# A weak inequality in fractional homogeneous Sobolev spaces

Lifeng Wang  Department of Mathematics, University of Pittsburgh, Pittsburgh,
PA 15260, USA

###### Abstract.

In this paper, we prove the following inequality

$\|\big{(}\int_{\mathbb{R}^{n}}\frac{|f(\cdot+y)-f(\cdot)|^{q}}{|y|^{n+sq}}dy\big{)}^{\frac{1}{q}}\|_{L^{p,\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})},$

where $\|\cdot\|_{L^{p,\infty}(\mathbb{R}^{n})}$ is the weak $L^{p}$ quasinorm
and $\|\cdot\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})}$ is the homogeneous Sobolev
norm, and parameters satisfy the condition that $1<p<q$, $2\leq q<\infty$, and
$0<s=n(\frac{1}{p}-\frac{1}{q})<1$. Furthermore, we prove the estimate
$\|\mathfrak{g}_{s,q}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}$
when $0<p,q<\infty$, $-1<s<1$, $\|\cdot\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}$
denotes the homogeneous Triebel-Lizorkin quasinorm and $\mathfrak{g}_{s,q}(f)$
is a generalization of the classical Littlewood-Paley $g$-function. Moreover,
we prove the weak type $(p,p)$ boundedness of the
$\mathcal{G}_{\lambda,q}$-function and the $\mathcal{R}_{s,q}$-function, where
the $\mathcal{G}_{\lambda,q}$-function is a generalization of the classical
Littlewood-Paley $g_{\lambda}^{*}$-function.

###### Key words and phrases:

Fractional Sobolev spaces, generalized Littlewood-Paley functions, Whitney
decomposition, homogeneous Triebel-Lizorkin spaces, Peetre-Fefferman-Stein
maximal functions.

###### 2020 Mathematics Subject Classification:

42B35.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proof of Theorem 5
  4. 4 Proof of Theorem 8
  5. 5 Proof of Theorem 9
  6. 6 Proof of Theorem 4

##  1\. Introduction

In this paper, we will flexibly use the following notations for inequalities.
Let $X,Y$ be two expressions, then “$X\lesssim Y$” means $X$ is dominated by a
constant multiple of $Y$ and the constant is determined by some fixed
parameters. In particular, “$X\lesssim 1$” means $X$ is bounded from above by
a positive finite constant whose value is determined by some fixed parameters.
If $X\lesssim Y$ and $Y\lesssim X$, then we consider $X$ and $Y$ are
equivalent and write $X\sim Y$. Furthermore, when we want to emphasize the
existence of a constant and the dependence of this constant on some fixed
parameters, we usually write it like “$X\leq C\cdot Y$ where $C=C(n,p)$” and
it means $X$ is dominated by the product of the constant $C$ and the
expression $Y$, and the value of the constant $C$ only depends on the
parameters $n$ and $p$. Moreover, unless otherwise stated, we will number
constants in different sections independently. For example, we may have
constants $C_{1}=C_{1}(n,p)$, $C_{2}=C_{2}(n,p,q)$, $C_{3}=C_{3}(q,s)$ in
section $1$ and we may also have constants $C_{1}=C_{1}(n,p)$,
$C_{2}=C_{2}(p,q,s)$, $C_{3}=C_{3}(s,\lambda)$ in section $2$. However the
constant $C_{1}$ of section $1$ may be different from the constant $C_{1}$ of
section $2$ even though they depend on the same parameters $n$ and $p$. And
the value of $C_{2}$ in section $1$ depends only on parameters $n,p,q$ while
the value of $C_{2}$ in section $2$ depends only on parameters $p,q,s$. In
this way, a change in the numbering of constants in one section will not
affect the numbering of constants in other sections of the paper.

We denote the $n$-dimensional Fourier transform of an integrable function
$f\in L^{1}(\mathbb{R}^{n})$ by

$\mathcal{F}_{n}f(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\xi}dx,$

and the $n$-dimensional inverse Fourier transform is denoted by

$\mathcal{F}^{-1}_{n}f(\xi)=\int_{\mathbb{R}^{n}}f(x)e^{2\pi ix\cdot\xi}dx,$

where for $x\in\mathbb{R}^{n}$ and $\xi\in\mathbb{R}^{n}$, $x\cdot\xi$ is the
inner product. Recall the definition of the gamma function below

$\varGamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt\quad\text{for a complex number
$z$ with $\operatorname{Re}z>0$,}$ (1)

and the gamma function has a meromorphic extension for all $z\in\mathbb{C}$
(cf. [5, Appendix A.5]). We denote the Poisson kernel by
$P(x)=C_{0}(1+|x|^{2})^{-\frac{n+1}{2}}$ for $x\in\mathbb{R}^{n}$, and we also
designate here that throughout the whole paper, we denote

$C_{0}:=\frac{\varGamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\quad\text{(cf.
\cite[cite]{[\@@bibref{}{14classical}{}{}, section 2.1.2]}).}$ (2)

Furthermore, we use the notation
$P_{t}(x)=\frac{1}{t^{n}}P(\frac{x}{t})=C_{0}t(t^{2}+|x|^{2})^{-\frac{n+1}{2}}$
for $x\in\mathbb{R}^{n}$ and $t>0$ then it is a well-known result that

$\mathcal{F}_{n}P_{t}(\xi)=\mathcal{F}_{n}P(t\xi)=e^{-2\pi t|\xi|}.$ (3)

Given an appropriate function $f$ defined on $\mathbb{R}^{n}$, the Poisson
integral of this function is a function defined on
$\mathbb{R}^{n+1}_{+}:=\\{(x,t):x\in\mathbb{R}^{n},t>0\\}$ and is denoted by

$\mathscr{P}(f;x,t):=f*P_{t}(x)=\int_{\mathbb{R}^{n}}f(y)\cdot\frac{C_{0}t}{(t^{2}+|x-y|^{2})^{\frac{n+1}{2}}}dy.$
(4)

When $f\in L^{p}(\mathbb{R}^{n})$ for $1\leq p\leq\infty$, we can use the
dominated convergence theorem and some direct calculations to obtain

$\displaystyle\partial_{k}\mathscr{P}(f;x,t):=\frac{\partial}{\partial
x_{k}}(\mathscr{P}(f;x,t))$
$\displaystyle=\int_{\mathbb{R}^{n}}f(y)\cdot\frac{(n+1)C_{0}t(y_{k}-x_{k})}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}dy,$
(5)

where $x_{k}$ and $y_{k}$ are the $k$-th coordinates of $x$ and $y$
respectively and $1\leq k\leq n$, and we also have

$\displaystyle\partial_{n+1}\mathscr{P}(f;x,t):=\frac{\partial}{\partial
t}(\mathscr{P}(f;x,t))$
$\displaystyle=\int_{\mathbb{R}^{n}}f(y)\cdot\big{(}\frac{C_{0}}{(t^{2}+|x-y|^{2})^{\frac{n+1}{2}}}-\frac{C_{0}(n+1)t^{2}}{(t^{2}+|x-y|^{2})^{\frac{n+3}{2}}}\big{)}dy.$
(6)

Furthermore, $\nabla_{n}\mathscr{P}(f;x,t)$ represents the $n$-dimensional
gradient vector of the Poisson integral $\mathscr{P}(f;x,t)$ with respect to
$x\in\mathbb{R}^{n}$, and $\nabla_{n+1}\mathscr{P}(f;x,t)$ represents the
$(n+1)$-dimensional gradient vector of the Poisson integral
$\mathscr{P}(f;x,t)$ with respect to both $x\in\mathbb{R}^{n}$ and $t>0$.

We review the definition of homogeneous Triebel-Lizorkin spaces. We fix
through out the whole paper the function $\psi\in\mathcal{S}(\mathbb{R}^{n})$
where $\mathcal{S}(\mathbb{R}^{n})$ is the class of Schwartz functions on
$\mathbb{R}^{n}$ and $\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is the space of
tempered distributions on $\mathbb{R}^{n}$, and $\psi$ satisfies the following
conditions

$0\leq\mathcal{F}_{n}\psi(\xi)\leq 1,\quad
spt.\mathcal{F}_{n}\psi\subseteq\\{\frac{1}{2}\leq|\xi|<2\\},$ (7)

and

$\sum_{j\in\mathbb{Z}}\mathcal{F}_{n}\psi(2^{-j}\xi)=1\quad\text{if }\xi\neq
0.$ (8)

Define for $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$, $j\in\mathbb{Z}$ and
$x\in\mathbb{R}^{n}$, the function $f_{j}(x):=\psi_{2^{-j}}*f(x)$ where
$\psi_{2^{-j}}(x)=2^{jn}\psi(2^{j}x)$ and thus we have the following
decomposition

$f=\sum_{j\in\mathbb{Z}}f_{j},$ (9)

where the sum in (9) converges in the space of tempered distributions modulo
polynomials (cf. [6, section 1.1.1]). In addition, given a sequence
$\\{f_{k}(x)\\}_{k\in\mathbb{Z}}$ of functions defined on $\mathbb{R}^{n}$ and
$0<p,q\leq\infty$, we use the notation
$\|\\{f_{k}\\}_{k\in\mathbb{Z}}\|_{L^{p}(l^{q})}$ to denote the
$\|\cdot\|_{L^{p}(\mathbb{R}^{n})}$-quasinorm of the
$\|\cdot\|_{l^{q}}$-quasinorm of the sequence
$\\{f_{k}(x)\\}_{k\in\mathbb{Z}}$.

######  Definition 1.

For $0<p<\infty$, $0<q\leq\infty$ and $s\in\mathbb{R}$, the homogeneous
Triebel-Lizorkin space $\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ is a subspace of
the space of tempered distributions modulo polynomials, and a tempered
distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is in
$\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ if and only if the quasinorm
$\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}:=\|\\{2^{ks}f_{k}\\}_{k\in\mathbb{Z}}\|_{L^{p}(l^{q})}$
is finite.

For an appropriate function $f$, $s,q\in\mathbb{R}$ and $q>0$, we use the
following notations

$\mathcal{D}_{s,q}f(x):=\big{(}\int_{\mathbb{R}^{n}}\frac{|f(x+y)-f(x)|^{q}}{|y|^{n+sq}}dy\big{)}^{\frac{1}{q}},$
(10)

and

$\mathcal{I}_{s}f(x):=\mathcal{F}^{-1}_{n}(|\xi|^{s}\mathcal{F}_{n}f(\xi))(x).$
(11)

In [12], the following known inequality is proven.

######  Theorem 2 (cf. Corollary 1.7 (i) of [12]).

If $0<p,q<\infty$, $\max\\{0,n(\frac{1}{p}-\frac{1}{q})\\}<s<1$, and
$f\in\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ has a function representative, then we
have

$\|\mathcal{D}_{s,q}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}.$
(12)

We also review the definition of homogeneous Sobolev spaces below.

######  Definition 3 (cf. section 1.3.3 of [6]).

Let $s\in\mathbb{R}$ and $1<p<\infty$. The homogeneous Sobolev space
$\dot{L}^{p}_{s}(\mathbb{R}^{n})$ is a subspace of the space of tempered
distributions modulo polynomials, and a tempered distribution
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ is in
$\dot{L}^{p}_{s}(\mathbb{R}^{n})$ if and only if the unique function
representative of the well-defined distribution
$\mathcal{I}_{s}f=\mathcal{F}^{-1}_{n}(|\xi|^{s}\mathcal{F}_{n}f)$ is in
$L^{p}(\mathbb{R}^{n})$ and the homogeneous Sobolev norm of $f$ is given by

$\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})}:=\big{(}\int_{\mathbb{R}^{n}}|\mathcal{I}_{s}f|^{p}dx\big{)}^{\frac{1}{p}}.$
(13)

We can use Definition 1 and the formula

$(\sum_{k\in\mathbb{Z}}a_{k})^{\theta}\leq\sum_{k\in\mathbb{Z}}a_{k}^{\theta}$
(14)

for a sequence $\\{a_{k}\\}_{k\in\mathbb{Z}}$ of nonnegative real numbers and
for $0\leq\theta\leq 1$ to deduce that
$\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}\leq\|f\|_{\dot{F}^{s}_{p,2}(\mathbb{R}^{n})}$
for any $s\in\mathbb{R}$ when $2\leq q<\infty$. And we can combine this
observation with the known equivalence relation

$\|f\|_{\dot{F}^{s}_{p,2}(\mathbb{R}^{n})}\sim\|\mathcal{I}_{s}f\|_{\dot{F}^{0}_{p,2}(\mathbb{R}^{n})}\sim\|\mathcal{I}_{s}f\|_{L^{p}(\mathbb{R}^{n})}=\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})}$
(15)

for $s\in\mathbb{R}$ and $1<p<\infty$ to further deduce from Theorem 2 the
following inequality

$\|\mathcal{D}_{s,q}f\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})},$
(16)

where $1<p<q$, $2\leq q<\infty$ and $n(\frac{1}{p}-\frac{1}{q})<s<1$ and $f$
is a function in $\dot{L}^{p}_{s}(\mathbb{R}^{n})$. In [4, Theorem 1’], C.
Fefferman stated that the following inequality

$\|\mathcal{D}_{s,2}f\|_{L^{p,\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})}$
(17)

is true when $1<p<2$ and $0<s=n(\frac{1}{p}-\frac{1}{2})<1$, and
$\|\cdot\|_{L^{p,\infty}(\mathbb{R}^{n})}$ denotes the weak $L^{p}$ quasinorm.
The proof of this inequality was not provided in [4]. We can consider that
inequality (17) is the case of inequality (16) under the critical condition
$s=n(\frac{1}{p}-\frac{1}{2})$ when $q=2$. In this paper, we will state and
prove the generalization of inequality (17) below.

######  Theorem 4.

If $1<p<q$, $2\leq q<\infty$, and $0<s=n(\frac{1}{p}-\frac{1}{q})<1$, and if
$f\in\dot{L}^{p}_{s}(\mathbb{R}^{n})$ is a bounded integrable real-valued
function defined on $\mathbb{R}^{n}$ and has first-order continuous
derivatives, then we have

$\|\mathcal{D}_{s,q}f\|_{L^{p,\infty}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{L}^{p}_{s}(\mathbb{R}^{n})}.$
(18)

In the above theorem, the condition that $f$ is a bounded real-valued function
and has first-order continuous derivatives, is necessary to guarantee the
subharmonicity of
$|\nabla_{n+1}\mathscr{P}(\varGamma(\frac{-s}{2})\mathcal{I}_{s}f;x,t)|^{q}$
when $q\neq 2$, where
$\mathscr{P}(\varGamma(\frac{-s}{2})\mathcal{I}_{s}f;x,t)$ is the Poisson
integral of the distribution $\varGamma(\frac{-s}{2})\mathcal{I}_{s}f$ and
$\varGamma(\frac{-s}{2})$ is the evaluation of the meromorphic extension of
the gamma function at $\frac{-s}{2}$. For more details about this requirement,
see Lemma 18, Remark 19, Lemma 20, Lemma 21, and inequality (252) in the proof
of Theorem 4 where we use this subharmonicity.

The integrability condition of Theorem 4 is needed, in conjunction with the
dominated convergence theorem, to deduce the identity

$\partial_{n+1}\mathscr{P}(f;x,t)=\mathcal{F}^{-1}_{n}[-2\pi|\xi|e^{-2\pi
t|\xi|}\mathcal{F}_{n}f(\xi)](x),$ (19)

which has been used in proving other indispensable results of this paper. And
by using the usual theory for tempered distributions on $\mathbb{R}^{n}$, we
can also obtain

$\partial_{k}\mathscr{P}(f;x,t)=\mathcal{F}^{-1}_{n}[2\pi i\xi_{k}e^{-2\pi
t|\xi|}\mathcal{F}_{n}f](x)\quad\text{for }1\leq k\leq n,$ (20)

where in (20), $f$ can be either a bounded tempered distribution (see [6,
section 2.1.1]) or an integrable function. Identity (19) cannot be achieved
simply by using the usual theory for tempered distributions on
$\mathbb{R}^{n}$, since the differentiation “$\partial_{n+1}$” is done with
respect to the independent $t$-variable. Therefore the integrability
assumption on the function $f$ is necessary whenever we want to use the
identity (19). In order to prove Theorem 4, we first investigate the
boundedness property of the generalized Littlewood-Paley
$\mathfrak{g}_{s,q}$-function given by

$\mathfrak{g}_{s,q}(f)(x)=\big{(}\int_{0}^{\infty}t^{q-sq-1}\big{|}\nabla_{n+1}\mathscr{P}(f;x,t)\big{|}^{q}dt\big{)}^{\frac{1}{q}},$
(21)

and we prove the following theorem.

######  Theorem 5.

Let $0<p,q<\infty$, $s\in\mathbb{R}$ and $|s|<1$. If
$f\in\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ is an integrable function, then we
have

$\|\mathfrak{g}_{s,q}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})},$
(22)

where $\mathfrak{g}_{s,q}(f)$ is the generalized Littlewood-Paley function
given in (21).

With a more general nonzero integrable kernel that satisfies some additional
conditions, replacing derivatives of the Poisson kernel in (21), D. Fan and F.
Zhao have proven the inequality (22) for $1<p,q<\infty$ and $s=0$ in their
recent development [3, Theorem 2.1]. When the kernel $\Phi$ is a compactly
supported function in $L^{q}(\mathbb{R}^{n})$ for some $q>1$ and satisfies
$\int_{\mathbb{R}^{n}}\Phi(x)dx=0$, H. Al-Qassem, L. Cheng and Y. Pan have
proven in [1, Theorem 1.1] the estimate

$\|\mathcal{S}_{\Phi}^{(\lambda)}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|\Phi\|_{L^{q}(\mathbb{R}^{n})}\cdot\|f\|_{\dot{F}^{0}_{p,\lambda}}(\mathbb{R}^{n}),$
(23)

where $\mathcal{S}_{\Phi}^{(\lambda)}(f)(x)$ is their generalized Littlewood-
Paley function given by

$\mathcal{S}_{\Phi}^{(\lambda)}(f)(x)=\big{(}\int_{0}^{\infty}|\Phi_{t}*f(x)|^{\lambda}\frac{dt}{t}\big{)}^{\frac{1}{\lambda}}\quad\text{for}\quad\lambda>1,$
(24)

and
$1/\big{(}1+\frac{1}{\lambda}-\max\\{\frac{1}{\lambda},\frac{1}{q}\\}\big{)}<p<\infty$.
In the same paper [1, Theorem 1.2 and Theorem 1.3], the authors also have
shown if additionally the kernel $\Phi$ has a radial majorant in
$L^{q}(\mathbb{R}^{n})$ for $1<q<\lambda$, then the operator
$\mathcal{S}_{\Phi}^{(\lambda)}(f)$ given in (24) is a bounded operator on
$L^{p}(\mathbb{R}^{n})$ for $(\lambda nq)/(\lambda nq-\lambda+q)<p<\infty$,
and in the case when $q=1$, the lower bound for the range of $p$ becomes
$(\lambda n)/(\lambda n-1)$. Imposing more conditions on the kernel in [1,
Theorem 1.4], the inequality (23) is proven for $1<\lambda,p<\infty$. An
immediate consequence of Theorem 5 is the following corollary.

######  Corollary 6.

Let $0<p<\infty$ and $2\leq q<\infty$. If
$f\in\dot{F}^{0}_{p,2}(\mathbb{R}^{n})$ is an integrable function, then we
have

$\|\mathfrak{g}_{0,q}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{0}_{p,2}(\mathbb{R}^{n})},$
(25)

where $\mathfrak{g}_{0,q}(f)$ is given by

$\mathfrak{g}_{0,q}(f)(x)=\big{(}\int_{0}^{\infty}t^{q-1}\big{|}\nabla_{n+1}\mathscr{P}(f;x,t)\big{|}^{q}dt\big{)}^{\frac{1}{q}}.$
(26)

###### Proof of Corollary 6.

We can use the formula (14) to deduce that

$\|f\|_{\dot{F}^{0}_{p,q}(\mathbb{R}^{n})}\leq\|f\|_{\dot{F}^{0}_{p,2}(\mathbb{R}^{n})}$

when $2\leq q<\infty$, and Corollary 6 is proved if we apply Theorem 5 with
$s=0$. Furthermore, if we use the equivalence relations,
$\|\cdot\|_{\dot{F}^{0}_{p,2}(\mathbb{R}^{n})}\sim\|\cdot\|_{L^{p}(\mathbb{R}^{n})}$
when $1<p<\infty$ and
$\|\cdot\|_{\dot{F}^{0}_{p,2}(\mathbb{R}^{n})}\sim\|\cdot\|_{H^{p}(\mathbb{R}^{n})}$
when $0<p\leq 1$, then we can obtain the following inequalities:

$\displaystyle\|\mathfrak{g}_{0,q}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{n})}\quad\text{if
$1<p<\infty$ and $2\leq q<\infty$,}$ (27)
$\displaystyle\|\mathfrak{g}_{0,q}(f)\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{H^{p}(\mathbb{R}^{n})}\quad\text{if
$0<p\leq 1$ and $2\leq q<\infty$,}$ (28)

where $\|\cdot\|_{H^{p}(\mathbb{R}^{n})}$ represents the Hardy quasinorm.
Moreover, when we let $2\leq p=q<\infty$ and $f$ is an integrable function in
$L^{q}(\mathbb{R}^{n})$, the inequality (27) yields

$\int_{\mathbb{R}^{n}}\int_{0}^{\infty}t^{q-1}|\nabla_{n+1}\mathscr{P}(f;x,t)|^{q}dtdx\lesssim\int_{\mathbb{R}^{n}}|f(x)|^{q}dx,$
(29)

and this inequality has been used in the proof of other important results in
this paper. Theorem 5 is also used in the significant estimate (200) in the
proof of Theorem 8. ∎

When $q=2$ and $s=0$, we recover from (21) the classical Littlewood-Paley
$g$-function introduced in [7, section 1 of chapter IV] by E. M. Stein, and
Corollary 6 and its consequential estimates (27), (28) turn out to be partial
generalizations of [7, Theorem 1 in section 1 of chapter IV]. The special case
of (29) when $p=q=2$ can be obtained by using routine calculations and
Plancherel’s identity, and it has been used to prove the weak type $(p,p)$
boundedness of the classical Littlewood-Paley $g_{\lambda}^{*}$-function under
the critical condition $\lambda=\frac{2}{p}$ in [4, Theorem 1]. To prove the
more challenging case of (29) when $p=q\neq 2$ and the more general Theorem 5,
where the absence of Plancherel’s identity poses a major difficulty, we use
the illuminating method introduced in [11, section 2.6.4] by H. Triebel.

We also study the weak type $(p,p)$ boundedness of the following operators.
Let $0<s,q,\lambda<\infty$ and $f$ be an appropriate function, then we define
the generalized Littlewood-Paley $\mathcal{G}_{\lambda,q}$-function is given
by

$\mathcal{G}_{\lambda,q}(f)(x):=\big{(}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{\mathbb{R}^{n}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot
t^{q-n-1}|\nabla_{n+1}\mathscr{P}(f;y,t)|^{q}dydt\big{)}^{\frac{1}{q}},$ (30)

and the $\mathcal{R}_{s,q}$-function is given by

$\mathcal{R}_{s,q}(f)(x):=\big{(}\\!\\!\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n+sq}}\cdot(\int_{0}^{2|x-y|}|\partial_{n+1}\mathscr{P}(f;y,t)|t^{s}dt)^{q}dy\big{)}^{\frac{1}{q}}.$
(31)

As another consequence of Theorem 5, the boundedness properties of the above
operators are given below.

######  Corollary 7.

For $2\leq q<\infty$, $1<\lambda<\infty$ and $0<s<1$, the operators
$\mathcal{G}_{\lambda,q}$ and $\mathcal{R}_{s,q}$ defined in (30) and (31) are
bounded from $L^{q}(\mathbb{R}^{n})\bigcap L^{1}(\mathbb{R}^{n})$ to
$L^{q}(\mathbb{R}^{n})$.

###### Proof of Corollary 7.

Let $f\in L^{q}(\mathbb{R}^{n})\bigcap L^{1}(\mathbb{R}^{n})$. By using the
defining expression (30) and the condition $1<\lambda<\infty$, we have

$\displaystyle\|\mathcal{G}_{\lambda,q}(f)\|_{L^{q}(\mathbb{R}^{n})}^{q}$
$\displaystyle=\int_{0}^{\infty}\\!\\!\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\big{(}\frac{1}{1+\frac{|x-y|}{t}}\big{)}^{\lambda
n}\cdot t^{-n}dx\cdot t^{q-1}|\nabla_{n+1}\mathscr{P}(f;y,t)|^{q}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\int_{\mathbb{R}^{n}}t^{q-1}|\nabla_{n+1}\mathscr{P}(f;y,t)|^{q}dydt\lesssim\|f\|_{L^{q}(\mathbb{R}^{n})}^{q},$
(32)

where the constants in (32) depend on $n$, $\lambda$, and some fixed
parameters due to the application of estimate (29). If we pick a positive
constant $A$ satisfying $\frac{1}{q^{\prime}}<A<s+\frac{1}{q^{\prime}}$ for
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$, then we can use Hölder’s inequality to
obtain

$\displaystyle(\int_{0}^{2|x-y|}|\partial_{n+1}\mathscr{P}(f;y,t)|t^{A}\cdot
t^{s-A}dt)^{q}$
$\displaystyle\lesssim\int_{0}^{2|x-y|}|\partial_{n+1}\mathscr{P}(f;y,t)|^{q}t^{qA}dt\cdot|x-y|^{q(1+s-A)-1},$
(33)

and the constant depends on $q,s,A$. Now we use the defining expression (31),
inequality (33), Fubini’s theorem, and estimate (29) in a sequence to obtain

$\displaystyle\|\mathcal{R}_{s,q}(f)\|_{L^{q}(\mathbb{R}^{n})}^{q}$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{1}{|x-y|^{n+sq}}\cdot(\int_{0}^{2|x-y|}|\partial_{n+1}\mathscr{P}(f;y,t)|t^{s}dt)^{q}dydx$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}|x-y|^{q-qA-n-1}\cdot\int_{0}^{2|x-y|}|\partial_{n+1}\mathscr{P}(f;y,t)|^{q}t^{qA}dtdydx$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{0}^{\infty}\int_{\frac{t}{2}\leq|x-y|}|x-y|^{q-qA-n-1}dx\cdot|\partial_{n+1}\mathscr{P}(f;y,t)|^{q}t^{qA}dtdy$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\int_{0}^{\infty}t^{q-1}|\partial_{n+1}\mathscr{P}(f;y,t)|^{q}dtdy\lesssim\|f\|_{L^{q}(\mathbb{R}^{n})}^{q},$
(34)

where the constants in (34) depend on $n,s,q,A$, and some other fixed
parameters due to the application of estimate (29). The integrability
condition in this corollary is also because of (29). ∎

We now propose the weak type $(p,p)$ boundedness theorems for the generalized
Littlewood-Paley $\mathcal{G}_{\lambda,q}$-function and the
$\mathcal{R}_{s,q}$-function.

######  Theorem 8.

For $1<p<q$, $2\leq q<\infty$, $\lambda=\frac{q}{p}>1$, and
$0<n(\frac{1}{p}-\frac{1}{q})<1$, the operator $\mathcal{G}_{\lambda,q}$
defined in (30) is of weak type $(p,p)$ for integrable functions in
$L^{p}(\mathbb{R}^{n})$ and satisfies the inequality

$\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\mathcal{G}_{\lambda,q}(f)(x)>\alpha\\})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(35)

where the constant only depends on some fixed parameters, $\alpha$ is a
positive number and $f$ is a function in $L^{p}(\mathbb{R}^{n})\bigcap
L^{1}(\mathbb{R}^{n})$.

We notice that when $q=2$ in (30), the function
$\mathcal{G}_{\lambda,2}(f)(x)$ is the classical Littlewood-Paley
$g_{\lambda}^{*}$-function, which was introduced and studied in [4], and
Theorem 8 is a generalization of [4, Theorem 1]. Comparing Theorem 8 with [4,
Theorem 1], we find that the extra condition $0<n(\frac{1}{p}-\frac{1}{q})<1$
in Theorem 8 is due to the requirement of Theorem 5, which has been used to
overcome the absence of Plancherel’s identity in the case $q\neq 2$. In order
to complete the proof of Theorem 4, we will need the following result.

######  Theorem 9.

For $1<p<q$, $2\leq q<\infty$, and $0<s=n(\frac{1}{p}-\frac{1}{q})<1$, the
operator $\mathcal{R}_{s,q}$ defined in (31) is of weak type $(p,p)$ for
integrable functions in $L^{p}(\mathbb{R}^{n})$ and satisfies the inequality

$\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\mathcal{R}_{s,q}(f)(x)>\alpha\\})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(36)

where the constant only depends on some fixed parameters, $\alpha$ is a
positive number and $f$ is a function in $L^{p}(\mathbb{R}^{n})\bigcap
L^{1}(\mathbb{R}^{n})$.

The integrability condition in Theorem 8 and Theorem 9 is because we use
Corollary 7 in the proofs of these theorems for the same reason. And from Step
6 of the proof of Theorem 4 given in section 6, we also see that the condition
$s=n(\frac{1}{p}-\frac{1}{q})$ in Theorem 4 is due to the weak type $(p,p)$
boundedness of $\mathcal{R}_{s,q}$-function, and is irrelevant to the weak
type $(p,p)$ boundedness of $\mathcal{G}_{\lambda,q}$-function.

Other useful notations in this paper include the following. For two nonempty
sets $A$ and $B$ in $\mathbb{R}^{n}$, we use $\mathcal{L}^{n}(A)$ to denote
the $n$-dimensional Lebesgue measure of $A$ and ${\rm dist\,}(A,B)=\inf_{x\in
A,y\in B}|x-y|$. We use $B^{n}(x,r)$ to denote the ball in $\mathbb{R}^{n}$
centered at $x\in\mathbb{R}^{n}$ with radius $r>0$, then the unit ball in
$\mathbb{R}^{n}$ is given as $B^{n}(0,1)$ and its volume is
$\nu_{n}=\mathcal{L}^{n}(B^{n}(0,1))$ and also
$\mathcal{L}^{n}(B^{n}(x,r))=\nu_{n}r^{n}$. The unit sphere in
$\mathbb{R}^{n}$ is written as $\mathbb{S}^{n-1}$ and its surface area is
denoted by $\omega_{n-1}$. If a ball in $\mathbb{R}^{n+1}$ is centered at
$(x,t)\in\mathbb{R}^{n+1}$ and has radius $r$, we denote such a ball by
$B^{n+1}((x,t),r)$, then the unit ball in $\mathbb{R}^{n+1}$ is given as
$B^{n+1}((0,0),1)$ and its volume is
$\nu_{n+1}=\mathcal{L}^{n+1}(B^{n+1}((0,0),1))$ and also
$\mathcal{L}^{n+1}(B^{n+1}((x,t),r))=\nu_{n+1}r^{n+1}$. A cone in
$\mathbb{R}^{n+1}_{+}$, whose vertex is the point $(x,0)\in\mathbb{R}^{n+1}$
for some $x\in\mathbb{R}^{n}$ and whose aperture is the positive number $a>0$,
is denoted by $\gamma_{a}(x):=\\{(z,t)\in\mathbb{R}^{n+1}_{+}:|z-x|<at\\}$. If
the set $I$ represents a cube in $\mathbb{R}^{n}$ with sides parallel to the
coordinate axes, then we denote the center of cube $I$ by $c(I)$ and the side
length of cube $I$ by $l(I)$. And “$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0pt}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits}$” denotes
the mean value integral.

The layout of this paper is given below. In section 1 we introduce the basic
definitions, notations and the main results. In section 2 we provide important
lemmas that will be useful in the proofs of our main results. In section 3, we
provide a proof for Theorem 5. And proofs for Theorem 8 and Theorem 9 can be
found in section 4 and section 5 respectively. Finally, Theorem 4 is proved in
section 6.

##  2\. Preliminaries

In order to write this paper in a more self-included fashion, we also cite
useful results directly from the literature and the proofs of cited results
can be found from their respective source. The following lemma is the famous
Whitney decomposition lemma of open sets in $\mathbb{R}^{n}$.

######  Lemma 10 (cf. Appendix J of [5]).

Let $\Omega$ be an open nonempty proper subset of $\mathbb{R}^{n}$. Then there
exists a family of closed dyadic cubes $\\{Q_{j}\\}_{j}$ (called the Whitney
cubes of $\Omega$) such that
(a) $\bigcup_{j}Q_{j}=\Omega$ and the $Q_{j}$’s have disjoint interiors.
(b) $\sqrt{n}l(Q_{j})\leq{\rm dist\,}(Q_{j},\Omega^{c})\leq
4\sqrt{n}l(Q_{j})$. Thus $10\sqrt{n}Q_{j}$ meets $\Omega^{c}$.
(c) If the boundaries of two cubes $Q_{j}$ and $Q_{k}$ touch, then

$\frac{1}{4}\leq\frac{l(Q_{j})}{l(Q_{k})}\leq 4.$

(d) For a given $Q_{j}$, there exist at most $12^{n}-4^{n}$ cubes $Q_{k}$ that
touch it.
(e) Let $0<\varepsilon<\frac{1}{4}$. If $Q_{j}^{*}$ has the same center as
$Q_{j}$ and $l(Q_{j}^{*})=(1+\varepsilon)l(Q_{j})$ then

$\chi_{\Omega}=\sum_{j}\chi_{Q_{j}^{*}}\leq 12^{n}-4^{n}+1.$

In the Lemma 11 below we show that if two Whitney cubes produced by Lemma 10
are sufficiently far away from each other, then the distance between one
arbitrary point in one cube and another arbitrary point in the other cube is
comparable to the distance between the two centers of these cubes.

######  Lemma 11.

Let $\Omega$ be an open nonempty proper subset of $\mathbb{R}^{n}$ and the
collection $\\{Q_{j}\\}_{j}$ of Whitney cubes of $\Omega$ satisfies the
conclusions of Lemma 10. Then for two cubes $Q_{j}$ and $Q_{m}$ in this
collection we further have the following properties.
(1) If $|c(Q_{m})-c(Q_{j})|>11(\sqrt{n}+1)l(Q_{m})$, and if $y\in Q_{m}$,
$z\in Q_{j}$ are arbitrary points in the respective cube, then we have

$|y-c(Q_{j})|\geq\frac{\sqrt{n}+1}{2}l(Q_{j}),$ (37)

and

$C_{1}|y-c(Q_{j})|\leq|y-z|\leq C_{2}|y-c(Q_{j})|,$ (38)

where $C_{1}=1-\frac{\sqrt{n}}{\sqrt{n}+1}$,
$C_{2}=1+\frac{\sqrt{n}}{\sqrt{n}+1}$ and $0<C_{1}<1<C_{2}$. Furthermore, we
have

$C_{3}|c(Q_{m})-c(Q_{j})|\leq|y-z|\leq C_{4}|c(Q_{m})-c(Q_{j})|,$ (39)

where
$C_{3}=(1-\frac{\sqrt{n}}{\sqrt{n}+1})(1-\frac{\sqrt{n}}{22(\sqrt{n}+1)})$,
$C_{4}=(1+\frac{\sqrt{n}}{\sqrt{n}+1})(1+\frac{\sqrt{n}}{22(\sqrt{n}+1)})$,
and $0<C_{3}<1<C_{4}$.
(2) If $|c(Q_{m})-c(Q_{j})|\leq 11(\sqrt{n}+1)l(Q_{m})$, then the cube $Q_{j}$
is contained in the $n$-dimensional ball that has the same center with $Q_{m}$
and has radius $11(\sqrt{n}+1)^{2}l(Q_{m})$.

###### Proof of Lemma 11.

First we prove Lemma 11 (1). If $|c(Q_{m})-c(Q_{j})|>11(\sqrt{n}+1)l(Q_{m})$
and $l(Q_{j})<20l(Q_{m})$, then for $y\in Q_{m}$ we have

$\displaystyle|y-c(Q_{j})|$
$\displaystyle\geq|c(Q_{m})-c(Q_{j})|-|c(Q_{m})-y|$
$\displaystyle>11(\sqrt{n}+1)l(Q_{m})-\frac{\sqrt{n}}{2}l(Q_{m})$
$\displaystyle>(\frac{21}{2}\sqrt{n}+11)\cdot\frac{l(Q_{j})}{20}>\frac{\sqrt{n}+1}{2}\cdot
l(Q_{j}).$ (40)

If $|c(Q_{m})-c(Q_{j})|>11(\sqrt{n}+1)l(Q_{m})$ and $l(Q_{j})\geq 20l(Q_{m})$,
then we pick $w\in\Omega^{c}$ and $y^{\prime}\in Q_{m}$ such that

${\rm dist\,}(Q_{m},\Omega^{c})\leq|w-y^{\prime}|<\frac{3}{2}{\rm
dist\,}(Q_{m},\Omega^{c})\leq 6\sqrt{n}l(Q_{m}),$ (41)

by Lemma 10 (b) and the definition of ${\rm dist\,}(Q_{m},\Omega^{c})$. Since
both $y$ and $y^{\prime}$ belong to $Q_{m}$, we have

$|w-y|\leq|w-y^{\prime}|+|y^{\prime}-y|<7\sqrt{n}l(Q_{m}).$ (42)

Because the ball centered at $c(Q_{j})$ and inscribed in the cube $Q_{j}$ has
radius $\frac{1}{2}l(Q_{j})$, we can obtain

$\displaystyle|y-c(Q_{j})|$ $\displaystyle\geq|w-c(Q_{j})|-|w-y|$
$\displaystyle>{\rm
dist\,}(Q_{j},\Omega^{c})+\frac{1}{2}l(Q_{j})-7\sqrt{n}l(Q_{m})$
$\displaystyle\geq(\sqrt{n}+\frac{1}{2})l(Q_{j})-\frac{7\sqrt{n}}{20}l(Q_{j})>\frac{\sqrt{n}+1}{2}\cdot
l(Q_{j}).$ (43)

Therefore (37) is proved. If $z\in Q_{j}$, then we use (37) to obtain

$\displaystyle|y-z|$ $\displaystyle\leq|y-c(Q_{j})|+|c(Q_{j})-z|$
$\displaystyle\leq|y-c(Q_{j})|+\frac{\sqrt{n}}{2}l(Q_{j})$
$\displaystyle\leq(1+\frac{\sqrt{n}}{1+\sqrt{n}})|y-c(Q_{j})|,$

and

$\displaystyle|y-z|$ $\displaystyle\geq|y-c(Q_{j})|-|c(Q_{j})-z|$
$\displaystyle\geq|y-c(Q_{j})|-\frac{\sqrt{n}}{2}l(Q_{j})$
$\displaystyle\geq(1-\frac{\sqrt{n}}{1+\sqrt{n}})|y-c(Q_{j})|.$

Thus (38) is proved. When $|c(Q_{m})-c(Q_{j})|>11(\sqrt{n}+1)l(Q_{m})$ and
$y\in Q_{m}$, we have

$\displaystyle|y-c(Q_{j})|$
$\displaystyle\leq|y-c(Q_{m})|+|c(Q_{m})-c(Q_{j})|$
$\displaystyle\leq\frac{\sqrt{n}}{2}l(Q_{m})+|c(Q_{m})-c(Q_{j})|$
$\displaystyle<(1+\frac{\sqrt{n}}{22(\sqrt{n}+1)})|c(Q_{m})-c(Q_{j})|,$ (44)

and

$\displaystyle|y-c(Q_{j})|$
$\displaystyle\geq|c(Q_{m})-c(Q_{j})|-|y-c(Q_{m})|$
$\displaystyle\geq|c(Q_{m})-c(Q_{j})|-\frac{\sqrt{n}}{2}l(Q_{m})$
$\displaystyle>(1-\frac{\sqrt{n}}{22(\sqrt{n}+1)})|c(Q_{m})-c(Q_{j})|.$ (45)

Applying (44) to the right side of (38) and applying (45) to the left side of
(38) yields the desired inequality (39).

Now we prove Lemma 11 (2). If $|c(Q_{m})-c(Q_{j})|\leq
11(\sqrt{n}+1)l(Q_{m})$, then the conclusion is trivial if $Q_{m}=Q_{j}$. If
$Q_{m}\neq Q_{j}$, then they must have disjoint interiors and hence the two
$n$-dimensional balls inscribed in $Q_{m}$ and $Q_{j}$, i.e. the balls
$B^{n}(c(Q_{m}),\frac{1}{2}l(Q_{m}))$ and
$B^{n}(c(Q_{j}),\frac{1}{2}l(Q_{j}))$, must be disjoint balls and we have

$\frac{1}{2}l(Q_{m})+\frac{1}{2}l(Q_{j})\leq|c(Q_{m})-c(Q_{j})|\leq
11(\sqrt{n}+1)l(Q_{m}),$ (46)

and thus

$l(Q_{j})<22(\sqrt{n}+1)l(Q_{m}).$ (47)

For $z\in Q_{j}$ we can obtain the following inequality

$\displaystyle|z-c(Q_{m})|$
$\displaystyle\leq|z-c(Q_{j})|+|c(Q_{j})-c(Q_{m})|$
$\displaystyle\leq\frac{\sqrt{n}}{2}l(Q_{j})+11(\sqrt{n}+1)l(Q_{m})$
$\displaystyle<11(\sqrt{n}+1)^{2}l(Q_{m}),$

therefore the cube $Q_{j}$ is contained in the ball
$B^{n}(c(Q_{m}),11(\sqrt{n}+1)^{2}l(Q_{m}))$. ∎

According to the above Lemma 11, for two Whitney cubes $Q_{j}$ and $Q_{m}$ of
the set $\Omega$ we define the following relations:

“$Q_{j}$ near $Q_{m}$” $\displaystyle\text{ means }|c(Q_{m})-c(Q_{j})|\leq
11(\sqrt{n}+1)l(Q_{m}),$ (48) “$Q_{j}$ far from $Q_{m}$” $\displaystyle\text{
means }|c(Q_{m})-c(Q_{j})|>11(\sqrt{n}+1)l(Q_{m}),$ (49)

where $c(Q_{j})$ and $c(Q_{m})$ represent the centers of cubes $Q_{j}$ and
$Q_{m}$ respectively and $l(Q_{m})$ is the side length of cube $Q_{m}$. Lemma
12 below is cited from [4] and it has been used frequently through out the
proofs of our main results so we also include a detailed proof for this lemma.
For a locally integrable function $f$, we denote the uncentered Hardy-
Littlewood maximal function by

$M(f)(x)=\sup_{x\in I}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}|f(y)|dy,$

where the supremum is taken over all the cubes containing $x$ in
$\mathbb{R}^{n}$ with sides parallel to the coordinate axes.

######  Lemma 12 (cf. [4]).

For $0<p<\infty$, let $f\in L^{p}(\mathbb{R}^{n})$ be a function and $\alpha$
is a real number satisfying $0<\alpha<\mathop{\rm
ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|$. Then the set
$\Omega:=\\{x\in\mathbb{R}^{n}:M(|f|^{p})(x)>\alpha^{p}\\}$ is an open
nonempty proper subset of $\mathbb{R}^{n}$, and there exist a collection of
cubes $\\{I_{j}\\}_{j}$ and two functions, the “good” function $g$ and the
“bad” function $b$, satisfying the following conditions.
(i) The $I_{j}$’s are pairwise disjoint, $\Omega=\bigcup_{j}I_{j}$ and

$\sum_{j}\mathcal{L}^{n}(I_{j})\leq
C_{5}\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}.$

(ii) For almost every $x\notin\Omega$, $|f(x)|\leq\alpha$.
(iii) For any one of the cubes $\\{I_{j}\\}_{j}$, $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|^{p}dy\leq
C_{6}\alpha^{p}$.
(iv) For any cube $I_{j}$ of the collection, if $\tilde{I_{j}}$ denotes the
cube concentric with $I_{j}$ but with twice the side length of $I_{j}$. Then
every point of $\mathbb{R}^{n}$ lies in at most $[\nu_{n}(40n)^{n}]+1$ cubes
of the collection $\\{\tilde{I_{j}}\\}_{j}$.
(v) $f=g+b$ and $b$ is supported in $\Omega$.
(vi) If $1\leq p<\infty$, then $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|b(y)|^{p}dy\leq
C_{7}\alpha^{p}$ and hence $\int_{I_{j}}|b(y)|dy\leq
C_{8}\alpha\mathcal{L}^{n}(I_{j})$ for each cube $I_{j}$.
(vii) $\int_{I_{j}}b(y)dy=0$ for each cube $I_{j}$.
(viii) If $1\leq p<\infty$, then for almost every $x\in\mathbb{R}^{n}$,
$|g(x)|\leq C_{9}\alpha$ and
$\|g\|_{L^{p}(\mathbb{R}^{n})}\leq\|f\|_{L^{p}(\mathbb{R}^{n})}$.
The dependence of the above constants on parameters is given as follows:
$C_{5}=C_{5}(n)$, $C_{6}=C_{6}(n)$, $C_{7}=C_{7}(n,p)$, $C_{8}=C_{8}(n,p)$ and
$C_{9}=C_{9}(n,p)$.

###### Proof of Lemma 12.

First we prove $\Omega=\\{x\in\mathbb{R}^{n}:M(|f|^{p})(x)>\alpha^{p}\\}$ is
an open nonempty proper subset of $\mathbb{R}^{n}$. If for all
$x\in\mathbb{R}^{n}$, $M(|f|^{p})(x)\leq\alpha^{p}$, then by Lebesgue
differentiation theorem, we have $|f(x)|^{p}=\lim_{x\in I,l(I)\rightarrow
0}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}|f(y)|^{p}dy\leq\alpha^{p}$
for almost every $x\in\mathbb{R}^{n}$. Thus $\mathop{\rm
ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|\leq\alpha<\mathop{\rm
ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|$ and this is a contradiction, and hence
$\Omega$ is nonempty. If $x\in\Omega$ and the uncentered Hardy-Littlewood
maximal function satisfies $M(|f|^{p})(x)>\alpha^{p}$, then there exists a
cube $I_{x}\ni x$ such that $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}|f(y)|^{p}dy>\alpha^{p}$.
And for every $z\in I_{x}$, we have
$M(|f|^{p})(z)\geq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{x}}}|f(y)|^{p}dy>\alpha^{p}$
and thus the cube $I_{x}$ is included in $\Omega$ and this shows the set
$\Omega$ is an open set. If $\Omega$ is all of $\mathbb{R}^{n}$ and
$\mathcal{L}^{n}(\Omega)=\infty$, because the uncentered Hardy-Littlewood
maximal function $M$ is a bounded operator from $L^{1}(\mathbb{R}^{n})$ into
$L^{1,\infty}(\mathbb{R}^{n})$, then by [5, Theorem 2.1.6] we have

$\alpha^{p}\mathcal{L}^{n}(\Omega)\leq
C_{5}\||f|^{p}\|_{L^{1}(\mathbb{R}^{n})}=C_{5}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}<\infty,$
(50)

where $C_{5}=C_{5}(n)$. This is a contradiction, therefore $\Omega$ must be a
proper subset of $\mathbb{R}^{n}$. Now because $\Omega$ satisfies all the
conditions of Lemma 10, we apply Lemma 10 to $\Omega$ and obtain a collection
of pairwise disjoint cubes $\\{I_{j}\\}_{j}$ and $\Omega=\bigcup_{j}I_{j}$. By
(50), we have

$\sum_{j}\mathcal{L}^{n}(I_{j})=\mathcal{L}^{n}(\Omega)\leq
C_{5}\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$ (51)

and Lemma 12 (i) is proved. To prove Lemma 12 (ii), since $|f|^{p}\in
L^{1}(\mathbb{R}^{n})$ and almost every point in $\mathbb{R}^{n}$ is a
Lebesgue point of $|f|^{p}$, thus if $x\notin\Omega$ is such a point then
$|f(x)|^{p}=\lim_{x\in I,l(I)\rightarrow 0}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I}}|f(y)|^{p}dy\leq
M(|f|^{p})(x)\leq\alpha^{p}$. To prove Lemma 12 (iii), let $I_{j}$ be a cube
in the collection, by Lemma 10 we have

$\sqrt{n}l(I_{j})\leq{\rm dist\,}(I_{j},\Omega^{c})\leq 4\sqrt{n}l(I_{j}),$
(52)

thus if $\bar{I_{j}}$ is the cube concentric with $I_{j}$ but with the side
length $l(\bar{I_{j}})=10\sqrt{n}l(I_{j})$ then the set
$\bar{I_{j}}\bigcap\Omega^{c}$ is nonempty. Otherwise $\bar{I_{j}}$ is
contained in $\Omega$ and hence

${\rm dist\,}(I_{j},\Omega^{c})\geq{\rm
dist\,}(I_{j},\bar{I_{j}}^{c})\geq\frac{1}{2}l(\bar{I_{j}})-\frac{1}{2}l(I_{j})>4\sqrt{n}l(I_{j}),$
(53)

and (53) poses a contradiction to (52). And if $w$ is a point in the set
$\bar{I_{j}}\bigcap\Omega^{c}$, then we have

$\frac{\int_{I_{j}}|f(y)|^{p}dy}{(10\sqrt{n})^{n}\mathcal{L}^{n}(I_{j})}\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0pt\bar{I_{j}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\bar{I_{j}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\bar{I_{j}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\bar{I_{j}}}}|f(y)|^{p}dy\leq
M(|f|^{p})(w)\leq\alpha^{p}.$ (54)

This inequality proves Lemma 12 (iii) with $C_{6}=(10\sqrt{n})^{n}$. To prove
Lemma 12 (iv), let $x\in\mathbb{R}^{n}$ and $x\in\tilde{I_{j}}$ for some $j$,
then $|x-c(I_{j})|\leq\sqrt{n}l(I_{j})$. This tells $x$ must belong to
$\Omega$, otherwise

$\sqrt{n}l(I_{j})\leq{\rm
dist\,}(I_{j},\Omega^{c})<|c(I_{j})-x|\leq\sqrt{n}l(I_{j}),$

and we obtain a contradiction. Therefore $x\in\Omega$ and ${\rm
dist\,}(\\{x\\},\Omega^{c})>0$. Let $y\in\Omega^{c}$, $z\in I_{j}$ satisfy

$\sqrt{n}l(I_{j})\leq{\rm dist\,}(I_{j},\Omega^{c})\leq|y-z|<\frac{3}{2}{\rm
dist\,}(I_{j},\Omega^{c})\leq 6\sqrt{n}l(I_{j}).$ (55)

Then
$|y-c(I_{j})|\leq|y-z|+|z-c(I_{j})|<6\sqrt{n}l(I_{j})+\frac{1}{2}\sqrt{n}l(I_{j})$,
and we have

${\rm
dist\,}(\\{x\\},\Omega^{c})\leq|x-y|\leq|x-c(I_{j})|+|y-c(I_{j})|<8\sqrt{n}l(I_{j}).$
(56)

Furthermore if $w\in\Omega^{c}$ satisfies

${\rm dist\,}(\\{x\\},\Omega^{c})\leq|w-x|<\frac{3}{2}{\rm
dist\,}(\\{x\\},\Omega^{c}),$ (57)

then because the ball centered at $c(I_{j})$ with radius $\frac{1}{2}l(I_{j})$
is inscribed in the cube $I_{j}$, we have

$\displaystyle|w-x|$ $\displaystyle\geq|w-c(I_{j})|-|x-c(I_{j})|$
$\displaystyle\geq{\rm
dist\,}(I_{j},\Omega^{c})+\frac{1}{2}l(I_{j})-\sqrt{n}l(I_{j})\geq\frac{1}{2}l(I_{j}).$
(58)

Inequalities (56), (57), (58) tell us that the condition $x\in\tilde{I_{j}}$
implies

$\frac{{\rm dist\,}(\\{x\\},\Omega^{c})}{8\sqrt{n}}\leq l(I_{j})\leq
3\cdot{\rm dist\,}(\\{x\\},\Omega^{c}).$ (59)

Hence the cube $I_{j}$ is contained in a ball centered at $x$ with radius
$5\sqrt{n}\cdot{\rm dist\,}(\\{x\\},\Omega^{c})$ and the side length
$l(I_{j})$ is at least $\frac{{\rm dist\,}(\\{x\\},\Omega^{c})}{8\sqrt{n}}$.
There can be at most $[\nu_{n}(40n)^{n}]+1$ such cubes that satisfy this
condition, and Lemma 12 (iv) is proved. Now we define

$\displaystyle g(x)=\begin{cases}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}f(y)dy&\text{
if }x\in I_{j}\\\ f(x)&\text{ if }x\notin\Omega\end{cases}$
$\displaystyle\quad\text{ and }b(x)=f(x)-g(x),$ (60)

then $b(x)$ is supported in $\Omega$ and Lemma 12 (vii) is obvious by
definition. To prove Lemma 12 (vi), we use Lemma 12 (iii) and Jensen’s
inequality for $1\leq p<\infty$ to obtain

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|b(y)|^{p}dy$
$\displaystyle\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}2^{p-1}(|f(y)|^{p}+|g(y)|^{p})dy$
$\displaystyle\leq 2^{p-1}(C_{6}\alpha^{p}+(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|dy)^{p})$
$\displaystyle\leq 2^{p-1}(C_{6}\alpha^{p}+\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|^{p}dy)\leq
C_{7}\alpha^{p},$

where $C_{7}=2^{p}C_{6}=2^{p}(10\sqrt{n})^{n}$. And the estimate for
$\int_{I_{j}}|b(y)|dy$ is obtained by Hölder’s inequality. To prove Lemma 12
(viii), when $x\notin\Omega$, we have $|g(x)|=|f(x)|\leq\alpha$ by (60) and
Lemma 12 (ii). When $x\in I_{j}$ for some $j$, we use (60), Jensen’s
inequality for $1\leq p<\infty$ and Lemma 12 (iii) to obtain

$|g(x)|\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|dy\leq(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|^{p}dy)^{\frac{1}{p}}\leq
C_{9}\alpha,$ (61)

where $C_{9}=(10\sqrt{n})^{\frac{n}{p}}$. Furthermore we have

$\displaystyle\|g\|_{L^{p}(\mathbb{R}^{n})}^{p}$
$\displaystyle=\int_{\Omega^{c}}|f(x)|^{p}dx+\sum_{j}\int_{I_{j}}|g(x)|^{p}dx$
$\displaystyle=\int_{\Omega^{c}}|f(x)|^{p}dx+\sum_{j}\mathcal{L}^{n}(I_{j})(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|dy)^{p}$
$\displaystyle\leq\int_{\Omega^{c}}|f(x)|^{p}dx+\sum_{j}\mathcal{L}^{n}(I_{j})\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-3.0ptI_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{I_{j}}}|f(y)|^{p}dy$
$\displaystyle=\int_{\Omega^{c}}|f(x)|^{p}dx+\int_{\bigcup_{j}I_{j}}|f(y)|^{p}dy=\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}.$

The proof of Lemma 12 is now complete. ∎

######  Lemma 13.

Let $\beta\in\mathbb{R}$ and $\beta>1$. Then for $1\leq k\leq n$, the
following quantities can be estimated from above by a positive finite constant
whose value only depends on $n$ and $\beta$,

$\displaystyle\sup_{x\in\mathbb{R}^{n}}\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{t(y_{k}-x_{k})}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{ty_{k}}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}dydt,$
(62)
$\displaystyle\sup_{x\in\mathbb{R}^{n}}\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{1}{(t^{2}+|y-x|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y|^{2})^{\frac{n+1}{2}}}\bigg{|}dydt,$
(63)
$\displaystyle\sup_{x\in\mathbb{R}^{n}}\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{t^{2}}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}dydt,$
(64)

where $x_{k}$ and $y_{k}$ are the $k$-th coordinates of $x,y\in\mathbb{R}^{n}$
respectively.

###### Proof of Lemma 13.

When $|y|>\beta|x|$, $\beta>1$ and $0\leq r\leq 1$, we have

$\displaystyle|y-rx|$
$\displaystyle\geq|y|-r|x|\geq|y|-|x|>(1-\frac{1}{\beta})|y|,$
$\displaystyle|y-rx|$
$\displaystyle\leq|y|+r|x|\leq|y|+|x|<(1+\frac{1}{\beta})|y|,$

hence we obtain

$(1-\frac{1}{\beta})|y|<|y-rx|<(1+\frac{1}{\beta})|y|.$ (65)

If we let $h_{1}(r)=(t^{2}+|y-rx|^{2})^{-\frac{n+3}{2}}$ and
$h_{2}(r)=(t^{2}+|y-rx|^{2})^{-\frac{n+1}{2}}$ and let $h_{1}^{\prime}(r)$ and
$h_{2}^{\prime}(r)$ denote their respective derivatives, then by the
fundamental theorem of calculus, we have the following estimates,

$\displaystyle\bigg{|}\frac{1}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{1}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}$
$\displaystyle\leq\int_{0}^{1}|h_{1}^{\prime}(r)|dr\leq\int_{0}^{1}\frac{(n+3)|y-rx|\cdot|x|}{(t^{2}+|y-rx|^{2})^{\frac{n+5}{2}}}dr$
$\displaystyle\leq\frac{C_{10}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}},$
(66)

and

$\displaystyle\bigg{|}\frac{1}{(t^{2}+|y-x|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y|^{2})^{\frac{n+1}{2}}}\bigg{|}$
$\displaystyle\leq\int_{0}^{1}|h_{2}^{\prime}(r)|dr\leq\int_{0}^{1}\frac{(n+1)|y-rx|\cdot|x|}{(t^{2}+|y-rx|^{2})^{\frac{n+3}{2}}}dr$
$\displaystyle\leq\frac{C_{11}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}},$
(67)

where $C_{10}=C_{10}(n,\beta)$ and $C_{11}=C_{11}(n,\beta)$. To prove (62) for
$1\leq k\leq n$, we use (65) and (66) and calculate as follows,

$\displaystyle\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{t(y_{k}-x_{k})}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{ty_{k}}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}dydt$
$\displaystyle\leq\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{t(y_{k}-x_{k})}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{ty_{k}}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}\bigg{|}$
$\displaystyle\quad+\bigg{|}\frac{ty_{k}}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{ty_{k}}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}dydt$
$\displaystyle\leq
C_{12}\int_{0}^{\infty}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}+\frac{t|y|^{2}\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt$
$\displaystyle\leq
C_{13}\int_{0}^{\infty}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt,$
(68)

where $C_{12}=C_{12}(n,\beta)$ and $C_{13}=C_{13}(n,\beta)$. We write the
double integral in the last line of (68) as below

$\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt+\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt.$
(69)

For the first term in (69), we apply the inequality
$(t^{2}+|y|^{2})^{\frac{n+3}{2}}\geq|y|^{n+3}$ and obtain

$\displaystyle\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle\leq\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t|x|}{|y|^{n+3}}dydt$
$\displaystyle=C_{14}|x|^{2}\cdot|x|\cdot|x|^{-3}\lesssim 1,$ (70)

where $C_{14}=C_{14}(n,\beta)$. For the second term in (69), we consider
$(t^{2}+|y|^{2})^{\frac{n+3}{2}}$ is the $(n+3)$-fold product of the term
$(t^{2}+|y|^{2})^{\frac{1}{2}}$ and in this product, we apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq t$ twice, apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq(2t|y|)^{\frac{1}{2}}$ once and apply the
inequality $(t^{2}+|y|^{2})^{\frac{1}{2}}\geq|y|$ for $n$ times and obtain

$\displaystyle\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle\leq\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t|x|}{t^{2}\cdot(2t|y|)^{\frac{1}{2}}\cdot|y|^{n}}dydt$
$\displaystyle=C_{15}|x|\cdot|x|^{-\frac{1}{2}}\cdot|x|^{-\frac{1}{2}}\lesssim
1,$ (71)

where $C_{15}=C_{15}(n,\beta)$. Combining (68), (69), (70) and (71) yields
(62) can be estimated from above by a positive finite constant whose value
only depends on $n$ and $\beta$. To prove (63), we use (67) and estimate as
below

$\displaystyle\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{1}{(t^{2}+|y-x|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y|^{2})^{\frac{n+1}{2}}}\bigg{|}dydt$
$\displaystyle\leq
C_{11}\int_{0}^{\infty}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle=C_{11}\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle\quad+C_{11}\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt.$
(72)

For the first term of (72), we use the inequality
$(t^{2}+|y|^{2})^{\frac{n+3}{2}}\geq|y|^{n+3}$ and obtain

$\displaystyle\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle\leq\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{|x|}{|y|^{n+2}}dydt$
$\displaystyle=C_{16}|x|\cdot|x|\cdot|x|^{-2}\lesssim 1,$ (73)

where $C_{16}=C_{16}(n,\beta)$. For the second term of (72), we consider
$(t^{2}+|y|^{2})^{\frac{n+3}{2}}$ is the $(n+3)$-fold product of the term
$(t^{2}+|y|^{2})^{\frac{1}{2}}$ and in this product, we apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq t$ once, apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq(2t|y|)^{\frac{1}{2}}$ once and apply the
inequality $(t^{2}+|y|^{2})^{\frac{1}{2}}\geq|y|$ for $(n+1)$ times and obtain

$\displaystyle\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}dydt$
$\displaystyle\leq\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{|y|\cdot|x|}{t\cdot(2t|y|)^{\frac{1}{2}}\cdot|y|^{n+1}}dydt$
$\displaystyle=C_{17}|x|\cdot|x|^{-\frac{1}{2}}\cdot|x|^{-\frac{1}{2}}\lesssim
1,$ (74)

where $C_{17}=C_{17}(n,\beta)$. Combining (72), (73) and (74) yields (63) can
be estimated from above by a positive finite constant whose value only depends
on $n$ and $\beta$. To prove (64), we use (66) and estimate as below

$\displaystyle\int_{0}^{\infty}\int_{|y|>\beta|x|}\bigg{|}\frac{t^{2}}{(t^{2}+|y-x|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y|^{2})^{\frac{n+3}{2}}}\bigg{|}dydt$
$\displaystyle\leq
C_{10}\int_{0}^{\infty}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt$
$\displaystyle=C_{10}\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt$
$\displaystyle\quad+C_{10}\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt.$
(75)

To estimate the first term of (75), we apply the inequality
$(t^{2}+|y|^{2})^{\frac{n+5}{2}}\geq|y|^{n+5}$ and obtain

$\displaystyle\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt$
$\displaystyle\leq\int_{0}^{|x|}\int_{|y|>\beta|x|}\frac{t^{2}|x|}{|y|^{n+4}}dydt$
$\displaystyle=C_{18}|x|\cdot|x|^{3}\cdot|x|^{-4}\lesssim 1,$ (76)

where $C_{18}=C_{18}(n,\beta)$. For the second term of (75), we consider
$(t^{2}+|y|^{2})^{\frac{n+5}{2}}$ is the $(n+5)$-fold product of the term
$(t^{2}+|y|^{2})^{\frac{1}{2}}$ and in this product, we apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq t$ for three times, apply the inequality
$(t^{2}+|y|^{2})^{\frac{1}{2}}\geq(2t|y|)^{\frac{1}{2}}$ once and apply the
inequality $(t^{2}+|y|^{2})^{\frac{1}{2}}\geq|y|$ for $(n+1)$ times and obtain

$\displaystyle\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{(t^{2}+|y|^{2})^{\frac{n+5}{2}}}dydt$
$\displaystyle\leq\int_{|x|}^{\infty}\int_{|y|>\beta|x|}\frac{t^{2}|y|\cdot|x|}{t^{3}\cdot(2t|y|)^{\frac{1}{2}}\cdot|y|^{n+1}}dydt$
$\displaystyle=C_{19}|x|\cdot|x|^{-\frac{1}{2}}\cdot|x|^{-\frac{1}{2}}\lesssim
1,$ (77)

where $C_{19}=C_{19}(n,\beta)$. Combining (75), (76) and (77) yields (64) can
be estimated from above by a positive finite constant whose value only depends
on $n$ and $\beta$. ∎

In the following Lemma 14, we cite the famous Hardy-Littlewood-Sobolev
inequality from [8].

######  Lemma 14.

For $0<\gamma<n$, $1<p<q<\infty$, and
$\frac{1}{q}=\frac{1}{p}-\frac{n-\gamma}{n}$, we have

$\|f*(|y|^{-\gamma})\|_{L^{q}(\mathbb{R}^{n})}\leq
C_{20}\cdot\|f\|_{L^{p}(\mathbb{R}^{n})},$

where $C_{20}=C_{20}(p,q)$.

######  Lemma 15.

Let $f\in L^{p}(\mathbb{R}^{n})$ for $1\leq p<\infty$, and assume
$0<\alpha<\mathop{\rm ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|$ and
$\Omega=\\{x\in\mathbb{R}^{n}:M(|f|^{p})(x)>\alpha^{p}\\}$ are given by Lemma
12. And let $\\{I_{j}\\}_{j}$ be the associated Whitney cubes and $f=g+b$ as
in Lemma 12. Let $\chi_{I_{j}}$ denote the characteristic function of the cube
$I_{j}$ and $b_{j}=b\cdot\chi_{I_{j}}$. Then the following estimates are true.
(1) For $1\leq k\leq n+1$ and $y\in\Omega^{c}$, we have

$\big{|}\partial_{k}\mathscr{P}(\sum_{j}b_{j};y,t)\big{|}\lesssim\frac{\alpha}{t},$
(78)

where the constant depends on $n,p$.
(2) For $1\leq k\leq n+1$, $I_{m}$ is a Whitney cube and $y\in I_{m}$, we have

$\big{|}\partial_{k}\mathscr{P}(\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far
from }I_{m}\end{subarray}}b_{j};y,t)\big{|}\lesssim\frac{\alpha}{t},$ (79)

where the constant depends on $n,p$.

###### Proof of Lemma 15.

(1) If $1\leq k\leq n$, $y\in\Omega^{c}$ and $z\in I_{j}$, by Lemma 10 (b) and
the fact that the $n$-dimensional ball centered at $c(I_{j})$ and inscribed in
the Whitney cube $I_{j}$ has radius $\frac{1}{2}l(I_{j})$, we can obtain

$|y-c(I_{j})|\geq{\rm
dist\,}(I_{j},\Omega^{c})+\frac{1}{2}l(I_{j})>\frac{\sqrt{n}+1}{2}\cdot
l(I_{j}),$ (80)

and thus

$(1-\frac{\sqrt{n}}{\sqrt{n}+1})\cdot|y-c(I_{j})|\leq|y-z|\leq(1+\frac{\sqrt{n}}{\sqrt{n}+1})\cdot|y-c(I_{j})|.$
(81)

Furthermore we also have

$\displaystyle\sum_{j}\mathcal{L}^{n}(I_{j})\cdot\frac{t|y-c(I_{j})|}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle=\sum_{j}\int_{I_{j}}\frac{t|y-c(I_{j})|}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz\lesssim
t^{-1},$ (82)

where we use the fact that $\\{I_{j}\\}_{j}$ have disjoint interiors and
$\bigcup_{j}I_{j}=\Omega\subseteq\mathbb{R}^{n}$, and the fact that the
integral

$\int_{\mathbb{R}^{n}}\frac{|z|}{(1+|z|^{2})^{\frac{n+3}{2}}}dz$

converges to a positive finite constant whose value only depends on the
dimension $n$. Applying (5), (81), (82), and Lemma 12 (vi) yields

$\displaystyle\big{|}\partial_{k}\mathscr{P}(\sum_{j}b_{j};y,t)\big{|}$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|\cdot\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|dz\cdot\frac{t|y-c(I_{j})|}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle\lesssim\sum_{j}\alpha\mathcal{L}^{n}(I_{j})\cdot\frac{t|y-c(I_{j})|}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}\lesssim\frac{\alpha}{t},$
(83)

where the constants in (83) depend on $n,p$. If $k=n+1$, $y\in\Omega^{c}$ and
$z\in I_{j}$, then we still use (80) and (81) to obtain

$\displaystyle\sum_{j}\mathcal{L}^{n}(I_{j})\cdot\big{(}\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{)}$
$\displaystyle=\sum_{j}\int_{I_{j}}\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz\lesssim
t^{-1},$ (84)

where we use the fact that the Whitney cubes $\\{I_{j}\\}_{j}$ have disjoint
interiors and $\bigcup_{j}I_{j}=\Omega\subseteq\mathbb{R}^{n}$, and the fact
that both integrals

$\int_{\mathbb{R}^{n}}\frac{1}{(1+|z|^{2})^{\frac{n+1}{2}}}dz\quad\text{and}\quad\int_{\mathbb{R}^{n}}\frac{1}{(1+|z|^{2})^{\frac{n+3}{2}}}dz$

converge to positive finite constants whose values only depend on the
dimension $n$. Applying (6), (81), (84), and Lemma 12 (vi) yields

$\displaystyle\big{|}\partial_{n+1}\mathscr{P}(\sum_{j}b_{j};y,t)\big{|}$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|\cdot\big{(}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{)}dz$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|dz\cdot\big{(}\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{)}$
$\displaystyle\lesssim\sum_{j}\alpha\mathcal{L}^{n}(I_{j})\cdot\big{(}\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{)}$
$\displaystyle\lesssim\frac{\alpha}{t},$ (85)

where the constants in (85) depend on $n,p$. The proof of Lemma 15 (1) is now
complete.

(2) Assume $1\leq k\leq n$, $I_{m}$ is a Whitney cube and $y\in I_{m}$, and if
$I_{j}$ is another Whitney cube satisfying the relation (49) between the two
cubes $I_{j}$ and $I_{m}$, then we can use inequality (39) of Lemma 11 for
$y\in I_{m}$ and $z\in I_{j}$ to obtain the following estimate

$\displaystyle\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\mathcal{L}^{n}(I_{j})\cdot\frac{t|c(I_{m})-c(I_{j})|}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle=\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}\frac{t|c(I_{m})-c(I_{j})|}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz\lesssim
t^{-1},$ (86)

where the constants in (86) depend on $n$. Applying (5), inequality (39) of
Lemma 11, Lemma 12 (vi), and (86) yields the following

$\displaystyle\big{|}\partial_{k}\mathscr{P}(\sum_{\begin{subarray}{c}j\\\
I_{j}\text{ far from }I_{m}\end{subarray}}b_{j};y,t)\big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|\cdot\frac{t|y-z|}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|dz\cdot\frac{t|c(I_{m})-c(I_{j})|}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\alpha\mathcal{L}^{n}(I_{j})\cdot\frac{t|c(I_{m})-c(I_{j})|}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}\lesssim\frac{\alpha}{t},$
(87)

where the constants in (87) depend on $n,p$. Assume $k=n+1$, $I_{m}$ is a
Whitney cube and $y\in I_{m}$, and if $I_{j}$ is another Whitney cube
satisfying the relation (49) between the two cubes $I_{j}$ and $I_{m}$, then
we can use inequality (39) of Lemma 11 for $y\in I_{m}$ and $z\in I_{j}$ to
obtain the following estimate

$\displaystyle\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\mathcal{L}^{n}(I_{j})\big{(}\frac{1}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+1}{2}}}$
$\displaystyle\quad+\frac{t^{2}}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}\frac{1}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+1}{2}}}$
$\displaystyle\quad+\frac{t^{2}}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}dz\lesssim
t^{-1},$ (88)

where the constants in (88) depend on the dimension $n$. Applying (6),
inequality (39) of Lemma 11, Lemma 12 (vi), and (88) yields

$\displaystyle\big{|}\partial_{n+1}\mathscr{P}(\sum_{\begin{subarray}{c}j\\\
I_{j}\text{ far from }I_{m}\end{subarray}}b_{j};y,t)\big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\\!\\!\\!\\!\int_{I_{j}}|b(z)|\big{\\{}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}+\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{\\}}dz$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|dz\cdot\big{\\{}\frac{1}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+1}{2}}}$
$\displaystyle\quad+\frac{t^{2}}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{\\}}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\alpha\mathcal{L}^{n}(I_{j})\cdot\big{\\{}\frac{1}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+1}{2}}}$
$\displaystyle\quad+\frac{t^{2}}{(t^{2}+|c(I_{m})-c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{\\}}\lesssim\frac{\alpha}{t},$
(89)

where the constants in (89) depend on $n,p$. The proof of Lemma 15 (2) is now
complete. ∎

######  Lemma 16.

Let $f\in L^{p}(\mathbb{R}^{n})$ for $1\leq p<\infty$, and assume
$0<\alpha<\mathop{\rm ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|$ and
$\Omega=\\{x\in\mathbb{R}^{n}:M(|f|^{p})(x)>\alpha^{p}\\}$ are given by Lemma
12. And let $\\{I_{j}\\}_{j}$ be the associated Whitney cubes and $f=g+b$ as
in Lemma 12. Let $\chi_{I_{j}}$ denote the characteristic function of the cube
$I_{j}$ and $b_{j}=b\cdot\chi_{I_{j}}$. Then for $1\leq k\leq n+1$, we have
the following estimates,

$\displaystyle\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\partial_{k}\mathscr{P}(\sum_{j}b_{j};y,t)\big{|}dydt$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|dz,$ (90)
$\displaystyle\sum_{m}\int_{0}^{\infty}\int_{I_{m}}\big{|}\partial_{k}\mathscr{P}(\sum_{\begin{subarray}{c}j\\\
I_{j}\text{ far from }I_{m}\end{subarray}}b_{j};y,t)\big{|}dydt$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|dz,$ (91)

and the constants in (90) and (91) only depend on $n$.

###### Proof of Lemma 16.

First we prove (90) and (91) when $1\leq k\leq n$. Due to Lemma 12 (vii) and
(5), $\int_{I_{j}}b(z)dz=0$ and we can write
$\partial_{k}\mathscr{P}(b_{j};y,t)$ identically as

$(n+1)C_{0}\int_{I_{j}}b(z)\big{\\{}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{\\}}dz,$
(92)

where $1\leq k\leq n$, and $y_{k}$, $z_{k}$, $c(I_{j})_{k}$ are the $k$-th
coordinates of $y$, $z$, $c(I_{j})$ in $\mathbb{R}^{n}$ respectively. Since
the Whitney cubes have disjoint interiors, then by applying (92) and by
exchanging the order of integration, we can rewrite the left side of (90)
equivalently as below

$\displaystyle\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\sum_{j}\partial_{k}\mathscr{P}(b_{j};y,t)\big{|}dydt$
$\displaystyle\lesssim\sum_{j}\int_{I_{j}}|b(z)|\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle\quad-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(93)

where the constant depends on $n$. If we use the change of variable
$w=y-c(I_{j})$ for $y\in\Omega^{c}$ and if $z\in I_{j}$, then from Lemma 10
(b) and the fact that the $n$-dimensional ball centered at $c(I_{j})$ and
inscribed in $I_{j}$ has radius $\frac{1}{2}l(I_{j})$, we can deduce the
inequality $|z-c(I_{j})|\leq\frac{\sqrt{n}}{2}l(I_{j})$ and furthermore we can
obtain

$|w|\geq{\rm
dist\,}(I_{j},\Omega^{c})+\frac{1}{2}l(I_{j})\geq(\sqrt{n}+\frac{1}{2})l(I_{j})>2|z-c(I_{j})|.$
(94)

Now we can invoke Lemma 13 (62) with $\beta=2$ and obtain the following
estimate for the inside double integral of (93) as below

$\displaystyle\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydt$
$\displaystyle\leq\int_{0}^{\infty}\\!\\!\\!\\!\int_{|w|>2|z-c(I_{j})|}\big{|}\frac{tw_{k}}{(t^{2}+|w|^{2})^{\frac{n+3}{2}}}-\frac{t(w_{k}-z_{k}+c(I_{j})_{k})}{(t^{2}+|w-z+c(I_{j})|^{2})^{\frac{n+3}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (95)

where the constant in (95) only depends on $n$. Inserting (95) into (93)
yields the estimate (90) for $1\leq k\leq n$. To prove (91) for $1\leq k\leq
n$, we can still use (92) and exchange the order of integrations and the
summation over $j$ to estimate the left side of (91) from above by the
following

$\displaystyle\sum_{m}\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{0}^{\infty}\int_{I_{m}}\big{|}\partial_{k}\mathscr{P}(b_{j};y,t)\big{|}dydt$
$\displaystyle\lesssim\sum_{m}\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far
from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|\int_{0}^{\infty}\int_{I_{m}}\big{|}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}$
$\displaystyle\quad-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(96)

and the constant depends on $n$. Notice that after exchanging the order of the
summation over $m$ and the summation over $j$ in the above expression (96), we
can rewrite (96) identically as the expression below

$\sum_{j}\\!\\!\int_{I_{j}}\\!\\!|b(z)|\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{U_{j}}\\!\\!\big{|}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(97)

where the set $U_{j}$ is the union of all the Whitney cube $I_{m}$’s that
satisfy the condition “$I_{j}$ far from $I_{m}$” for some fixed $j$. By the
definition (49), when $y\in U_{j}$, $y$ belongs to some Whitney cube $I_{m}$
and the centers of $I_{m}$ and $I_{j}$ satisfy the condition
$|c(I_{m})-c(I_{j})|>11(\sqrt{n}+1)l(I_{m})$. By Lemma 11 (1), we can further
deduce that $|y-c(I_{j})|\geq\frac{\sqrt{n}+1}{2}\cdot l(I_{j})$. Moreover, if
$z\in I_{j}$ for this $j$ then $|z-c(I_{j})|\leq\frac{\sqrt{n}}{2}\cdot
l(I_{j})$. From the above analysis, we can conclude that

$|y-c(I_{j})|\geq(1+\frac{1}{\sqrt{n}})|z-c(I_{j})|$ (98)

if $y\in U_{j}$ and $z\in I_{j}$ for some fixed $j$. Therefore applying the
change of variable $w=y-c(I_{j})$, the above conclusion (98) and Lemma 13 (62)
with $\beta=1+\frac{1}{\sqrt{n}}$, we can estimate the inside double integral
of (97) as follows

$\displaystyle\int_{0}^{\infty}\\!\\!\\!\\!\int_{U_{j}}\big{|}\frac{t(y_{k}-c(I_{j})_{k})}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t(y_{k}-z_{k})}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\\!\int_{\bar{U}_{j}}\big{|}\frac{tw_{k}}{(t^{2}+|w|^{2})^{\frac{n+3}{2}}}-\frac{t[w_{k}-(z_{k}-c(I_{j})_{k})]}{(t^{2}+|w-(z-c(I_{j}))|^{2})^{\frac{n+3}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (99)

where
$\bar{U}_{j}=\\{w\in\mathbb{R}^{n}:|w|\geq(1+\frac{1}{\sqrt{n}})\cdot|z-c(I_{j})|\\}$
and the constants in (99) depend on $n$. Inserting (99) into (97) and
combining the result with (96) yield the estimate (91) for $1\leq k\leq n$.

Second, we prove (90) and (91) when $k=n+1$. Since $\int_{I_{j}}b(z)dz=0$, we
can use formula (6) to rewrite $\partial_{n+1}\mathscr{P}(b_{j};y,t)$
equivalently as the sum of the following two terms,

$\int_{I_{j}}b(z)\cdot\big{(}\frac{C_{0}}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{C_{0}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{)}dz$
(100)

and

$\int_{I_{j}}b(z)\cdot\big{(}\frac{(n+1)C_{0}t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{(n+1)C_{0}t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{)}dz.$
(101)

Inserting (100) and (101) into the left side of (90) yields that the term

$\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\partial_{n+1}\mathscr{P}(\sum_{j}b_{j};y,t)\big{|}dydt$

can be estimated from above by a constant multiple of the sum of the following
two terms,

$\sum_{j}\\!\\!\int_{I_{j}}\\!\\!|b(z)|\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}\\!\\!\big{|}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{|}dydtdz$
(102)

and

$\sum_{j}\\!\\!\int_{I_{j}}\\!\\!|b(z)|\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}\\!\\!\big{|}\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(103)

and the constant depends on $n$. If we use the change of variable
$w=y-c(I_{j})$ for $y\in\Omega^{c}$ and if $z\in I_{j}$, then with the same
reason we still have the inequality (94). Applying Lemma 13 (63) with
$\beta=2$, we can estimate the inside double integral of (102) as follows

$\displaystyle\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{|}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\\!\int_{|w|>2|z-c(I_{j})|}\big{|}\frac{1}{(t^{2}+|w-(z-c(I_{j}))|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|w|^{2})^{\frac{n+1}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (104)

where the constants in (104) only depend on $n$. Applying Lemma 13 (64) with
$\beta=2$, we can estimate the inside double integral of (103) as follows

$\displaystyle\int_{0}^{\infty}\int_{\Omega^{c}}\big{|}\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\\!\\!\\!\\!\int_{|w|>2|z-c(I_{j})|}\big{|}\frac{t^{2}}{(t^{2}+|w|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|w-(z-c(I_{j}))|^{2})^{\frac{n+3}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (105)

and the constants in (105) depend on $n$. Inserting (104), (105) into (102),
(103) respectively and combining the results with the left side of (90) yield
the estimate (90) for $k=n+1$. To prove estimate (91) when $k=n+1$, we still
rewrite $\partial_{n+1}\mathscr{P}(b_{j};y,t)$ equivalently as the sum of
(100) and (101), and estimate the term

$\sum_{m}\int_{0}^{\infty}\int_{I_{m}}\big{|}\partial_{n+1}\mathscr{P}(\sum_{\begin{subarray}{c}j\\\
I_{j}\text{ far from }I_{m}\end{subarray}}b_{j};y,t)\big{|}dydt$

from above by a constant multiple of the sum of the following two terms,

$\displaystyle\sum_{m}\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|$
$\displaystyle\cdot\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\big{|}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{|}dydtdz$
(106)

and

$\displaystyle\sum_{m}\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}\int_{I_{j}}|b(z)|$
$\displaystyle\cdot\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\big{|}\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(107)

and the constant depends on $n$. Notice that after exchanging the order of the
summation over $m$ and the summation over $j$ in the two expressions above, we
can rewrite (106) and (107) identically as the expressions below,

$\sum_{j}\\!\\!\int_{I_{j}}\\!\\!|b(z)|\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{U_{j}}\\!\\!\big{|}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{|}dydtdz$
(108)

and

$\sum_{j}\\!\\!\int_{I_{j}}\\!\\!|b(z)|\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{U_{j}}\\!\\!\big{|}\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydtdz,$
(109)

where $U_{j}$ is the union of all the Whitney cube $I_{m}$’s that satisfy the
condition “$I_{j}$ far from $I_{m}$” for some fixed $j$. By the definition
(49), when $y\in U_{j}$, $y$ belongs to some Whitney cube $I_{m}$ and the
centers of $I_{m}$ and $I_{j}$ satisfy the condition
$|c(I_{m})-c(I_{j})|>11(\sqrt{n}+1)l(I_{m})$. By inequality (37) of Lemma 11,
we can deduce that $|y-c(I_{j})|\geq\frac{\sqrt{n}+1}{2}\cdot l(I_{j})$.
Moreover, if $z\in I_{j}$ for this $j$ then
$|z-c(I_{j})|\leq\frac{\sqrt{n}}{2}\cdot l(I_{j})$. From the above analysis,
we can conclude that (98) is still true for the variables $y$ and $z$
appearing in (108) and (109). Recall that

$\bar{U}_{j}=\\{w\in\mathbb{R}^{n}:|w|\geq(1+\frac{1}{\sqrt{n}})|z-c(I_{j})|\\}.$

Applying the change of variable $w=y-c(I_{j})$ and Lemma 13 (63) with
$\beta=1+\frac{1}{\sqrt{n}}$, we can estimate the inside double integral of
(108) as below

$\displaystyle\int_{0}^{\infty}\int_{U_{j}}\big{|}\frac{1}{(t^{2}+|y-z|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+1}{2}}}\big{|}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\int_{\bar{U}_{j}}\big{|}\frac{1}{(t^{2}+|w-(z-c(I_{j}))|^{2})^{\frac{n+1}{2}}}-\frac{1}{(t^{2}+|w|^{2})^{\frac{n+1}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (110)

where the constants in (110) depend on $n$. Applying the change of variable
$w=y-c(I_{j})$ and Lemma 13 (64) with $\beta=1+\frac{1}{\sqrt{n}}$, we can
estimate the inside double integral of (109) as below

$\displaystyle\int_{0}^{\infty}\int_{U_{j}}\big{|}\frac{t^{2}}{(t^{2}+|y-c(I_{j})|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|y-z|^{2})^{\frac{n+3}{2}}}\big{|}dydt$
$\displaystyle\lesssim\int_{0}^{\infty}\int_{\bar{U}_{j}}\big{|}\frac{t^{2}}{(t^{2}+|w|^{2})^{\frac{n+3}{2}}}-\frac{t^{2}}{(t^{2}+|w-(z-c(I_{j}))|^{2})^{\frac{n+3}{2}}}\big{|}dwdt$
$\displaystyle\lesssim 1,$ (111)

and the constants in (111) depend on $n$. Inserting (110), (111) into (108),
(109) respectively and combining the results with (106), (107) yield the
estimate (91) for $k=n+1$. Now we finish the proof of Lemma 16. ∎

######  Lemma 17.

Let $f$ be an integrable function in the space of tempered distributions
modulo polynomials and $f_{j}=\psi_{2^{-j}}*f$, where
$\psi\in\mathcal{S}(\mathbb{R}^{n})$ satisfies the conditions (7) and (8).
Then we have

$\partial_{n+1}\mathscr{P}(f;x,t)=\sum_{j\in\mathbb{Z}}\partial_{n+1}\mathscr{P}(f_{j};x,t),$
(112)

and the series converges in the space of tempered distributions modulo
polynomials on $\mathbb{R}^{n}$.

###### Proof of Lemma 17.

Let $\varphi\in\mathcal{S}(\mathbb{R}^{n})$ satisfy the condition

$\int_{\mathbb{R}^{n}}x^{\gamma}\varphi(x)dx=0\quad\text{for all multi-indices
}\gamma.$ (113)

By using Young’s inequality (cf. [5, Theorem 1.2.12]), we see that $f_{j}\in
L^{1}(\mathbb{R}^{n})$ for all $j\in\mathbb{Z}$ whenever $f\in
L^{1}(\mathbb{R}^{n})$ and hence the identity (19) holds with $f_{j}$ in place
of $f$. We would like to prove the following limit

$\displaystyle\lim_{N\rightarrow+\infty}<\partial_{n+1}\mathscr{P}(f;\cdot,t)-\sum_{|j|\leq
N}\partial_{n+1}\mathscr{P}(f_{j};\cdot,t),\varphi>$
$\displaystyle=\lim_{N\rightarrow+\infty}<f,\mathcal{F}_{n}[-2\pi|\xi|e^{-2\pi
t|\xi|}\cdot(1-\sum_{|j|\leq
N}\mathcal{F}_{n}\psi(2^{-j}\xi))\mathcal{F}^{-1}_{n}\varphi(\xi)]>$

equals zero. Since $f$ is a tempered distribution, it suffices to prove the
Schwartz seminorms of the function

$x\in\mathbb{R}^{n}\mapsto\mathcal{F}_{n}[-2\pi|\xi|e^{-2\pi
t|\xi|}\cdot(1-\sum_{|j|\leq
N}\mathcal{F}_{n}\psi(2^{-j}\xi))\mathcal{F}^{-1}_{n}\varphi(\xi)](x)$ (114)

converge to zero as $N\rightarrow+\infty$. Let
$\alpha,\beta,\gamma^{\prime},\gamma^{\prime\prime}$ be multi-indices, and we
have

$\displaystyle\sup_{x\in\mathbb{R}^{n}}\big{|}x^{\alpha}\cdot\partial^{\beta}_{x}\\{\mathcal{F}_{n}[|\xi|e^{-2\pi
t|\xi|}\cdot(1-\sum_{|j|\leq
N}\mathcal{F}_{n}\psi(2^{-j}\xi))\mathcal{F}^{-1}_{n}\varphi(\xi)](x)\\}\big{|}$
$\displaystyle\lesssim\int_{\mathbb{R}^{n}}\big{|}\partial^{\alpha}_{\xi}[\xi^{\beta}|\xi|e^{-2\pi
t|\xi|}\cdot(1-\sum_{|j|\leq
N}\mathcal{F}_{n}\psi(2^{-j}\xi))\mathcal{F}^{-1}_{n}\varphi(\xi)]\big{|}d\xi,$
(115)

where the constant depends on $\alpha,\beta$. We denote

$\Phi_{N}(\xi)=\begin{cases}1-\sum_{|j|\leq
N}\mathcal{F}_{n}\psi(2^{-j}\xi)&\quad\text{if }\xi\neq 0,\\\ 1&\quad\text{if
}\xi=0,\end{cases}$ (116)

then due to (7) and (8), we have

$0\leq\Phi_{N}(\xi)\leq 1,\text{
}spt.\Phi_{N}\subseteq\\{\xi\in\mathbb{R}^{n}:|\xi|\leq 2^{-N}\text{ or
}|\xi|\geq 2^{N}\\},$ (117)

and

$\Phi_{N}(\xi)=1\quad\text{if }|\xi|<2^{-N-1}\text{ or }|\xi|>2^{N+1}.$ (118)

Some direct and trivial calculations show that we can estimate (115) from
above by a finite linear combination, with coefficients depending on
$\alpha,\beta,t$, of integrals of the following form

$\int_{\mathbb{R}^{n}}|\xi|^{L}e^{-2\pi
t|\xi|}\cdot|(\partial^{\gamma^{\prime}}\Phi_{N})(\xi)|\cdot|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|d\xi,$
(119)

where $L$ is an integer. The crucial cases are the following. In the first
case when $|\gamma^{\prime}|>0$, then $\partial^{\gamma^{\prime}}\Phi_{N}$ is
supported in the set

$\\{\xi\in\mathbb{R}^{n}:2^{-N-1}\leq|\xi|\leq 2^{-N},2^{N}\leq|\xi|\leq
2^{N+1}\\},$

and by the support condition of $\mathcal{F}_{n}\psi$ and the definition of
$\Phi_{N}$, we have

$\displaystyle(\partial^{\gamma^{\prime}}\Phi_{N})(\xi)$
$\displaystyle=\partial^{\gamma^{\prime}}(\sum_{j<-N}\mathcal{F}_{n}\psi(2^{-j}\xi)+\sum_{j>N}\mathcal{F}_{n}\psi(2^{-j}\xi))$
$\displaystyle=\partial^{\gamma^{\prime}}(\mathcal{F}_{n}\psi(2^{N+1}\xi)+\mathcal{F}_{n}\psi(2^{-N-1}\xi))$
$\displaystyle=2^{(N+1)|\gamma^{\prime}|}\cdot(\partial^{\gamma^{\prime}}\mathcal{F}_{n}\psi)(2^{N+1}\xi)+2^{-(N+1)|\gamma^{\prime}|}\cdot(\partial^{\gamma^{\prime}}\mathcal{F}_{n}\psi)(2^{-N-1}\xi).$
(120)

Notice that $(\partial^{\gamma^{\prime}}\mathcal{F}_{n}\psi)(2^{N+1}\xi)$ is
supported in the set $\\{\xi\in\mathbb{R}^{n}:2^{-N-2}\leq|\xi|<2^{-N}\\}$ and
$(\partial^{\gamma^{\prime}}\mathcal{F}_{n}\psi)(2^{-N-1}\xi)$ is supported in
the set $\\{\xi\in\mathbb{R}^{n}:2^{N}\leq|\xi|<2^{N+2}\\}$, then we insert
(120) into (119) and estimate (119) from above by a constant multiple of the
sum of the following two terms,

$2^{(N+1)|\gamma^{\prime}|}\int_{2^{-N-2}\leq|\xi|\leq
2^{-N}}|\xi|^{L}e^{-2\pi
t|\xi|}\cdot|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|d\xi$
(121)

and

$2^{-(N+1)|\gamma^{\prime}|}\int_{2^{N}\leq|\xi|\leq 2^{N+2}}|\xi|^{L}e^{-2\pi
t|\xi|}\cdot|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|d\xi,$
(122)

and the constant depends on $\gamma^{\prime},\psi$. To estimate (121), we use
Taylor’s formula of
$(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)$ at $0$
with the Peano remainder. Since the condition (113) implies
$\partial^{\gamma}\mathcal{F}_{n}\varphi(0)=0$ for all multi-indices $\gamma$,
then we have

$|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|\lesssim|\xi|^{M}$
(123)

for an arbitrarily large positive integer $M$. Inserting (123) into (121)
yields

$(\ref{eq2-45})\lesssim 2^{N(|\gamma^{\prime}|+|L|-n-M)}\rightarrow 0\text{ as
}N\rightarrow+\infty,$

if we pick $M$ satisfying $M>|\gamma^{\prime}|+|L|-n$. To estimate (122), we
use the fast decay of the exponential factor and obtain the estimate

$(\ref{eq2-46})\lesssim 2^{N(n+|L|-|\gamma^{\prime}|)}e^{-2\pi t\cdot
2^{N}}\cdot\|\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi\|_{L^{\infty}(\mathbb{R}^{n})}\rightarrow
0\text{ as }N\rightarrow+\infty.$

The above analysis shows that when $|\gamma^{\prime}|>0$, (119) approaches $0$
as $N\rightarrow+\infty$. The second case is when $|\gamma^{\prime}|=0$, we
use the support condition of $\Phi_{N}$ and the fact that
$\|\Phi_{N}\|_{L^{\infty}(\mathbb{R}^{n})}\leq 1$ to estimate (119) from above
by a constant multiple of the sum of the following two terms,

$\int_{|\xi|\leq 2^{-N}}|\xi|^{L}e^{-2\pi
t|\xi|}\cdot|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|d\xi$
(124)

and

$\int_{|\xi|\geq 2^{N}}|\xi|^{L}e^{-2\pi
t|\xi|}\cdot|(\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi)(\xi)|d\xi,$
(125)

and the constant depends on $\psi$. To estimate (124), we use (123) again and
(124) is controlled by $2^{N(|L|-n-M)}$, which approaches $0$ as
$N\rightarrow+\infty$ if we pick $M$ satisfying $M>|L|-n$. To estimate (125),
we use the fact that
$\|\partial^{\gamma^{\prime\prime}}\mathcal{F}^{-1}_{n}\varphi\|_{L^{\infty}(\mathbb{R}^{n})}<\infty$
and the fact that

$\int_{|\xi|\geq 1}|\xi|^{L}e^{-2\pi t|\xi|}d\xi$

is a convergent integral due to the fast decay of $e^{-2\pi t|\xi|}$ for
$t>0$, and then we can conclude that (125) approaches zero as
$N\rightarrow+\infty$. This analysis shows (119) approaches zero as
$N\rightarrow+\infty$ when $|\gamma^{\prime}|=0$. Therefore we have proven the
limit of (115) as $N\rightarrow+\infty$ is zero, and so are the limits of the
Schwartz seminorms of the function (114). The proof of Lemma 17 is now
complete. ∎

The following lemma is cited from [6, section 2.4.4] and lays the foundation
for the justification of subharmonicity needed in this paper.

######  Lemma 18 (cf. Lemma 2.4.9 of [6]).

Let $u_{j}$ be real-valued harmonic functions on $\mathbb{R}^{n+1}$ satisfying
the system of equations

$\displaystyle\sum_{j=1}^{n+1}\frac{\partial u_{j}}{\partial x_{j}}=0,$ (126)
$\displaystyle\frac{\partial u_{j}}{\partial x_{k}}-\frac{\partial
u_{k}}{\partial x_{j}}=0,\quad\text{for }k,j\in\\{1,\cdots,n+1\\}\text{ and
}k\neq j,$ (127)

and let $F=(u_{1},\cdots,u_{n+1})$. Then the function

$|F|^{q}=(\sum_{j=1}^{n+1}|u_{j}|^{2})^{\frac{q}{2}}$

is subharmonic when $q\geq\frac{n-1}{n}$, i.e. it satisfies
$\Delta(|F|^{q})\geq 0$, on $\mathbb{R}^{n+1}_{+}$.

######  Remark 19.

From the proof of Lemma 18 given in [6], we know the condition that $u_{j}$ is
a real-valued function on $\mathbb{R}^{n+1}$ for every $1\leq j\leq n+1$, is a
necessary condition. This is because the proof relies on the diagonalization
of the $(n+1)\times(n+1)$ matrix

$A:=\bigg{\\{}\frac{\partial u_{k}}{\partial x_{j}}\bigg{\\}}^{n+1}_{k,j=1},$
(128)

and the condition (127) shows that the matrix $A$ is real symmetric and thus
is a Hermitian matrix if $u_{j}$’s ($1\leq j\leq n+1$) are all real-valued
functions on $\mathbb{R}^{n+1}$, hence the diagonalization of a Hermitian
matrix is justified. However if $u_{j}$’s ($1\leq j\leq n+1$) are complex-
valued functions, then (127) only tells us that the matrix $A$ defined in
(128) is a complex symmetric matrix, and may not be a Hermitian matrix since a
Hermitian matrix is a complex conjugate symmetric matrix.

######  Lemma 20.

If $1<p<\infty$ and $f\in L^{p}(\mathbb{R}^{n})$ is a real-valued function,
and if $F(x,t)=\nabla_{n+1}\mathscr{P}(f;x,t)$, then
$|F|^{q}=|\nabla_{n+1}\mathscr{P}(f;x,t)|^{q}$ is subharmonic on
$\mathbb{R}^{n+1}_{+}$ when $q\geq\frac{n-1}{n}$.

###### Proof of Lemma 20.

If $f$ is real-valued, then the component functions of $F(x,t)$ are all real-
valued harmonic functions on $\mathbb{R}^{n+1}_{+}$. To apply Lemma 18, we
need to check the component functions of $F(x,t)$ satisfy the system of
equations (126) and (127). Because the Poisson integral $\mathscr{P}(f;x,t)$
is harmonic on $\mathbb{R}^{n+1}_{+}$, we have

$\sum_{k=1}^{n+1}\partial_{k}[\partial_{k}\mathscr{P}(f;x,t)]=\Delta\mathscr{P}(f;x,t)=0,$
(129)

and thus condition (126) is satisfied. If $k,j\in\\{1,\cdots,n\\}$ and $k\neq
j$, we can use the dominated convergence theorem and the assumption that $f\in
L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$, and the formula (5) to obtain

$\displaystyle\partial_{j}\partial_{k}\mathscr{P}(f;x,t)$
$\displaystyle=\int_{\mathbb{R}^{n}}f(y)\cdot\frac{(n+1)(n+3)C_{0}t(y_{k}-x_{k})(y_{j}-x_{j})}{(t^{2}+|x-y|^{2})^{\frac{n+5}{2}}}dy,$
(130)

and similarly, $\partial_{k}\partial_{j}\mathscr{P}(f;x,t)$ has the same
expression as given in (130). Therefore we have
$\partial_{j}\partial_{k}\mathscr{P}(f;x,t)=\partial_{k}\partial_{j}\mathscr{P}(f;x,t)$
for $k,j\in\\{1,\cdots,n\\}$ and $k\neq j$. If $1\leq k\leq n$, we use the
dominated convergence theorem and the assumption that $f\in
L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$, and the formula (6) to obtain

$\displaystyle\partial_{k}\partial_{n+1}\mathscr{P}(f;x,t)$
$\displaystyle=\int_{\mathbb{R}^{n}}f(y)\cdot\big{[}\frac{(n+1)(n+3)C_{0}t^{2}(x_{k}-y_{k})}{(t^{2}+|x-y|^{2})^{\frac{n+5}{2}}}$
$\displaystyle\quad-\frac{(n+1)C_{0}(x_{k}-y_{k})}{(t^{2}+|x-y|^{2})^{\frac{n+3}{2}}}\big{]}dy,$
(131)

and similarly, $\partial_{n+1}\partial_{k}\mathscr{P}(f;x,t)$ has the same
expression as given in (131). Hence we also have
$\partial_{k}\partial_{n+1}\mathscr{P}(f;x,t)=\partial_{n+1}\partial_{k}\mathscr{P}(f;x,t)$
for $1\leq k\leq n$. By now we have shown
$F(x,t)=\nabla_{n+1}\mathscr{P}(f;x,t)$ satisfies all the conditions of Lemma
18 and thus $|\nabla_{n+1}\mathscr{P}(f;x,t)|^{q}$ is subharmonic on
$\mathbb{R}^{n+1}_{+}$ when $q\geq\frac{n-1}{n}$. ∎

######  Lemma 21.

Let $0<s<1<p<\infty$. Assume that $f\in\dot{L}^{p}_{s}(\mathbb{R}^{n})$ is a
bounded real-valued function defined on $\mathbb{R}^{n}$ and has first-order
continuous derivatives, then $\varGamma(\frac{-s}{2})\mathcal{I}_{s}f$ has a
real-valued function representative in $L^{p}(\mathbb{R}^{n})$, where
$\mathcal{I}_{s}f=\mathcal{F}^{-1}_{n}(|\xi|^{s}\mathcal{F}_{n}f)$ in the
distributional sense and $\varGamma(\frac{-s}{2})$ is the evaluation of the
meromorphic extension of the gamma function at $\frac{-s}{2}$.

###### Proof of Lemma 21.

From Definition 3, we know that $f$ is an element of
$\dot{L}^{p}_{s}(\mathbb{R}^{n})$ if and only if the distribution
$\mathcal{I}_{s}f$ has a unique function representative in
$L^{p}(\mathbb{R}^{n})$. Now we show that the product
$\varGamma(\frac{-s}{2})\mathcal{I}_{s}f$ has a real-valued function
representative. Let $\varphi\in\mathcal{S}(\mathbb{R}^{n})$ satisfy the
condition (113), then we have

$<\mathcal{I}_{s}f,\varphi>=<f,\mathcal{F}_{n}(|\xi|^{s}\mathcal{F}^{-1}_{n}\varphi(\xi))>=<f,\mathcal{F}_{n}(|\xi|^{s})*\varphi>$
(132)

in the distributional sense. We recall the distribution $u_{z}$ defined in [5,
Definition 2.4.5]. If $z\in\mathbb{C}$ is a complex number satisfying
$\operatorname{Re}z>-n$, then

$<u_{z},\varphi>=\int_{\mathbb{R}^{n}}\frac{\pi^{\frac{z+n}{2}}}{\varGamma(\frac{z+n}{2})}\cdot|x|^{z}\varphi(x)dx.$
(133)

Given an arbitrary nonnegative integer $N$, [5, expression (2.4.8)] gives an
analytic extension of $<u_{z},\varphi>$ to the case where $z\in\mathbb{C}$
satisfies $\operatorname{Re}z>-N-n-1$, and we have

$\displaystyle<u_{z},\varphi>=$ $\displaystyle\int_{|x|\geq
1}\frac{\pi^{\frac{z+n}{2}}}{\varGamma(\frac{z+n}{2})}\cdot|x|^{z}\varphi(x)dx$
$\displaystyle+\sum_{|\alpha|\leq
N}\frac{\pi^{\frac{z+n}{2}}}{\varGamma(\frac{z+n}{2})}\cdot\frac{\frac{1}{\alpha!}\int_{\mathbb{S}^{n-1}}\theta^{\alpha}d\theta}{|\alpha|+z+n}\cdot(\partial^{\alpha}\varphi)(0)$
$\displaystyle+\int_{|x|<1}\frac{\pi^{\frac{z+n}{2}}}{\varGamma(\frac{z+n}{2})}\cdot\big{\\{}\varphi(x)-\sum_{|\alpha|\leq
N}\frac{(\partial^{\alpha}\varphi)(0)}{\alpha!}\cdot
x^{\alpha}\big{\\}}\cdot|x|^{z}dx,$ (134)

where $\alpha!=\alpha_{1}!\cdots\alpha_{n}!$ for the multi-index
$\alpha=(\alpha_{1},\cdots,\alpha_{n})$ and $\varGamma(\frac{z+n}{2})$ in the
expression (134) is the evaluation of the meromorphic extension of the gamma
function at $\frac{z+n}{2}$. By [5, Theorem 2.4.6], we have the distributional
identity $\mathcal{F}_{n}u_{z}=u_{-n-z}$ for all $z\in\mathbb{C}$. Now we
apply this identity to the case where $z=s\in(0,1)$ then we have

$\frac{\pi^{\frac{s+n}{2}}}{\varGamma(\frac{s+n}{2})}\cdot\mathcal{F}_{n}(|\xi|^{s})=\mathcal{F}_{n}(u_{s})=u_{-n-s}.$
(135)

Since $-n>-n-s>-n-1$, choosing $N=0$ in (134) will suffice. By doing this, we
obtain

$\displaystyle\mathcal{F}_{n}(|\xi|^{s})*\varphi(x)$
$\displaystyle=<\mathcal{F}_{n}(|\xi|^{s}),\varphi(x-\cdot)>$
$\displaystyle=\frac{\varGamma(\frac{s+n}{2})}{\pi^{\frac{s+n}{2}}}\cdot<u_{-n-s},\varphi(x-\cdot)>$
$\displaystyle=\frac{\varGamma(\frac{s+n}{2})}{\varGamma(\frac{-s}{2})\pi^{s+\frac{n}{2}}}\cdot\big{\\{}\int_{|y|\geq
1}\varphi(x-y)\cdot|y|^{-n-s}dy-\frac{\varphi(x)}{s}\cdot\omega_{n-1}$
$\displaystyle\quad+\int_{|y|<1}(\varphi(x-y)-\varphi(x))\cdot|y|^{-n-s}dy\big{\\}}.$
(136)

Inserting (136) into (132) and applying a proper change of variables yield

$\displaystyle<\varGamma(\frac{-s}{2})\mathcal{I}_{s}f,\varphi>$
$\displaystyle=\varGamma(\frac{-s}{2})\cdot<f,\mathcal{F}_{n}(|\xi|^{s})*\varphi>$
$\displaystyle=\int_{\mathbb{R}^{n}}f(x)\cdot\varGamma(\frac{-s}{2})\mathcal{F}_{n}(|\xi|^{s})*\varphi(x)dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\frac{\varGamma(\frac{s+n}{2})}{\pi^{s+\frac{n}{2}}}\cdot\big{\\{}\int_{|y|\geq
1}f(x+y)\cdot|y|^{-n-s}dy-\frac{\omega_{n-1}f(x)}{s}$
$\displaystyle\quad+\int_{|y|<1}(f(x+y)-f(x))\cdot|y|^{-n-s}dy\big{\\}}\cdot\varphi(x)dx,$
(137)

where we also notice that $\varGamma(\frac{-s}{2})$ has been canceled out in
the process. By the arbitrariness of $\varphi$, (137) shows that the
distribution $\varGamma(\frac{-s}{2})\mathcal{I}_{s}f$ can be identified with
the following function

$\frac{\varGamma(\frac{s+n}{2})}{\pi^{s+\frac{n}{2}}}\cdot\big{\\{}\\!\\!\\!\int_{|y|\geq
1}\\!\\!\\!\frac{f(x+y)}{|y|^{n+s}}dy-\frac{\omega_{n-1}f(x)}{s}+\\!\\!\int_{|y|<1}\\!\\!\\!\\!\\!\\!\frac{f(x+y)-f(x)}{|y|^{n+s}}dy\big{\\}}.$
(138)

Because $f$ is a bounded real-valued function and has first-order continuous
derivatives, both integrals in (138) converge absolutely in the Riemann sense
for almost every $x\in\mathbb{R}^{n}$ and have real values. Because
$\frac{s+n}{2}>0$, the definition (1) tells us that $\varGamma(\frac{s+n}{2})$
is a well-defined positive finite real number. Therefore the function
representative of $\varGamma(\frac{-s}{2})\mathcal{I}_{s}f$ given by (138) is
a real-valued function. ∎

##  3\. Proof of Theorem 5

###### Proof.

Step 1: First we notice that it suffices to prove

$\|(\int_{0}^{\infty}t^{q-sq-1}|\partial_{k}\mathscr{P}(f;\cdot,t)|^{q}dt)^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})},$
(139)

for $1\leq k\leq n+1$, $0<p,q<\infty$, $-1<s<1$, and
$f\in\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ is an integrable function, and we will
only provide a detailed proof for (139) in the case of $k=n+1$, and the proof
for (139) in the case where $k=1,2,\cdots,n$ will be very much alike and hence
is omitted. We begin with the estimate

$\displaystyle\int_{0}^{\infty}t^{q-sq-1}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|)^{q}dt$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{-k(q-sq-1)}\int_{2^{-k-1}}^{2^{-k}}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|)^{q}dt$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{-k(q-sq)}\mathop{\rm
ess\,sup\,}_{2^{-k-1}\leq t\leq
2^{-k}}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|)^{q}.$
(140)

Since $f$ is an integrable function, so is

$f_{j}(\cdot)=\psi_{2^{-j}}*f(\cdot)=\mathcal{F}^{-1}_{n}[\mathcal{F}_{n}\psi(2^{-j}\xi)\mathcal{F}_{n}f(\xi)](\cdot)$

for every $j\in\mathbb{Z}$, where $\psi\in\mathcal{S}(\mathbb{R}^{n})$
satisfies conditions (7) and (8). We can use the identity (19) to write for
each $k\in\mathbb{Z}$ and $2^{-k-1}\leq t\leq 2^{-k}$ in the last line of
(140) the following

$\displaystyle\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|$
$\displaystyle=\sum_{k>j}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|+\sum_{k\leq
j}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|$
$\displaystyle=\sum_{k>j}2^{j}|\mathcal{F}^{-1}_{n}[-2\pi e^{-2\pi
t|\xi|}\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x)|$
$\displaystyle\quad+\sum_{k\leq
j}2^{-j}|\mathcal{F}^{-1}_{n}[-2\pi|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x)|.$ (141)

For every $j\in\mathbb{Z}$ in (141) we use the following notations,

$f_{j}^{\mathfrak{n}}(x)=\mathcal{F}^{-1}_{n}[2^{-j}|\xi|\mathcal{F}_{n}\psi(2^{-j}\xi)\mathcal{F}_{n}f(\xi)](x)$
(142)

where “$\mathfrak{n}$” means the factor $2^{-j}|\xi|$ appears in the
numerator, and

$f_{j}^{\mathfrak{d}}(x)=\mathcal{F}^{-1}_{n}[\frac{\mathcal{F}_{n}\psi(2^{-j}\xi)\mathcal{F}_{n}f(\xi)}{2^{-j}|\xi|}](x)$
(143)

where “$\mathfrak{d}$” means the factor $2^{-j}|\xi|$ appears in the
denominator. Now we pick a function $\eta\in\mathcal{S}(\mathbb{R}^{n})$
satisfying

$0\leq\mathcal{F}_{n}\eta\leq 1,\quad
spt.\mathcal{F}_{n}\eta\subseteq\\{\xi\in\mathbb{R}^{n}:\frac{1}{4}\leq|\xi|<4\\},$
(144)

and

$\mathcal{F}_{n}\eta(\xi)=1\text{ on the annulus
}\\{\xi\in\mathbb{R}^{n}:\frac{1}{2}\leq|\xi|<2\\},$ (145)

then $\mathcal{F}_{n}\eta=1$ on the support set of $\mathcal{F}_{n}\psi$ and
we have

$\displaystyle\mathcal{F}^{-1}_{n}[-2\pi e^{-2\pi
t|\xi|}\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x)$
$\displaystyle=\mathcal{F}^{-1}_{n}[-2\pi e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x),$
(146)

and

$\displaystyle\mathcal{F}^{-1}_{n}[-2\pi|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x)$
$\displaystyle=\mathcal{F}^{-1}_{n}[-2\pi|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x).$
(147)

Inserting (146) and (147) into (141) and combining the result with (140) yield
that the term
$\int_{0}^{\infty}t^{q-sq-1}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|)^{q}dt$
can be estimated from above by a constant multiple of the sum of the following
two terms,

$\sum_{k\in\mathbb{Z}}2^{ksq}\\!\\!\\!\\!\mathop{\rm ess\,sup\,}_{2^{-k-1}\leq
t\leq 2^{-k}}\\!\\!\big{\\{}\sum_{k>j}2^{j-k}|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x)|\big{\\}}^{q}$
(148)

and

$\sum_{k\in\mathbb{Z}}2^{ksq}\\!\\!\\!\\!\mathop{\rm ess\,sup\,}_{2^{-k-1}\leq
t\leq 2^{-k}}\\!\\!\big{\\{}\sum_{k\leq
j}2^{-k-j}|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x)|\big{\\}}^{q},$
(149)

and the constant only depends on $s,q$.

Step 2: We estimate the term (148). When $k>j$ and $2^{-k-1}\leq t\leq
2^{-k}$, we recall [12, Definition 2.2] of the Peetre-Fefferman-Stein maximal
function and observe that the $n$-dimensional Fourier transform of the
function $f_{j}^{\mathfrak{n}}$ is supported in the annulus
$\\{\xi\in\mathbb{R}^{n}:2^{j-1}\leq|\xi|\leq 2^{j+1}\\}$ and the associated
Peetre-Fefferman-Stein maximal function is given by

$\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x)=\mathop{\rm
ess\,sup\,}_{z\in\mathbb{R}^{n}}\frac{|f_{j}^{\mathfrak{n}}(x-z)|}{(1+2^{j+1}|z|)^{\frac{n}{r}}},$

where in the proof of this theorem, $r$ is a positive real number chosen to
satisfy $0<r<\min\\{p,q\\}$. Then we have

$\displaystyle|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x)|$
$\displaystyle=|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)]*f_{j}^{\mathfrak{n}}(x)|$
$\displaystyle\leq\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)](z)|\cdot|f_{j}^{\mathfrak{n}}(x-z)|dz$
$\displaystyle\leq\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(\frac{t\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}|z|)^{\frac{n}{r}}dz\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](\frac{z}{t})|\cdot(1+2^{j+1}t\cdot|\frac{z}{t}|)^{\frac{n}{r}}\frac{dz}{t^{n}}\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}t|z|)^{\frac{n}{r}}dz\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x),$
(150)

where we also use the formula

$\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(\frac{t\xi}{2^{j}t})](z)=t^{-n}\cdot\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](\frac{z}{t})$

and an appropriate change of variable. We can split the integral in the last
line of (150) into two parts, one is an integral over the domain
$\\{z\in\mathbb{R}^{n}:2^{j}t|z|\leq 1\\}$ and the other is an integral over
the domain $\\{z\in\mathbb{R}^{n}:2^{j}t|z|\geq 1\\}$. For the first part, we
estimate as follows

$\displaystyle\int_{2^{j}t|z|\leq
1}|\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}t|z|)^{\frac{n}{r}}dz$
$\displaystyle\lesssim\int_{2^{j}t|z|\leq
1}\int_{\mathbb{R}^{n}}e^{-2\pi|\xi|}\cdot|\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})|d\xi
dz$ $\displaystyle\lesssim
2^{-jn}t^{-n}\cdot\int_{2^{j-2}t\leq|\xi|<2^{j+2}t}e^{-2\pi|\xi|}d\xi\cdot\|\mathcal{F}_{n}\eta\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim e^{-\frac{\pi}{2}\cdot 2^{j}t}\lesssim 1,$ (151)

where the implicit constants in (151) depend on $n,r,\eta$ and we also use the
support condition of $\mathcal{F}_{n}\eta$ given in (144). For the second
part, when $k>j$, $2^{-k-1}\leq t\leq 2^{-k}$, and $2^{j}t|z|\geq 1$, we can
assume without loss of generality that
$z=(z_{1},z_{2},\cdots,z_{n})=(z_{1},z^{\prime})$,
$z^{\prime}=(z_{2},\cdots,z_{n})$, and
$|z_{1}|=\max\\{|z_{1}|,|z_{2}|,\cdots,|z_{n}|\\}$. Therefore we have
$|z_{1}|\cdot\sqrt{n}\geq|z|>0$ and we can use integration by parts with
respect to $\xi_{1}$ repeatedly to obtain the following

$\displaystyle|\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|$
$\displaystyle=|\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{\infty}\frac{\partial^{M}}{\partial\xi_{1}^{M}}\big{(}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})\big{)}\cdot\frac{e^{2\pi
i\xi_{1}z_{1}}\cdot e^{2\pi i\xi^{\prime}\cdot z^{\prime}}}{(-2\pi
iz_{1})^{M}}d\xi_{1}d\xi^{\prime}|$
$\displaystyle\lesssim|z|^{-M}\int_{\mathbb{R}^{n}}\big{|}\frac{\partial^{M}}{\partial\xi_{1}^{M}}\big{(}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})\big{)}\big{|}d\xi,$
(152)

where $M$ is a sufficiently large positive integer and the constant in (152)
depends on $n,M$. Combining some direct and trivial calculations with the fact
that the support condition of $\mathcal{F}_{n}\eta$ given in (144) restricts
the range of $\xi$ in the integral in the last line of (152) to a domain where
$|\xi|\sim 2^{j}t$, we can obtain

$(\ref{eq3.14})\lesssim\frac{e^{-\frac{\pi}{2}\cdot
2^{j}t}}{|z|^{M}}\cdot\sum_{l=0}^{M}(2^{j}t)^{n-l},$ (153)

where the constant in (153) depends on $n,M,\eta$. Combining (152) and (153)
and inserting the result into the expression below yield

$\displaystyle\int_{2^{j}t|z|\geq
1}|\mathcal{F}^{-1}_{n}[e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}t|z|)^{\frac{n}{r}}dz$
$\displaystyle\lesssim\int_{2^{j}t|z|\geq
1}\frac{(1+2^{j+1}t|z|)^{\frac{n}{r}}}{|z|^{M}}dz\cdot e^{-\frac{\pi}{2}\cdot
2^{j}t}\cdot\sum_{l=0}^{M}(2^{j}t)^{n-l}$
$\displaystyle\lesssim\int_{2^{j}t|z|\geq
1}\frac{(2^{j}t|z|)^{\frac{n}{r}}}{(2^{j}t|z|)^{M}}dz\cdot
e^{-\frac{\pi}{2}\cdot 2^{j}t}\cdot\sum_{l=0}^{M}(2^{j}t)^{n+l}$
$\displaystyle\lesssim e^{-\frac{\pi}{4}\cdot
2^{j-k}}\cdot\sum_{l=0}^{M}2^{(j-k)l}\lesssim(M+1)e^{-\frac{\pi}{4}\cdot
2^{j-k}}\lesssim 1,$ (154)

where we pick $M$ so that $\frac{n}{r}+n<M$ and thus the integral
$\int_{|z|\geq 1}|z|^{\frac{n}{r}-M}dz$ converges and we also use the
conditions that $k>j$ and $2^{-k-1}\leq t\leq 2^{-k}$. The implicit constants
in (154) depend on $n,r,M,\eta$ and are independent of $k,j,t$. Combining
estimates (151) and (154), we can conclude that the integral in the last line
of (150) can be bounded from above by a constant solely depending on the
parameters $n,r,M,\eta$, therefore we have obtained the following pointwise
estimate

$|\mathcal{F}^{-1}_{n}[e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{n}}(\xi)](x)|\lesssim\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x),$
(155)

for $k>j$ and $2^{-k-1}\leq t\leq 2^{-k}$. Inserting (155) into (148) yields

$\displaystyle(\ref{eq3.10})$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\big{\\{}\sum_{k>j}2^{j-k}\mathcal{P}_{n}f_{j}^{\mathfrak{n}}(x)\big{\\}}^{q}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\big{(}\sum_{l<0}2^{l}\cdot
2^{ks}\mathcal{P}_{n}f_{k+l}^{\mathfrak{n}}(x)\big{)}^{q}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\big{(}\sum_{l<0}2^{l(1-s)}\cdot
2^{(k+l)s}\mathcal{P}_{n}f_{k+l}^{\mathfrak{n}}(x)\big{)}^{q}.$ (156)

If $0<q<1$, then we can have the following inequality directly and exchange
the order of summations to obtain

$\displaystyle(\ref{eq3.18})$
$\displaystyle\leq\sum_{k\in\mathbb{Z}}\sum_{l<0}2^{l(1-s)q}\cdot
2^{(k+l)sq}\mathcal{P}_{n}f_{k+l}^{\mathfrak{n}}(x)^{q}$
$\displaystyle=\sum_{l<0}2^{l(1-s)q}\sum_{k\in\mathbb{Z}}2^{(k+l)sq}\mathcal{P}_{n}f_{k+l}^{\mathfrak{n}}(x)^{q}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{n}}(x)^{q}.$
(157)

If $1\leq q<\infty$, then we use the Minkowski’s inequality for
$\|\cdot\|_{l^{q}}$-norm with respect to $k\in\mathbb{Z}$ to obtain

$\displaystyle(\ref{eq3.18})$
$\displaystyle=\|\sum_{l<0}2^{l(1-s)}\cdot\\{2^{(k+l)s}\mathcal{P}_{n}f_{k+l}^{\mathfrak{n}}(x)\\}_{k\in\mathbb{Z}}\|_{l^{q}}^{q}$
$\displaystyle\leq\big{(}\sum_{l<0}2^{l(1-s)}\cdot(\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{n}}(x)^{q})^{\frac{1}{q}}\big{)}^{q}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{n}}(x)^{q}.$
(158)

In (157) and (158), we use the condition $s<1$ and the constants depend on
$s,q$. Combining (156), (157), and (158), we have obtained

$(\ref{eq3.10})\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{n}}(x)^{q}\quad\text{for
$s<1$ and $0<q<\infty$},$ (159)

and the constant only depends on $n,r,M,\eta,s,q$.

Step 3: We estimate the term (149). When $k\leq j$ and $2^{-k-1}\leq t\leq
2^{-k}$, we notice that the $n$-dimensional Fourier transform of
$f_{j}^{\mathfrak{d}}$ is supported in the annulus
$\\{\xi\in\mathbb{R}^{n}:2^{j-1}\leq|\xi|<2^{j+1}\\}$ and hence the associated
Peetre-Fefferman-Stein maximal function is given by

$\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)=\mathop{\rm
ess\,sup\,}_{z\in\mathbb{R}^{n}}\frac{|f_{j}^{\mathfrak{d}}(x-z)|}{(1+2^{j+1}|z|)^{\frac{n}{r}}},$

where in the proof of this theorem, $r$ is a positive real number chosen to
satisfy $0<r<\min\\{p,q\\}$. Then we have

$\displaystyle|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x)|$
$\displaystyle=|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)]*f_{j}^{\mathfrak{d}}(x)|$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\\!\\!\\!\\!|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)](z)|\cdot|f_{j}^{\mathfrak{d}}(x-z)|dz$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\\!\\!\\!\\!|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)](z)|(1+2^{j+1}|z|)^{\frac{n}{r}}dz\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\\!\\!\\!\\!t^{-2}|\mathcal{F}^{-1}_{n}[|t\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(\frac{t\xi}{2^{j}t})](z)|(1+2^{j+1}|z|)^{\frac{n}{r}}dz\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\\!\\!\\!\\!t^{-2}|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](\frac{z}{t})|(1+2^{j+1}t\cdot|\frac{z}{t}|)^{\frac{n}{r}}\frac{dz}{t^{n}}\cdot\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\\!\\!\\!\\!|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|(1+2^{j+1}t|z|)^{\frac{n}{r}}dz\cdot
t^{-2}\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x),$ (160)

where we also use the formula

$\mathcal{F}^{-1}_{n}[|t\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(\frac{t\xi}{2^{j}t})](z)=t^{-n}\cdot\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](\frac{z}{t})$

and an appropriate change of variable. We can split the integral in the last
line of (160) into two parts, one is an integral over the domain
$\\{z\in\mathbb{R}^{n}:2^{j}t|z|\leq 1\\}$ and the other is an integral over
the domain $\\{z\in\mathbb{R}^{n}:2^{j}t|z|\geq 1\\}$. For the first part, we
estimate as follows

$\displaystyle\int_{2^{j}t|z|\leq
1}|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}t|z|)^{\frac{n}{r}}dz$
$\displaystyle\lesssim\int_{2^{j}t|z|\leq
1}\int_{\mathbb{R}^{n}}|\xi|^{2}e^{-2\pi|\xi|}\cdot|\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})|d\xi
dz$ $\displaystyle\lesssim
2^{-jn}t^{-n}\cdot\int_{2^{j-2}t\leq|\xi|<2^{j+2}t}|\xi|^{2}e^{-2\pi|\xi|}d\xi\cdot\|\mathcal{F}_{n}\eta\|_{L^{\infty}(\mathbb{R}^{n})}$
$\displaystyle\lesssim 2^{2(j-k)}e^{-\frac{\pi}{4}\cdot 2^{j-k}}\lesssim 1,$
(161)

where the implicit constants in (161) depend on $n,r,\eta$ and we also use the
support condition of $\mathcal{F}_{n}\eta$ given in (144). The last inequality
of (161) is due to the fact that
$\lim_{x\rightarrow+\infty}\frac{2^{2x}}{e^{\frac{\pi}{4}\cdot 2^{x}}}=0$,
which can be easily verified by repeatedly applying L’Hospital’s rule. For the
second part, when $k\leq j$, $2^{-k-1}\leq t\leq 2^{-k}$, and $2^{j}t|z|\geq
1$, we can assume without loss of generality that
$z=(z_{1},z_{2},\cdots,z_{n})=(z_{1},z^{\prime})$,
$z^{\prime}=(z_{2},\cdots,z_{n})$, and
$|z_{1}|=\max\\{|z_{1}|,|z_{2}|,\cdots,|z_{n}|\\}$. Therefore we have
$|z_{1}|\cdot\sqrt{n}\geq|z|>0$ and we can use integration by parts with
respect to $\xi_{1}$ repeatedly to obtain the following

$\displaystyle|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|$
$\displaystyle=|\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{\infty}\frac{\partial^{M}}{\partial\xi_{1}^{M}}\big{(}|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})\big{)}\cdot\frac{e^{2\pi
i\xi_{1}z_{1}}\cdot e^{2\pi i\xi^{\prime}\cdot z^{\prime}}}{(-2\pi
iz_{1})^{M}}d\xi_{1}d\xi^{\prime}|$
$\displaystyle\lesssim|z|^{-M}\int_{\mathbb{R}^{n}}\big{|}\frac{\partial^{M}}{\partial\xi_{1}^{M}}\big{(}|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})\big{)}\big{|}d\xi,$
(162)

where $M$ is a sufficiently large positive integer satisfying the condition
$\frac{n}{r}+n<M$ and the constant in (162) depends on $n,M$. Combining some
direct and trivial calculations with the fact that the support condition of
$\mathcal{F}_{n}\eta$ given in (144) restricts the range of $\xi$ in the
integral in the last line of (162) to a domain where $|\xi|\sim 2^{j}t$, we
can obtain

$(\ref{eq3.24})\lesssim\frac{e^{-\frac{\pi}{2}\cdot
2^{j}t}}{|z|^{M}}\cdot\sum_{l=0}^{M}(2^{j}t)^{n+2-l},$ (163)

where the constant in (163) depends on $n,M,\eta$. Combining (162) and (163)
and inserting the result into the expression below yield

$\displaystyle\int_{2^{j}t|z|\geq
1}|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi|\xi|}\mathcal{F}_{n}\eta(\frac{\xi}{2^{j}t})](z)|\cdot(1+2^{j+1}t|z|)^{\frac{n}{r}}dz$
$\displaystyle\lesssim\int_{2^{j}t|z|\geq
1}\frac{(1+2^{j+1}t|z|)^{\frac{n}{r}}}{|z|^{M}}dz\cdot e^{-\frac{\pi}{2}\cdot
2^{j}t}\cdot\sum_{l=0}^{M}(2^{j}t)^{n+2-l}$
$\displaystyle\lesssim\int_{2^{j}t|z|\geq
1}\frac{(2^{j}t|z|)^{\frac{n}{r}}}{(2^{j}t|z|)^{M}}dz\cdot
e^{-\frac{\pi}{2}\cdot 2^{j}t}\cdot\sum_{l=0}^{M}(2^{j}t)^{n+2+l}$
$\displaystyle\lesssim(M+1)2^{(j-k)(M+2)}\cdot e^{-\frac{\pi}{4}\cdot
2^{j-k}}\lesssim 1,$ (164)

where we use the conditions that $k\leq j$, $2^{-k-1}\leq t\leq 2^{-k}$, and
the fact that the integral $\int_{|z|\geq 1}|z|^{\frac{n}{r}-M}dz$ converges
since $\frac{n}{r}+n<M$, and the implicit constants in (164) depend on
$n,r,M,\eta$. The last inequality of (164) is due to the fact that
$\lim_{x\rightarrow+\infty}\frac{2^{(M+2)x}}{e^{\frac{\pi}{4}\cdot 2^{x}}}=0$,
which can be verified by repeatedly applying L’Hospital’s rule. Combining
(161) and (164), we can conclude that the integral in the last line of (160)
can be bounded from above by a constant whose value only depends on the
parameters $n,r,M,\eta$ and in particular, is independent of $k,j,t$.
Therefore we have obtained the following pointwise estimate

$|\mathcal{F}^{-1}_{n}[|\xi|^{2}e^{-2\pi
t|\xi|}\mathcal{F}_{n}\eta(2^{-j}\xi)\mathcal{F}_{n}f_{j}^{\mathfrak{d}}(\xi)](x)|\lesssim
t^{-2}\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)\lesssim
2^{2k}\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x),$ (165)

for $k\leq j$ and $2^{-k-1}\leq t\leq 2^{-k}$. Inserting (165) into (149)
yields

$\displaystyle(\ref{eq3.11})$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\big{\\{}\sum_{k\leq
j}2^{k-j}\mathcal{P}_{n}f_{j}^{\mathfrak{d}}(x)\big{\\}}^{q}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\big{(}\sum_{l\geq 0}2^{-l}\cdot
2^{ks}\mathcal{P}_{n}f_{k+l}^{\mathfrak{d}}(x)\big{)}^{q}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\big{(}\sum_{l\geq 0}2^{-l(1+s)}\cdot
2^{(k+l)s}\mathcal{P}_{n}f_{k+l}^{\mathfrak{d}}(x)\big{)}^{q}.$ (166)

If $0<q<1$, then we can have the following inequality directly and exchange
the order of summations to obtain

$\displaystyle(\ref{eq3.28})$
$\displaystyle\leq\sum_{k\in\mathbb{Z}}\sum_{l\geq 0}2^{-l(1+s)q}\cdot
2^{(k+l)sq}\mathcal{P}_{n}f_{k+l}^{\mathfrak{d}}(x)^{q}$
$\displaystyle=\sum_{l\geq
0}2^{-l(1+s)q}\sum_{k\in\mathbb{Z}}2^{(k+l)sq}\mathcal{P}_{n}f_{k+l}^{\mathfrak{d}}(x)^{q}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{d}}(x)^{q}.$
(167)

If $1\leq q<\infty$, then we use the Minkowski’s inequality for
$\|\cdot\|_{l^{q}}$-norm with respect to $k\in\mathbb{Z}$ to obtain

$\displaystyle(\ref{eq3.28})$ $\displaystyle=\|\sum_{l\geq
0}2^{-l(1+s)}\cdot\\{2^{(k+l)s}\mathcal{P}_{n}f_{k+l}^{\mathfrak{d}}(x)\\}_{k\in\mathbb{Z}}\|_{l^{q}}^{q}$
$\displaystyle\leq\big{(}\sum_{l\geq
0}2^{-l(1+s)}\cdot(\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{d}}(x)^{q})^{\frac{1}{q}}\big{)}^{q}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{d}}(x)^{q}.$
(168)

In (167) and (168), we use the condition $-1<s$ and the constants there depend
on $s,q$. Combining (166), (167), and (168), we have obtained

$(\ref{eq3.11})\lesssim\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{d}}(x)^{q}\quad\text{for
$-1<s$ and $0<q<\infty$},$ (169)

and the constant only depends on $n,r,M,\eta,s,q$.

Step 4: Now we combine the estimates (140), (148), (149), (159), (169) all
together, raise both sides of the resulting inequality to the power of
$\frac{1}{q}$, and put the subsequent result into
$\|\cdot\|_{L^{p}(\mathbb{R}^{n})}$-quasinorm, then we can obtain the estimate
below

$\displaystyle\|(\int_{0}^{\infty}t^{q-sq-1}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};\cdot,t)|)^{q}dt)^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|(\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{n}}(x)^{q})^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}+\|(\sum_{k\in\mathbb{Z}}2^{ksq}\mathcal{P}_{n}f_{k}^{\mathfrak{d}}(x)^{q})^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|\\{2^{ks}f_{k}^{\mathfrak{n}}\\}_{k\in\mathbb{Z}}\|_{L^{p}(l^{q})}+\|\\{2^{ks}f_{k}^{\mathfrak{d}}\\}_{k\in\mathbb{Z}}\|_{L^{p}(l^{q})},$
(170)

where the constants in (170) depend on $n,r,M,\eta,s,p,q$, and the last
inequality in (170) is due to [12, Remark 2.13] since we choose
$0<r<\min\\{p,q\\}$ and the $n$-dimensional distributional Fourier transforms
of $f_{k}^{\mathfrak{n}}$ and $f_{k}^{\mathfrak{d}}$ are both compactly
supported in the annulus $\\{\xi\in\mathbb{R}^{n}:2^{k-1}\leq|\xi|<2^{k+1}\\}$
for each $k\in\mathbb{Z}$. Recall the defining expressions of
$f_{k}^{\mathfrak{n}}$ and $f_{k}^{\mathfrak{d}}$ given in (142) and (143),
the condition that $\mathcal{F}_{n}\eta=1$ on the support set of
$\mathcal{F}_{n}\psi$ due to (144) and (145), and the notation
$f_{k}=\psi_{2^{-k}}*f$ for every $k\in\mathbb{Z}$, then we have

$\displaystyle|f_{k}^{\mathfrak{n}}(x)|$
$\displaystyle=|\mathcal{F}^{-1}_{n}[2^{-k}|\xi|\mathcal{F}_{n}\eta(2^{-k}\xi)\mathcal{F}_{n}\psi(2^{-k}\xi)\mathcal{F}_{n}f(\xi)](x)|$
$\displaystyle=|\mathcal{F}^{-1}_{n}[2^{-k}|\xi|\mathcal{F}_{n}\eta(2^{-k}\xi)]*f_{k}(x)|$
$\displaystyle\leq\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[2^{-k}|\xi|\mathcal{F}_{n}\eta(2^{-k}\xi)](z)|\cdot|f_{k}(x-z)|dz$
$\displaystyle\leq\int_{\mathbb{R}^{n}}2^{kn}|\mathcal{F}^{-1}_{n}[|\xi|\mathcal{F}_{n}\eta(\xi)](2^{k}z)|\cdot(1+2^{k+1}|z|)^{\frac{n}{r}}dz\mathcal{P}_{n}f_{k}(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}|\mathcal{F}^{-1}_{n}[|\xi|\mathcal{F}_{n}\eta(\xi)](z)|\cdot(1+2|z|)^{\frac{n}{r}}dz\mathcal{P}_{n}f_{k}(x)\lesssim\mathcal{P}_{n}f_{k}(x),$
(171)

where we recall that the Peetre-Fefferman-Stein maximal function for the
function $f_{k}$ whose $n$-dimensional distributional Fourier transform is
compactly supported in the annulus
$\\{\xi\in\mathbb{R}^{n}:2^{k-1}\leq|\xi|<2^{k+1}\\}$, is given by

$\mathcal{P}_{n}f_{k}(x)=\mathop{\rm
ess\,sup\,}_{z\in\mathbb{R}^{n}}\frac{|f_{k}(x-z)|}{(1+2^{k+1}|z|)^{\frac{n}{r}}}.$

In (171), we also use the formula

$\mathcal{F}^{-1}_{n}[2^{-k}|\xi|\mathcal{F}_{n}\eta(2^{-k}\xi)](z)=2^{kn}\mathcal{F}^{-1}_{n}[|\xi|\mathcal{F}_{n}\eta(\xi)](2^{k}z),$

and the integral in the last line of (171) converges to a positive finite
constant whose value depends on $n,r,\eta$, due to the fact that the mapping

$z\in\mathbb{R}^{n}\rightarrow\mathcal{F}^{-1}_{n}[|\xi|\mathcal{F}_{n}\eta(\xi)](z)$

defines a Schwartz function on $\mathbb{R}^{n}$. In a similar way as given by
(171), since the mapping

$z\in\mathbb{R}^{n}\rightarrow\mathcal{F}^{-1}_{n}[\frac{\mathcal{F}_{n}\eta(\xi)}{|\xi|}](z)$

also defines a Schwartz function on $\mathbb{R}^{n}$, we have

$|f_{k}^{\mathfrak{d}}(x)|\lesssim\mathcal{P}_{n}f_{k}(x)\quad\text{for every
$k\in\mathbb{Z}$ and $x\in\mathbb{R}^{n}$,}$ (172)

where the implicit constant depends only on $n,r,\eta$ and is independent of
$k$ and $x$. Inserting (171) and (172) into (170) and invoking [12, Remark
2.13] again, we can obtain

$\displaystyle\|(\int_{0}^{\infty}t^{q-sq-1}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};\cdot,t)|)^{q}dt)^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\lesssim\|\\{2^{ks}\mathcal{P}_{n}f_{k}\\}_{k\in\mathbb{Z}}\|_{L^{p}(l^{q})}\lesssim\|f\|_{\dot{F}^{s}_{p,q}(\mathbb{R}^{n})}<\infty,$
(173)

where $0<p,q<\infty$, $-1<s<1$ and $f\in\dot{F}^{s}_{p,q}(\mathbb{R}^{n})$ is
an integrable function. Inequality (173) implies that the series
$\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};x,t)|$ converges for
almost every $x\in\mathbb{R}^{n}$ and $0<t<\infty$. Combining this result with
Lemma 17, we see that the equation (112) is true in the sense of tempered
distributions modulo polynomials for almost every $x\in\mathbb{R}^{n}$ and
$0<t<\infty$, and the tempered distribution $\partial_{n+1}\mathscr{P}(f;x,t)$
has a function representative which is the pointwise limit of the series
$\sum_{j\in\mathbb{Z}}\partial_{n+1}\mathscr{P}(f_{j};x,t)$. Furthermore, we
have justified the inequality

$\displaystyle\|(\int_{0}^{\infty}t^{q-sq-1}|\partial_{n+1}\mathscr{P}(f;\cdot,t)|^{q}dt)^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}$
$\displaystyle\leq\|(\int_{0}^{\infty}t^{q-sq-1}(\sum_{j\in\mathbb{Z}}|\partial_{n+1}\mathscr{P}(f_{j};\cdot,t)|)^{q}dt)^{\frac{1}{q}}\|_{L^{p}(\mathbb{R}^{n})}.$
(174)

Inequalities (173) and (174) together complete the proof of (139) when
$k=n+1$. The proof of (139) where $1\leq k\leq n$ can be obtained in a similar
way with minor differences generated from using identity (20) in place of
(19). We now conclude the proof of Theorem 5. ∎

##  4\. Proof of Theorem 8

###### Proof.

Step 1: Assume $f$ is an integrable function in $L^{p}(\mathbb{R}^{n})$. If

$\mathop{\rm ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|\leq\alpha<\infty,$

then since $2\leq q<\infty$ and $\lambda>1$, we can use Corollary 7 to obtain
the following inequality

$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\mathcal{G}_{\lambda,q}(f)(x)>\alpha\\})$
$\displaystyle\leq\alpha^{-q}\|\mathcal{G}_{\lambda,q}(f)\|_{L^{q}(\mathbb{R}^{n})}^{q}$
$\displaystyle\lesssim\alpha^{-q}\int_{\mathbb{R}^{n}}|f(x)|^{q-p}\cdot|f(x)|^{p}dx\leq\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(175)

and hence inequality (35) is proved in this case. If

$0<\alpha<\mathop{\rm ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|,$

we apply Lemma 12 to the function $f\in L^{p}(\mathbb{R}^{n})$ and the set
$\Omega=\\{x\in\mathbb{R}^{n}:M(|f|^{p})(x)>\alpha^{p}\\}$, and obtain a
collection of Whitney cubes $\\{I_{j}\\}_{j}$ and two functions, the “good”
function $g$ and the “bad” function $b$, where $\\{I_{j}\\}_{j}$, $g$ and $b$
satisfy all the conclusions of Lemma 12. Then by Lemma 12 (v), we have the
pointwise estimate
$\mathcal{G}_{\lambda,q}(f)(x)\lesssim\mathcal{G}_{\lambda,q}(g)(x)+\mathcal{G}_{\lambda,q}(b)(x)$
and hence the following inequality

$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\mathcal{G}_{\lambda,q}(f)(x)>\alpha\\})$
$\displaystyle\leq\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(g)(x)\\})+\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(b)(x)\\}),$

where the implicit constants in the above inequalities only depend on $q$.
Therefore to prove (35), it suffices to prove the following two inequalities,

$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(g)(x)\\})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(176)
$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(b)(x)\\})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(177)

where the values of implicit constants only depend on some fixed parameters.
From Lemma 12 (viii), (v), (vi) and (i), we see that when
$0<\alpha<\mathop{\rm ess\,sup\,}_{x\in\mathbb{R}^{n}}|f(x)|$ and $f\in
L^{p}(\mathbb{R}^{n})\bigcap L^{1}(\mathbb{R}^{n})$, we have

$\|g\|_{L^{q}(\mathbb{R}^{n})}^{q}\lesssim\alpha^{q-p}\|g\|_{L^{p}(\mathbb{R}^{n})}^{p}\lesssim\alpha^{q-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(178)

where the constants in (178) depend on $n,p,q$. And we also have

$\displaystyle\|g\|_{L^{1}(\mathbb{R}^{n})}$
$\displaystyle\leq\|f\|_{L^{1}(\mathbb{R}^{n})}+\sum_{j}\int_{I_{j}}|b(x)|dx$
$\displaystyle\lesssim\|f\|_{L^{1}(\mathbb{R}^{n})}+\sum_{j}\alpha\cdot\mathcal{L}^{n}(I_{j})$
$\displaystyle\lesssim\|f\|_{L^{1}(\mathbb{R}^{n})}+\alpha^{1-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}<\infty,$
(179)

and the constants in (179) depend on $n,p$. Inequalities (178) and (179) show
that $g$ is an integrable function in $L^{q}(\mathbb{R}^{n})$, and hence we
can apply Corollary 7 to obtain

$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(g)(x)\\})$
$\displaystyle\lesssim(\frac{2}{\alpha})^{q}\|\mathcal{G}_{\lambda,q}(g)\|_{L^{q}(\mathbb{R}^{n})}^{q}\lesssim\alpha^{-q}\|g\|_{L^{q}(\mathbb{R}^{n})}^{q}\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(180)

where the constants in (180) depend on $n,p,q$ and the implicit constant
coming from Corollary 7. Inequality (176) is proved. Now we prove inequality
(177). We consider the following equation

$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\frac{\alpha}{2}\lesssim\mathcal{G}_{\lambda,q}(b)(x)\\})$
$\displaystyle=\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim\mathcal{G}_{\lambda,q}(b)(x)^{q}\\}),$
(181)

where the implicit constants depend on $q$. We recall the relations between
two Whitney cubes given in (48) and (49), and we also use the decomposition

$\mathbb{R}^{n}=\Omega^{c}\bigcup\Omega=\Omega^{c}\bigcup\bigg{(}\bigcup_{m}I_{m}\bigg{)}.$
(182)

By Lemma 12 (v), the function $b$ is supported in $\Omega$ and if
$\chi_{I_{j}}$ denotes the characteristic function of the cube $I_{j}$ and if
$b_{j}=b\cdot\chi_{I_{j}}$, then we have the equation

$b=\sum_{j}b_{j}=b^{(n.m.)}+b^{(f.m.)},$ (183)

and for each $m$, we denote

$b^{(n.m.)}=\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ near
}I_{m}\end{subarray}}b_{j}\quad\text{and}\quad
b^{(f.m.)}=\sum_{\begin{subarray}{c}j\\\ I_{j}\text{ far from
}I_{m}\end{subarray}}b_{j},$ (184)

where “$(n.m.)$” comes from the phrase “near $I_{m}$” while “$(f.m.)$” comes
from the phrase “far from $I_{m}$”. By using the defining expression (30), we
can estimate $\mathcal{G}_{\lambda,q}(b)(x)^{q}$ from above by a constant
multiple of the sum of the following $(2n+3)$ terms,

$\displaystyle
A_{k}(x):=\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot t^{q-n-1}|\partial_{k}\mathscr{P}(b;y,t)|^{q}dydt,$ (185)
$\displaystyle
B_{k}(x):=\sum_{m}\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot t^{q-n-1}|\partial_{k}\mathscr{P}(b^{(f.m.)};y,t)|^{q}dydt,$ (186)
$\displaystyle
D(x):=\sum_{m}\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot t^{q-n-1}|\nabla_{n+1}\mathscr{P}(b^{(n.m.)};y,t)|^{q}dydt,$ (187)

where $1\leq k\leq n+1$ in (185) and (186), and the constant depends on $n,q$.
Combining the above pointwise estimate with (181), we can obtain

$\alpha^{q}\leq
C_{1}\cdot(\sum_{k=1}^{n+1}A_{k}(x)+\sum_{k=1}^{n+1}B_{k}(x)+D(x)),$ (188)

where $C_{1}=C_{1}(n,q)$ and $x$ belongs to the set
$\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim\mathcal{G}_{\lambda,q}(b)(x)^{q}\\}$.
Inequality (188) says that at least one of the terms $D(x)$, $A_{k}(x)$ and
$B_{k}(x)$ for $1\leq k\leq n+1$ is greater than or equal to
$\frac{\alpha^{q}}{C_{1}(2n+3)}$, therefore we can estimate (181) from above
by the sum of the following,

$\displaystyle\sum_{k=1}^{n+1}\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim
A_{k}(x)\\}),$ (189)
$\displaystyle\sum_{k=1}^{n+1}\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim
B_{k}(x)\\}),$ (190)
$\displaystyle\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim
D(x)\\}),$ (191)

and the implicit constants in (189), (190), (191) depend only on $n,q$.
Therefore to prove (177), it suffices to prove that each and every one of
(189), (190), (191) is dominated by a constant multiple of
$\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p}$. We use the following notations
for $1\leq k\leq n+1$,

$\displaystyle\bar{A}_{k}(x):=\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot t^{-n}|\partial_{k}\mathscr{P}(b;y,t)|dydt,$ (192)
$\displaystyle\bar{B}_{k}(x):=\sum_{m}\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\\!\\!\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\cdot t^{-n}|\partial_{k}\mathscr{P}(b^{(f.m.)};y,t)|dydt.$ (193)

Step 2: We estimate the term (189). From (185) and (192), we can apply Lemma
15 (1) to the term $|\partial_{k}\mathscr{P}(b;y,t)|^{q-1}$ and obtain

$\alpha^{q}\lesssim A_{k}(x)\lesssim\alpha^{q-1}\bar{A}_{k}(x),$

where $x$ belongs to the set $\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim
A_{k}(x)\\}$ and $1\leq k\leq n+1$, and the constants depend on $n,p,q$.
Therefore invoking Chebyshev’s inequality, Fubini’s theorem, Lemma 16 (90),
Lemma 12 (i) and (vi) in a sequence yields the following estimate

$\displaystyle(\ref{eq4.15})$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha\lesssim\bar{A}_{k}(x)\\})$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\frac{1}{\alpha}\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}\\!\\!\int_{\mathbb{R}^{n}}\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\frac{dx}{t^{n}}\cdot|\partial_{k}\mathscr{P}(b;y,t)|dydt$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\frac{1}{\alpha}\int_{0}^{\infty}\\!\\!\\!\\!\int_{\Omega^{c}}|\partial_{k}\mathscr{P}(b;y,t)|dydt$
$\displaystyle\lesssim\frac{1}{\alpha}\cdot\sum_{j}\int_{I_{j}}|b(z)|dz\lesssim\sum_{j}\mathcal{L}^{n}(I_{j})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(194)

where the constants in (194) depend on $n,p,q,\lambda$, and we also use the
fact that

$\int_{\mathbb{R}^{n}}(1+|x|)^{-n\lambda}dx<\infty,$

since $\lambda=\frac{q}{p}>1$ for $1<p<q$.

Step 3: We estimate the term (190). From (186) and (193), we can apply Lemma
15 (2) to the term $|\partial_{k}\mathscr{P}(b^{(f.m.)};y,t)|^{q-1}$ and
obtain

$\alpha^{q}\lesssim B_{k}(x)\lesssim\alpha^{q-1}\bar{B}_{k}(x),$

where $x$ belongs to the set $\\{x\in\mathbb{R}^{n}:\alpha^{q}\lesssim
B_{k}(x)\\}$ and $1\leq k\leq n+1$, and the constants depend on $n,p,q$.
Therefore invoking Chebyshev’s inequality, Fubini’s theorem, Lemma 16 (91),
Lemma 12 (i) and (vi) in a sequence yields the following estimate

$\displaystyle(\ref{eq4.16})$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\mathcal{L}^{n}(\\{x\in\mathbb{R}^{n}:\alpha\lesssim\bar{B}_{k}(x)\\})$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\frac{1}{\alpha}\sum_{m}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}\\!\\!\int_{\mathbb{R}^{n}}\big{(}\frac{t}{t+|x-y|}\big{)}^{\lambda
n}\frac{dx}{t^{n}}|\partial_{k}\mathscr{P}(b^{(f.m.)};y,t)|dydt$
$\displaystyle\lesssim\sum_{k=1}^{n+1}\frac{1}{\alpha}\sum_{m}\int_{0}^{\infty}\\!\\!\\!\\!\int_{I_{m}}|\partial_{k}\mathscr{P}(b^{(f.m.)};y,t)|dydt$
$\displaystyle\lesssim\frac{1}{\alpha}\cdot\sum_{j}\int_{I_{j}}|b(z)|dz\lesssim\sum_{j}\mathcal{L}^{n}(I_{j})\lesssim\alpha^{-p}\|f\|_{L^{p}(\mathbb{R}^{n})}^{p},$
(195)

where the constants in (195) depend on $n,p,q,\lambda$, and we also use the
fact that