k,d)\sum_{l:|l|\leq
k}\int_{0}^{\infty}\int_{\frac{1}{2}<|\xi|<2}\big{|}2^{j|l|}t^{|l|}D_{\xi}^{l}m(2^{j}t\xi)\big{|}^{2}d\xi\frac{dt}{t}$
$\displaystyle=\sum_{l:|l|\leq
k}\int_{\mathbb{R}^{d}}\big{|}D_{\xi}^{l}m(\xi)\big{|}^{2}\bigg{(}\int_{\frac{|\xi|}{2^{j+1}}<t<\frac{|\xi|}{2^{j-1}}}(2^{j}t)^{2|l|-d}\frac{dt}{t}\bigg{)}d\xi$
$\displaystyle=N(k,d)\sum_{l:|l|\leq
k}\int_{\mathbb{R}^{d}}\big{|}D_{\xi}^{l}m(\xi)\big{|}^{2}|\xi|^{2|l|-d}d\xi$
(5.4) $\displaystyle\leq N(k,d)\big{\|}m\big{\|}_{\Sigma^{2}(L_{k}^{2})}^{2}.$

In (5.4) we choose $k\in\mathbb{N}_{0}$, so one can perform interpolation to
obtain the following result:

$\displaystyle\sup_{j\in\mathbb{Z}}\big{\|}m(t2^{j}\xi)\widehat{\psi}(\xi)\big{\|}_{L_{\beta}^{2}(\mathbb{R}^{d};\mathcal{H}(\mathbb{R}_{+}))}\lesssim\|m\|_{\Sigma^{2}(L_{\beta}^{2})},\quad\beta\geq
0.$

Thus we conclude that for any $\varepsilon>0$ if
$m\in\Sigma^{2}(L_{d/2+\varepsilon}^{2})$, then

(5.5)
$\displaystyle\big{\|}G_{m}(f)\big{\|}_{L^{1}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L_{d/2+\varepsilon}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})}.$

For $L^{\infty}\to BMO$ estimates, it is well-known (by duality) that

$\|G_{m}\|_{L^{\infty}\to BMO}\lesssim\|G_{m}\|_{L^{2}\to
L^{2}}+\|G_{m}\|_{H^{1}\to L^{1}},$

which is a special case of [16, Theorem 4.2]. Since $\|G_{m}\|_{L^{2}\to
L^{2}}\lesssim\|m\|_{\Sigma^{2}(L^{\infty})}$ by (5.3), we make use of
$\|f\|_{L^{\infty}(\mathbb{R}^{d})}\lesssim\|f\|_{L_{s}^{2}(\mathbb{R}^{d})}$
for $s>\frac{d}{2}$ and obtain

$\|m\|_{\Sigma^{2}(L^{\infty})}\lesssim\|m\|_{\Sigma^{2}(L_{s}^{2})}.$

Thus together with $\|G_{m}\|_{H^{1}\to
L^{1}}\lesssim\|m\|_{\Sigma^{2}(L_{s}^{2})}$ for $s>\frac{d}{2}$, we have

(5.6)
$\displaystyle\big{\|}G_{m}(f)\big{\|}_{BMO(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L_{d/2+\varepsilon}^{2})}\|f\|_{L^{\infty}(\mathbb{R}^{d})},$

which proves Lemma 3.2 together with (5.2) and (5.5).

###  5.2. Interpolation argument

We introduce a complex interpolation result for multilinear operators due to
Calderón [4].

######  Lemma 5.2.

[[4], paragraph 10.1] Let $L(x_{1},\dots,x_{n})$ be a multilinear mapping
defined for $x_{i}\in A_{i}\cap B_{i},i=1,\dots,n$ with values in $A\cap B$
and such that

$\displaystyle\big{\|}L(x_{1},x_{2},\dots,x_{n})\big{\|}_{A}\leq
M_{0}\prod_{i=1}^{n}\|x_{i}\|_{A_{i}},$
$\displaystyle\big{\|}L(x_{1},x_{2},\dots,x_{n})\big{\|}_{B}\leq
M_{1}\prod_{i=1}^{n}\|x_{i}\|_{B_{i}}\,.$

Then we have

$\displaystyle\big{\|}L(x_{1},x_{2},\dots,x_{n})\big{\|}_{C}\leq
M_{0}^{1-\theta}M_{1}^{\theta}\prod_{i=1}^{n}\|x_{i}\|_{C_{i}},$

where $C=[A,B]_{\theta}$, $C_{i}=[A_{i},B_{i}]_{\theta}$, $i=1,\dots,n$ and
thus $L$ can be extended continuously to a multilinear mapping of
$C_{1}\times\ldots\times C_{n}$ into $C$ .

To make use of Lemma 5.2, it suffices to check $x_{i}\in A_{i}\cap B_{i}$ for
$i=2,\dots,n$. In our case, we want $f\in\mathscr{S}^{\infty}$, where

$\mathscr{S}^{\infty}:=\\{u\in\mathscr{S}(\mathbb{R}^{d}):\int
x^{\alpha}u(x)dx=0\,\,\text{for all multi-indices $\alpha$}\\}$

Then $\mathscr{S}^{\infty}$ is a dense subset of $H^{p}$ for $0<p<\infty$
(p.128 in [21]) and $f$ can be an element of $L^{2}\cap H^{1}$. For
$(L^{2},L^{\infty})$-interpolation, we choose
$f\in\mathscr{S}(\mathbb{R}^{d})$ so that $f$ can be an element of $L^{2}\cap
L^{\infty}$.

Now we consider $G_{\widetilde{m}}(f)(x)$ as an $L^{2}(\frac{dt}{t})$-norm of
a bilinear operator

$B(m,f)(x,t):=T_{\widetilde{m}(t\cdot)}f(x)\,.$

The novelty of bilinear approach is that we do not need to construct an
analytic family for $m$ to obtain desired interpolated norms of $m$. Then the
strategy is to apply the multilinear interpolation of Lemma 5.2 on the
following bilinear estimates given by Lemma 3.1:

$\displaystyle\begin{array}[]{lll}\big{\|}B(m,f)\big{\|}_{L^{2}(\mathbb{R}^{d},\mathcal{H})}&\lesssim\sup_{\xi\neq
0}\Big{(}\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}\Big{)}^{\frac{1}{2}}\|f\|_{L^{2}(\mathbb{R}^{d})}&\lesssim\|m\|_{\Sigma^{2}(C^{0,1/2+3\varepsilon})}\|f\|_{L^{2}(\mathbb{R}^{d})},\\\
\big{\|}B(m,f)\big{\|}_{L^{1}(\mathbb{R}^{d},\mathcal{H})}&\lesssim\|\widetilde{m}\|_{\Sigma^{2}(L_{d/2+\varepsilon}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})}&\lesssim\|m\|_{\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})}.\end{array}$

Choose $\theta=2\bigl{(}\frac{1}{p}-\frac{1}{2}\bigr{)}\in(0,1)$, then we will
show that

(5.7)
$\displaystyle\big{[}L^{1}\big{(}\mathbb{R}^{d};\mathcal{H}\big{)},L^{2}\bigl{(}\mathbb{R}^{d};\mathcal{H}\bigr{)}\big{]}_{\theta}=L^{p}(\mathbb{R}^{d};\mathcal{H}),$
(5.8)
$\displaystyle\bigl{[}\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2}),\Sigma^{2}(C^{0,1/2+3\varepsilon})\bigr{]}_{\theta}=\Sigma^{2}(B_{p_{0}}^{s}),\,\,\,p_{0}=\frac{2p}{2-p},\,s=d\Big{(}\frac{1}{p}-\frac{1}{2}\Big{)}+\frac{1}{2}+3\varepsilon,$
(5.9)
$\displaystyle\bigl{[}H^{1}(\mathbb{R}^{d}),L^{2}(\mathbb{R}^{d})\bigr{]}_{\theta}=L^{p}(\mathbb{R}^{d}).$

Note that the second interpolation follows from (2.3). Now, taking (5.7),
(5.8), (5.9) for granted, by Lemma 5.2 we can show that

$\displaystyle\big{\|}B(m,f)\big{\|}_{L^{p}(\mathbb{R}^{d},\mathcal{H})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})},\quad
s\geq d\Big{(}\frac{1}{p}-\frac{1}{2}\Big{)}+\frac{1}{2}+3\varepsilon.$

When $p\geq 2$ we replace $(H^{1}\to L^{1})$-estimate into $(L^{\infty}\to
BMO)$-estimate, which is (5.6). That is,

$\displaystyle\big{[}BMO\bigl{(}\mathbb{R}^{d};\mathcal{H}\bigr{)},L^{2}\big{(}\mathbb{R}^{d};\mathcal{H}\big{)}\big{]}_{\theta}=L^{p}(\mathbb{R}^{d};\mathcal{H}),$
$\displaystyle\bigl{[}\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2}),\Sigma^{2}(C^{0,1/2+3\varepsilon})\bigr{]}_{\theta}=\Sigma^{2}(B_{p_{0}}^{s}),\quad
p_{0}=\frac{2p}{p-2}\,,\,\,s=d\Big{(}\frac{1}{2}-\frac{1}{p}\Big{)}+\frac{1}{2}+3\varepsilon,$
$\displaystyle\bigl{[}L^{\infty}(\mathbb{R}^{d}),L^{2}(\mathbb{R}^{d})\bigr{]}_{\theta}=L^{p}(\mathbb{R}^{d}).$

Then by Lemma 5.2 we obtain

$\displaystyle\big{\|}B(m,f)\big{\|}_{L^{p}(\mathbb{R}^{d},\mathcal{H})}\lesssim\|m\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{L^{p}(\mathbb{R}^{d})},\quad
s\geq d\Big{(}\frac{1}{2}-\frac{1}{p}\Big{)}+\frac{1}{2}+3\varepsilon,$

which yields $\frac{1}{p_{0}}=|\frac{1}{p}-\frac{1}{2}|$ and $s\geq
d|\frac{1}{p}-\frac{1}{2}|+\frac{1}{2}+3\varepsilon$ in general.

For the Hardy space interpolation, it is given in [8, Section 5] that
$[H^{1},L^{p}]_{\theta}=[L^{1},L^{p}]_{\theta}$ for $\theta\in(0,1)$ and
$p>1$. For $BMO(\mathbb{R}^{d})$, it is well known in the literature such as
[9, Theorem 3.4.7] that $[BMO,L^{p}]_{\theta}=[L^{\infty},L^{p}]_{\theta}$ for
$\theta\in(0,1)$ and $1\leq p<\infty$, and we refer to [2] for a vector-valued
analogue. Therefore we end this section by verifying the vector-valued
interpolation and (5.8).

For the interpolation in the vector-valued setting, it is known in [22,
Theorem 1.18.4] and [2] that for an interpolation couple $\\{A_{0},A_{1}\\}$
and $\theta,\gamma\in(0,1)$

$\displaystyle[L^{1}(A_{0}),L^{2}(A_{1})]_{\theta}$
$\displaystyle=L^{p}\big{(}[A_{0},A_{1}]_{\theta}\big{)},\quad\frac{1}{p}=1-\theta+\frac{\theta}{2},$
$\displaystyle[BMO(A_{0}),L^{2}(A_{1})]_{\gamma}$
$\displaystyle=L^{q}\big{(}[A_{0},A_{1}]_{\gamma}\big{)},\quad\frac{1}{q}=\frac{\gamma}{2}.$

In our case, $A_{0}=A_{1}=\mathcal{H}$ and $\mathcal{H}$ is the Hilbert space.
Thus it follows that

$[L^{1}(\mathcal{H}),L^{2}(\mathcal{H})]_{\theta}=L^{p}(\mathcal{H})\quad\text{and}\quad[BMO(\mathcal{H}),L^{2}(\mathcal{H})]_{\gamma}=L^{q}(\mathcal{H}).$

For (5.8) together with
$\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2})=\Sigma^{2}(B_{2}^{(d+1)/2+3\varepsilon})$,
we perform the complex interpolation to obtain

$\displaystyle\bigl{[}\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2}),\Sigma^{2}(C^{0,1/2+3\varepsilon})\bigr{]}_{\theta}\,$
$\displaystyle=\bigl{[}\Sigma^{2}(B_{2}^{(d+1)/2+3\varepsilon}),\Sigma^{2}(B_{\infty}^{1/2+3\varepsilon})\bigr{]}_{\theta}$
$\displaystyle=\Sigma^{2}\big{(}[B_{2}^{(d+1)/2+3\varepsilon},B_{\infty}^{1/2+3\varepsilon}]_{\theta}\big{)}.$

By the interpolation of the Besov space, we have

$\bigl{[}\Sigma^{2}(L_{(d+1)/2+3\varepsilon}^{2}),\Sigma^{2}(C^{0,1/2+3\varepsilon})\bigr{]}_{\theta}=\Sigma^{2}(B_{p_{0}}^{s})$

for $p_{0}=2p/2-p$, $s=d(\frac{1}{p}-\frac{1}{2})+\frac{1}{2}+3\varepsilon$.
Interpolation procedure for (5.7) and (5.8) is given in Theorem 1.18.1 and
Theorem 2.4.1 of [22] and Theorem 6.4.5 of [1].

##  6\. Applications of Theorem 1.1

In this section, we provide proofs of Corollaries 1.2, 1.3, 1.4, 1.6 and
Propositions 1.5, 1.7.

###  6.1. Proof of Corollary 1.2

We use the following properties of $B_{p_{0}}^{s}$:

  * •

Since $s>\frac{d}{p_{0}}+\frac{1}{2}$, $B_{p_{0}}^{s}(\mathbb{R}^{d})$ is
continuously embedded in
$C^{0,1/2}(\mathbb{R}^{d})=B_{\infty}^{1/2}(\mathbb{R}^{d})$.

  * •

For $f\in B_{p_{0}}^{s}(\mathbb{R}^{d})$,
$\|f\|_{B_{p_{0}}^{s}(\mathbb{R}^{d})}\simeq\|f\|_{L^{p_{0}}(\mathbb{R}^{d})}+\|f\|_{\dot{B}_{p_{0}}^{s}(\mathbb{R}^{d})}$,
and
$\|f(\lambda\,\cdot\,)\|_{\dot{B}_{p_{0}}^{s}(\mathbb{R}^{d})}\simeq\lambda^{s-\frac{d}{p_{0}}}\|f\|_{\dot{B}_{p_{0}}^{s}(\mathbb{R}^{d})}$.
Here, $\dot{B}_{p_{0}}^{s}$ is the homogeneous Besov space whose norm is given
by

$\|f\|_{\dot{B}_{p_{0}}^{s}}=\bigg{(}\sum_{j\in\mathbb{Z}}\big{(}2^{js}\|\psi_{j}\ast
f\|_{L^{p_{0}}(\mathbb{R}^{d})}\big{)}^{p_{0}}\bigg{)}^{1/p_{0}}\,.$

The embedding $B_{p_{0}}^{s}(\mathbb{R}^{d})\subset
C^{0,1/2}(\mathbb{R}^{d})=B_{\infty}^{1/2}(\mathbb{R}^{d})$ implies that

$|m(0)|+\frac{|m(\xi)-m(0)|}{|\xi|^{1/2}}\lesssim\|m\|_{B_{p_{0}}^{s}}\qquad\text{for
any}\,\,\,\,\xi\in\mathbb{R}^{d}\setminus\\{0\\}\,.$

Let $\phi_{0}$ supported on $B(0,2^{N})$. Then denote $m_{0}=m(0)\phi_{0}$ and
$m_{1}=m\phi_{0}-m(0)\phi_{0}$. Since $\phi_{0}\in
C_{c}^{\infty}(\mathbb{R}^{d})$, we obtain

(6.1)
$\displaystyle\big{|}\mathcal{M}_{m_{0}}f\big{|}=|m(0)|\big{|}\mathcal{M}_{\phi_{0}}f\big{|}\lesssim\|m\|_{B_{p_{0}}^{s}}|\mathcal{M}(f)|\,,$

where $\mathcal{M}(f)$ is the Hardy-littlewood maximal function of $f$.

For $\mathcal{M}_{m_{1}}$, first we consider the case of $p\in(1,\infty)$.
Observe that for $j\leq N$

$\displaystyle\|m_{1}(2^{j}\,\cdot\,)\widehat{\psi}\|_{L^{p_{0}}}\lesssim\|\big{(}m(2^{j}\,\cdot\,)-m(0)\big{)}\widehat{\psi}\|_{L^{\infty}}\lesssim
2^{j/2}\|m\|_{B_{p_{0}}^{s}},\quad\text{and}$
$\displaystyle\|m_{1}(2^{j}\,\cdot\,)\widehat{\psi}\|_{\dot{B}_{p_{0}}^{s}}\lesssim\|m_{1}(2^{j}\,\cdot\,)\|_{\dot{B}_{p_{0}}^{s}}$
$\displaystyle\qquad\qquad\qquad\,\,\,\,\,\simeq
2^{j(s-d/p_{0})}\|m_{1}\|_{\dot{B}_{p_{0}}^{s}}\lesssim
2^{j/2}\big{(}\|m\|_{B_{p_{0}}^{s}}+|m(0)|\big{)}\lesssim
2^{j/2}\|m\|_{B_{p_{0}}^{s}},$

which yield

$\displaystyle\|m_{1}\|_{\Sigma^{2}(B_{p_{0}}^{s})}^{2}\,\simeq\sum_{j\leq
N}\Big{(}\|m_{1}(2^{j}\,\cdot\,)\widehat{\psi}\|_{p_{0}}+\|m_{1}(2^{j}\,\cdot\,)\widehat{\psi}\|_{\dot{B}_{p_{0}}^{s}}\Big{)}^{2}\lesssim\Big{(}\sum_{j\leq
N}2^{j}\Big{)}\|m\|_{B_{p_{0}}^{s}}^{2}.$

Therefore it follows by Theorem 1.1 that

(6.2)
$\displaystyle\big{\|}\mathcal{M}_{m_{1}}f\big{\|}_{p}\lesssim\|m_{1}\|_{\Sigma^{2}(B_{p_{0}}^{s})}\|f\|_{p}\lesssim\|m\|_{B_{p_{0}}^{s}}\|f\|_{p}\,.$

By combining (6.1) and (6.2) we obtain that

$\big{\|}\mathcal{M}_{m}f\big{\|}_{p}\leq\big{\|}\mathcal{M}_{m_{0}}f\big{\|}_{p}+\big{\|}\mathcal{M}_{m_{1}}f\big{\|}_{p}\lesssim\|m\|_{B_{p_{0}}^{s}}\|f\|_{p}.$

The case of $p=1$ and $p=\infty$ are proved by the same argument.

###  6.2. Proof of Corollary 1.3

Due to $\mathcal{M}_{m}(f)\leq G_{\widetilde{m}}(f)$ in (3.2), it suffices to
show

$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{p}(\mathbb{R}^{d})}\lesssim\|h\|_{\Sigma^{2}(L_{s}^{2}(\mathbb{R}_{+}))}\|f\|_{L^{p}(\mathbb{R}^{d})},$

where $s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}$ and
$m(\xi)=h(|\xi|)$. By direct calculation, we obtain

$\displaystyle\widetilde{m}(\xi)=\widetilde{h}(|\xi|)\quad\text{where}\quad\widetilde{h}(r):=h(r)+(1/2+\varepsilon)\int_{0}^{\infty}(1-s)^{-\frac{3}{2}-\varepsilon}\big{(}h(r)-h(sr)\big{)}ds.$

For the $L^{2}$ boundedness of $G_{\widetilde{m}}$, note that

(6.3)
$\displaystyle\int_{0}^{\infty}|\widetilde{m}(t\xi)|^{2}\frac{dt}{t}=\int_{0}^{\infty}|\widetilde{h}(t|\xi|)|^{2}\frac{dt}{t}=\int_{0}^{\infty}|\widetilde{h}(t)|^{2}\frac{dt}{t}\simeq\|\widetilde{h}\|^{2}_{\Sigma^{2}(L^{2}(\mathbb{R}_{+}))}.$

Then by Lemma 3.1 we have

(6.4)
$\displaystyle\|\widetilde{h}\|^{2}_{\Sigma^{2}(L^{2}(\mathbb{R}_{+}))}\lesssim\|h\|^{2}_{\Sigma(L^{2}_{1/2+2\varepsilon}(\mathbb{R}_{+}))}\lesssim\|h\|^{2}_{\Sigma(L^{2}_{1/2+3\varepsilon}(\mathbb{R}_{+}))}.$

Making use of Lemma 4.1 we also have for a nonnegative integer $k$

$\displaystyle\|m\|_{\Sigma^{2}(L^{2}_{k})}^{2}\simeq\sum_{l\leq
k}\int_{\mathbb{R}^{d}}|m(\xi)|^{2}|\xi|^{2l-d}d\xi\simeq\sum_{l\leq
k}\int_{0}^{\infty}|h(r)|^{2}|r|^{2l-1}dr\simeq\|h\|_{\Sigma^{2}(L^{2}_{k}(\mathbb{R}_{+}))}^{2},$

which yields

$\displaystyle\|m\|_{\Sigma^{2}(L^{2}_{s})}\lesssim\|h\|_{\Sigma^{2}(L^{2}_{s}(\mathbb{R}_{+}))}$

for real $s\geq 0$. Thus by the equality of (5.2), (6.3) and (6.4), we have

(6.5)
$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{2}(\mathbb{R}^{d})}\lesssim\|h\|_{\Sigma^{2}(L_{1/2+3\varepsilon}^{2}(\mathbb{R}_{+}))}\|f\|_{L^{2}(\mathbb{R}^{d})},$

and by Theorem 1.1 we have

(6.6)
$\displaystyle\big{\|}G_{\widetilde{m}}(f)\big{\|}_{L^{1}(\mathbb{R}^{d})}\lesssim\|m\|_{\Sigma^{2}(L_{\frac{d+1}{2}+3\varepsilon}^{2})}\|f\|_{H^{1}(\mathbb{R}^{d})}\lesssim\|h\|_{\Sigma^{2}(L_{\frac{d+1}{2}+3\varepsilon}^{2}(\mathbb{R}_{+}))}\|f\|_{H^{1}(\mathbb{R}^{d})}.$

Corollary 1.3 follows from applying Lemma 5.2 on (6.5), (6.6) and duality.

###  6.3. Proof of Corollary 1.4

Before applying Theorem 1.1, we recall embedding property of $B_{p,q}^{s}$.
Since we use $B_{p_{0},p_{0}}^{s}$ for $p_{0}\geq 2$,
$s\in\big{(}d\bigl{|}\frac{1}{p}-\frac{1}{2}\bigr{|}+\frac{1}{2},\frac{\beta}{\alpha})$,
it follows that

(6.7) $\displaystyle L_{s}^{p_{0}}=F_{p_{0},2}^{s}\subseteq
F_{p_{0},p_{0}}^{s}=B_{p_{0},p_{0}}^{s}.$

Thus for $m_{\alpha,\beta}$ it suffices to check

$\displaystyle\big{\|}m_{\alpha,\beta}\big{\|}_{\Sigma^{2}(L_{s}^{p_{0}}(\mathbb{R}^{d}))}<\infty.$

Morevoer, $\ell^{2}$-summation over $\mathbb{Z}$ can be reduced to summation
over non-negative integers since $m_{\alpha,\beta}$ vanishes near the origin.
Therefore we have

(6.8)
$\displaystyle\big{\|}m_{\alpha,\beta}\big{\|}_{\Sigma^{2}(B_{p_{0},p_{0}}^{s}(\mathbb{R}^{d}))}^{2}\leq\sum_{j\geq
N}\big{\|}m_{\alpha,\beta}(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}^{2}$

for some $N\in\mathbb{Z}$. To compute $L_{s}^{p_{0}}$-norm of
$m_{\alpha,\beta}$, recall that $|\partial^{\gamma}m_{\alpha,\beta}(\xi)|\leq
C|\xi|^{-\beta-|\gamma|(1-\alpha)}$, which yields

(6.9)
$\displaystyle\big{\|}m_{\alpha,\beta}(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\leq
C2^{js}2^{-j(\beta+s(1-\alpha))}.$

Since we choose
$\beta/\alpha>s>d\bigl{|}\frac{1}{p}-\frac{1}{2}\bigr{|}+\frac{1}{2}$, the
summation in (6.8) is bounded and we have

(6.10)
$\displaystyle\frac{\beta}{\alpha}>d\biggl{|}\frac{1}{p}-\frac{1}{2}\biggr{|}+\frac{1}{2}.$

Note that (6.10) is equivalent to

$\bigg{|}\frac{1}{p}-\frac{1}{2}\bigg{|}<\frac{2\beta-\alpha}{2\alpha
d}=\frac{\beta/\alpha}{d}-\frac{1}{2d},$

which proves Corollary 1.4.

###  6.4. Proof of Proposition 1.5

Note that the symbol of $U_{\alpha,\beta}$ is

(6.11) $\displaystyle\frac{e^{it|\xi|^{\alpha}}-1}{t^{\beta}}.$

We define

$m_{\alpha,\alpha\beta}(t^{1/\alpha}\xi)=\frac{e^{it|\xi|^{\alpha}}-1}{t^{\beta}|\xi|^{\alpha\beta}}$

in the sense of (1.8).

Let $0<\beta\leq 1$ and $|\xi|\leq 1$. By Taylor’s expansion we have

$m_{\alpha,\alpha\beta}(\xi)=\sum_{k=1}^{\infty}\frac{i^{k}}{k!}|\xi|^{\alpha(k-\beta)}\quad\text{and}\quad
m_{\alpha,\alpha}(\xi)=i+\sum_{k=2}^{\infty}\frac{i^{k}}{k!}|\xi|^{\alpha(k-1)}\,.$

For each $k\in\mathbb{N}$

$\displaystyle\||2^{j}\xi|^{\alpha(k-\beta)}\widehat{\psi}(\xi)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\lesssim
2^{j\alpha(k-\beta)}\||\xi|^{\alpha(k-\beta)}\widehat{\psi}\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\lesssim
2^{j\alpha(k-\beta)}k^{d}8^{k}.$

Then we have for $\beta\in(0,1)$

$\displaystyle\big{(}\sum_{j\leq
0}\big{\|}m_{\alpha,\alpha\beta}(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}^{2}\big{)}^{1/2}$
$\displaystyle\leq\sum_{j\leq
0}\big{\|}m_{\alpha,\alpha\beta}(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}$
$\displaystyle\leq\sum_{j\leq
0}\sum_{k=1}^{\infty}\frac{1}{k!}\||2^{j}\xi|^{\alpha(k-\beta)}\widehat{\psi}(\xi)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{j\leq
0}\sum_{k=1}^{\infty}\frac{1}{k!}2^{j\alpha(k-\beta)}k^{d}8^{k}$
$\displaystyle\leq N\sum_{k=1}^{\infty}\frac{1}{k!}k^{d}8^{k}<\infty.$

For $\beta=1$ since we have
$m_{\alpha,\alpha}\varphi_{0}=i\varphi_{0}+\sum_{k=2}^{\infty}\frac{i^{k}}{k!}|\xi|^{\alpha(k-1)}\varphi_{0}$
and $\varphi_{0}$ is of class $C_{c}^{\infty}(\mathbb{R}^{d})$, we only
consider the second term.

$\displaystyle\bigg{(}\sum_{j\leq
0}\bigg{\|}\sum_{k=2}^{\infty}\frac{i^{k}}{k!}|2^{j}\xi|^{\alpha(k-1)}\widehat{\psi}(\xi)\bigg{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}^{2}\bigg{)}^{1/2}$
$\displaystyle\leq\sum_{j\leq
0}\sum_{k=2}^{\infty}\frac{1}{k!}\||2^{j}\xi|^{\alpha(k-1)}\widehat{\psi}(\xi)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\sum_{j\leq
0}\sum_{k=2}^{\infty}\frac{1}{k!}2^{j\alpha(k-1)}k^{d}8^{k}$
$\displaystyle\leq N\sum_{k=2}^{\infty}\frac{1}{k!}k^{d}8^{k}<\infty.$

For $|\xi|\geq 1$ we have

$|\partial^{\gamma}m_{\alpha,\alpha\beta}(\xi)|\lesssim|\xi|^{-\alpha\beta-(1-\alpha)|\gamma|},$

so $m_{\alpha,\alpha\beta}$ is a slowly decaying Fourier multiplier. Thus we
make use of (6.7), (6.8), (6.9) to obtain

$\big{\|}m_{\alpha,\alpha\beta}(2^{j}\cdot)\widehat{\psi}(\cdot)\big{\|}_{L_{s}^{p_{0}}(\mathbb{R}^{d})}\leq
C2^{js}2^{-j(\alpha\beta+s(1-\alpha))},$

which yields $\beta>d|\frac{1}{p}-\frac{1}{2}|+\frac{1}{2}$ by Theorem 1.1.
Then it follows that

$\displaystyle\big{\|}\sup_{t>0}\big{|}U_{\alpha,\beta}f(\cdot,t)\big{|}\big{\|}_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle=\big{\|}\sup_{t>0}\big{|}U_{\alpha,\beta}f(\cdot,t^{\alpha})\big{|}\big{\|}_{L^{p}(\mathbb{R}^{d})}$
(6.12)
$\displaystyle=\big{\|}\sup_{t>0}\big{|}T_{m_{\alpha,\alpha\beta}(t\,\cdot\,)}\big{(}(-\Delta)^{\alpha\beta}f\big{)}\big{|}\big{\|}_{L^{p}(\mathbb{R}^{d})}$
$\displaystyle\lesssim\|f\|_{\dot{L}_{\alpha\beta}^{p}(\mathbb{R}^{d})},$

whenever $|\frac{1}{p}-\frac{1}{2}|<\frac{\beta}{d}-\frac{1}{2d}$. Together
with $\beta\leq 1$, we have
$\frac{d-2\beta+1}{2d}<\frac{1}{p}<\frac{d+2\beta-1}{2d}$. Finally, (6.4)
yields that

$\displaystyle\lim_{t\to
0}\big{\|}U_{\alpha,\beta}f(\cdot,t)\big{\|}_{L^{p}(\mathbb{R}^{d})}=\big{\|}\lim_{t\to
0}U_{\alpha,\beta}f(\cdot,t)\big{\|}_{L^{p}(\mathbb{R}^{d})}=0,$

since $\lim_{t\to 0}\frac{e^{itA}-1}{t^{\beta}}=0$ for a constant $A$ and
$\beta\in(0,1)$. This proves the proposition.

###  6.5. Proof of Corollary 1.6

Since we assume that $m_{a,b}\in L^{2}_{(d+1)/2+\varepsilon,loc}$ for some
$\varepsilon>0$, it follows that $m\phi\in B_{2}^{s}(\mathbb{R}^{d})$ for any
$\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Thus by Corollary 1.2 it suffices to
check

(6.13) $\displaystyle\sum_{j\geq
0}\|m_{a,b}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{B_{p_{0}}^{s}(\mathbb{R}^{d})}^{2}<\infty.$

To show (6.13), we make use of (6.7) and (1.9) to obtain

(6.14) $\displaystyle\sum_{j\geq
0}\|m_{a,b}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{B_{p_{0}}^{s}(\mathbb{R}^{d})}^{2}\leq\sum_{j\geq
0}\|m_{a,b}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}^{2}\lesssim\sum_{j\geq
0}2^{-j2\min(a,b-s)}.$

Since we choose $a,b>0$ and
$s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2}$, the RHS of (6.14)
converges if $b-s>0$. That is, we have

$\displaystyle b>s>d\big{|}\frac{1}{p}-\frac{1}{2}\big{|}+\frac{1}{2},$

which yields $\frac{d+1-2b}{2d}<\frac{1}{p}<\frac{d-1+2b}{2d}$. This proves
the corollary.

###  6.6. Proof of Proposition 1.7

By the Fourier transform, we know that

$\displaystyle\frac{f(x)-T_{m(t\cdot)}f(x)}{t^{\alpha}}=\int_{\mathbb{R}^{d}}e^{i\langle
x,\xi\rangle}\frac{1-m(t\xi)}{|t\xi|^{\alpha}}|\xi|^{\alpha}\widehat{f}(\xi)d\xi.$

We define $m_{\alpha}(\xi):=\frac{1-m(\xi)}{|\xi|^{\alpha}}$. Since
$|1-m(\xi)|\lesssim|\xi|$ due to $m(0)=1$ and the mean value theorem,
$m_{\alpha}$ is well defined and
$|m_{\alpha}(\xi)|\lesssim(1+|\xi|)^{-\alpha}$. Also, due to
$|\partial^{\gamma}m(\xi)|\lesssim(1+|\xi|)^{-\beta}$ and
$|1-m(\xi)|\lesssim|\xi|$, it follows that for $|\xi|\leq 1$

$\displaystyle|\partial^{\gamma}m_{\alpha}(\xi)|$
$\displaystyle\lesssim|\xi|^{-\alpha-|\gamma|+1}$

Then for $|\xi|\leq 1$, we have for any $s\geq 0$

$\displaystyle\sum_{j\leq
0}\|m_{\alpha}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}^{2}\lesssim\sum_{j\leq
0}2^{-2j(\alpha-1)}<\infty.$

For $|\xi|\geq 1$, it follows that for $1/p_{0}=|1/p-1/2|$ and
$s>d|1/p-1/2|+1/2$ as in (1.9)

$\displaystyle\|m_{\alpha}(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}(\mathbb{R}^{d})}$
$\displaystyle\lesssim
2^{-j\alpha}(1+\|m(2^{j}\cdot)\widehat{\psi}(\cdot)\|_{L_{s}^{p_{0}}})$
$\displaystyle\lesssim 2^{-j\alpha}+2^{-j\alpha}2^{-j(\beta-s)}\leq
2^{-j\min(\alpha,\alpha+\beta-s)}.$

Thus we can conclude that $m_{\alpha}$ is a Fourier multiplier of type
$m_{\alpha,\alpha+\beta}$ given in (1.9). Therefore, Proposition 1.7 can be
proved by Corollary 1.6.

## Acknowledgement

The authors are sincerely grateful to the refree for his/her careful reading
and valuable comments. We also thank to Jae-Hwan Choi and Kwan Woo for useful
conversation.

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