\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 185, pp. 1--37.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/185\hfil Modified Navier-Stokes equations] {Global mild solutions to modified Navier-Stokes equations with small initial data in critical Besov-Q spaces} \author[P. Li, J. Xiao, Q. Yang \hfil EJDE-2014/185\hfilneg] {Pengtao Li, Jie Xiao, Qixiang Yang} % in alphabetical order \address{Pengtao Li \newline College of Mathematics, Qingdao University, Qingdao, Shandong 515063, China} \email{ptli@qdu.edu.cn} \address{Jie Xiao \newline Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C, 5S7, Canada} \email{jxiao@mun.ca} \address{Qixiang Yang \newline School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China} \email{qxyang@whu.edu.cn} \thanks{Submitted April 29, 2014. Published September 2, 2014.} \subjclass[2000]{35Q30, 76D03, 42B35, 46E30} \keywords{Modified Navier-Stokes equations; Besov-Q spaces; mild solutions; \hfill\break\indent existence; uniqueness} \begin{abstract} This article is devoted to establishing the global existence and uniqueness of a mild solution of the modified Navier-Stokes equations with a small initial data in the critical Besov-Q space. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Statement of the main results} \label{intro} For $\beta>1/2$, the Cauchy problem of the modified Navier-Stokes equations on the half-space $\mathbb{R}^{1+n}_{+}= (0,\infty) \times \mathbb{R}^{n}$, $n\geq 2$, consists of studying the existence of a solution $u$ to \begin{equation}\label{eqn:ns} \begin{gathered} \frac{\partial u} {\partial t} +(-\Delta)^{\beta} u + u \cdot \nabla u -\nabla p=0, \quad \text{in } \mathbb{R}^{1+n}_{+}; \\ \nabla \cdot u=0, \quad \text{in } \mathbb{R}^{1+n}_{+}; \\ u|_{t=0}= a, \quad \text{in } \mathbb{R}^{n}, \end{gathered} \end{equation} where $(-\Delta)^{\beta}$ represents the $\beta$-order Laplace operator defined by the Fourier transform in the space variable: $$ \widehat{(-\Delta)^{\beta}u}(\cdot,\xi)= |\xi|^{2\beta} \hat{u}(\cdot,\xi). $$ Here, we point out that \eqref{eqn:ns} is a generalization of the classical Navier-Stokes system and two-dimensional quasi-geostrophic equation which have continued to attract attention extensively, and that the dissipation $(-\Delta)^{\beta}u$ still retains the physical meaning of the nonlinearity $u\cdot\nabla u+\nabla p$ and the divergence-free condition $\nabla\cdot u=0$. Upon letting $R_j$, $j=1,2,\dots n$, be the Riesz transforms, writing \begin{equation} \begin{gathered} \mathbb{P}= \{\delta_{l,l'}+ R_lR_{l'}\},\quad l,l'=1,\dots,n;\\ \mathbb{P}\nabla (u\otimes u)= \sum_l \frac{\partial}{\partial x_l} (u_lu) - \sum_l \sum_{l'} R_lR_{l'} \nabla (u_l u_{l'});\\ \widehat{e^{-t(-\Delta)^{\beta}}f}(\xi) = e^{-t|\xi|^{2\beta}}\hat{f}(\xi), \end{gathered} \end{equation} and using $\nabla \cdot u=0$, we can see that a solution of the above Cauchy problem is then obtained via the integral equation \begin{equation}\label{eqn:mildsolution} \begin{gathered} u(t,x)= e^{-t (-\Delta)^{\beta}} a(x) - B(u,u)(t,x);\\ B(u,u)(t,x)\equiv\int^{t}_0 e^{-(t-s)(-\Delta)^{\beta}} \mathbb{P}\nabla (u\otimes u) ds, \end{gathered} \end{equation} which can be solved by a fixed-point method whenever the convergence is suitably defined in a function space. Solutions of \eqref{eqn:mildsolution} are called mild solutions of \eqref{eqn:ns}. The notion of such a mild solution was pioneered by Kato-Fujita \cite{KF} in 1960s. During the latest decades, many important results about the mild solutions to \eqref{eqn:ns} have been established; see for example, Cannone \cite{C1, C2}, Germin-Pavlovic-Staffilani \cite{GPS}, Giga-Miyakawa \cite{GM}, Kato \cite{Kat}, Koch-Tataru \cite{KT}, Wu \cite{W1,W2,W3,W4}, and their references including Kato-Ponce \cite{KP} and Taylor \cite{Ta}. The main purpose of this paper is to establish the following global existence and uniqueness of a mild solution to \eqref{eqn:ns} with a small initial data in the critical Besov-Q space. \begin{theorem}\label{mthmain} Assume that $\beta>1/2$; $1< p, q<\infty$; $\gamma_1=\gamma_2-2\beta+1$; $m>\max \{p,\frac{n}{2\beta}\}$; $0\max\{1,1+\frac{n}{p}-2\beta\}$; see also \cite{W4} concerning the corresponding regularity. Importantly, Koch-Tataru \cite{KT} studied the global existence and uniqueness of \eqref{eqn:ns} with $\beta=1$ via introducing $BMO^{-1}(\mathbb{R}^{n})$. Extending Koch-Tataru's work \cite{KT}, Xiao \cite{X, X1} introduced the Q-spaces $Q^{-1}_{0<\alpha<1}(\mathbb{R}^{n})$ to investigate the global existence and uniqueness of the classical Navier-Stokes system. The ideas of \cite{X} were developed by Li-Zhai \cite{LZ} to study the global existence and uniqueness of \eqref{eqn:ns} with small data in a class of Q-type spaces $Q_{\alpha}^{\beta,-1}(\mathbb{R}^{n})$ under $\beta\in(\frac{1}{2}, 1)$. Recently, Lin-Yang \cite{LY} got the global existence and uniqueness of \eqref{eqn:ns} with initial data being small in a diagonal Besov-Q space for $\beta\in(\frac{1}{2}, 1)$. In fact, the above historical citations lead us to make a decisive two-fold observation. On the one hand, thanks to that \eqref{eqn:ns} is invariant under the scaling \begin{gather*} u_{\lambda}(t,x) = \lambda^{2\beta-1} u(\lambda^{2\beta}t, \lambda x);\\ p_{\lambda}(t,x) = \lambda^{4\beta-2} p(\lambda^{2\beta}t, \lambda x), \end{gather*} the initial data space $\dot{B}^{\gamma_1,\gamma_2}_{p,q}$ is critical for \eqref{eqn:ns} in the sense that the space is invariant under the scaling \begin{equation}\label{eq2} f_{\lambda}(x)=\lambda^{2\beta-1}f(\lambda x). \end{equation} A simple computation, along with letting $\beta=1$ in \eqref{eq2}, indicates that the function spaces \begin{gather*} \dot{L}^{2}_{\frac{n}{2}-1}(\mathbb{R}^{n}) =\dot{B}^{-1+\frac{n}{2},2}_2(\mathbb{R}^{n});\quad L^{n}(\mathbb{R}^{n});\\ \dot{B}^{-1+\frac{n}{p},q}_{p}(\mathbb{R}^{n});\quad BMO^{-1}(\mathbb{R}^{n}), \end{gather*} are critical for \eqref{eqn:ns} with $\beta=1$. Moreover, \eqref{eq2} under $\beta>1/2$ is valid for functions in the homogeneous Besov spaces $\dot{B}^{1+\frac{n}{2}-2\beta,1}_2(\mathbb{R}^{n})$ and $\dot{B}^{1+\frac{n}{2}-2\beta,\infty}_2(\mathbb{R}^{n})$ attached to \eqref{eqn:ns}. On the other hand, it is suitable to mention the following relations: \begin{gather*} \dot{B}^{\gamma_1, \frac{n}{p}}_{p,q}=\dot{B}^{\gamma_1,q}_{p}(\mathbb{R}^{n})\quad \text{for } 1\leq p,\; q<\infty, -\infty<\gamma_1<\infty;\\ \dot{B}^{1+\frac{n}{p}-2\beta,\frac{n}{p}}_{p_0,q_0}\supseteq \dot{B}^{1+\frac{n}{p}-2\beta,q}_{p}(\mathbb{R}^{n})\quad \text{for } 10;\\ \dot{B}^{\alpha-\beta+1,\alpha+\beta-1}_{2,2}=Q^{\beta}_{\alpha}(\mathbb{R}^{n}) \quad \text{for } \alpha\in(0,1),\; \beta\in(1/2,1),\; \alpha+\beta-1\geq0. \end{gather*} To briefly describe the argument for Theorem \ref{mthmain}, we should point out that the function spaces used in \cite{KT,X,LZ} have a common trait in the structure; i.e., these spaces can be seen as the Q-spaces with $L^{2}$ norm, and the advantage of such spaces is that Fourier transform plays an important role in estimating the bilinear term on the corresponding solution spaces. Nevertheless, for the global existence and uniqueness of a mild solution to \eqref{eqn:ns} with a small initial data in $\dot{B}^{\gamma_1,\gamma_2}_{p,q}$, we have to seek a new approach. Generally speaking, a mild solution of \eqref{eqn:ns} is obtained by using the following method. Assume that the initial data belongs to $\dot{B}^{\gamma_1,\gamma_2}_{p,q}$. Via the iteration process: \begin{gather*} u^{(0)}(t,x)=e^{-t(-\Delta)^{\beta}} a(x);\\ u^{(j+1)}(t,x)=u^{(0)}(t,x)-B(u^{(j)},u^{(j)})(t,x)\quad\text{for}\quad j=0,1,2,\dots , \end{gather*} we construct a contraction mapping on a space in $\mathbb{R}^{1+n}_{+}$, denoted by $X(\mathbb{R}^{1+n}_{+})$. With the initial data being small, the fixed point theorem implies that there exists a unique mild solution of \eqref{eqn:ns} in $X(\mathbb{R}^{1+n}_{+})$. In this paper, we choose $X(\mathbb{R}^{1+n}_{+})=\mathbb{B}^{\gamma_1,\gamma_2}_{p, q, m, m'}$ associated with $\dot{B}^{\gamma_1,\gamma_2}_{p, q}$. Owing to Theorem \ref{th1}, we know that if $f\in \dot{B}^{\gamma_1,\gamma_2}_{p,q}$ then $e^{-t(-\Delta)^{\beta}}f(x)\in X(\mathbb{R}^{1+n}_{+})$. Hence the construction of contraction mapping comes down to prove the following assertion: The bilinear operator $$ B(u,v)=\int^{t}_0 e^{-(t-s)(-\Delta)^{\beta}} \mathbb{P}\nabla (u\otimes v) ds $$ is bounded from $(X(\mathbb{R}^{1+n}_{+}))^n \times (X(\mathbb{R}^{1+n}_{+}))^n $ to $(X(\mathbb{R}^{1+n}_{+}))^n $. For this purpose, using multi-resolution analysis, we decompose $B_l(u,v)$ into several parts based on the relation between $t$ and $2^{-2j\beta}$, and expanse every part in terms of $\{\Phi^{\varepsilon}_{j,k}\}$. More importantly, Lemmas \ref{lem53} \& \ref{lem54} enable us to obtain an estimate from $(\mathbb{B}^{\gamma_1,\gamma_2}_{p,q, m,m'})^n \times (\mathbb{B}^{\gamma_1,\gamma_2}_{p,q, m,m'})^n$ to $(\mathbb{B}^{\gamma_1,\gamma_2}_{p,q, m,m'})^n$. \begin{remark} \label{rmk1.2} \rm (i) Our initial spaces in Theorem \ref{mthmain} include both $\dot{B}^{1+\frac{n}{p}-2\beta,q}_{p}(\mathbb{R}^{n})$ in Wu \cite{W1,W2,W3,W4}, $Q_{\alpha}^{\beta,-1}(\mathbb{R}^{n})$ in Xiao \cite{X} and Li-Zhai \cite{LZ}. Moreover, in \cite{LZ, LY}, the scope of $\beta$ is $(\frac{1}{2}, 1)$. Our method is valid for $\beta>1/2$. (ii) We point out that $\dot{B}^{\gamma_1, \gamma_2}_{p,q}$ provide a lot of new critical initial spaces where the well-posedness of equations \eqref{eqn:ns} holds. By Lemma \ref{lem:2.9}, for $\beta=1$, Theorem \ref{mthmain} holds for the initial spaces $\dot{B}^{\gamma_1, \gamma_2}_{p,q}$ satisfying $$ \dot{B}^{-1+w, w}_{2,q'}\subset Q^{-1}_{\alpha}(\mathbb{R}^{n}),\quad q'<2,\; w>0 $$ or $$ Q^{-1}_{\alpha}(\mathbb{R}^{n})\subset \dot{B}^{-1+w, w}_{2,q''} \subset BMO^{-1}(\mathbb{R}^{n}),\quad 20. $$ In some sense, $\dot{B}^{\gamma_1, \gamma_2}_{p,q}$ fill the gap between the spaces $Q^{-1}_{\alpha}(\mathbb{R}^{n})$ and $BMO^{-1}(\mathbb{R}^{n})$. See also Lemma \ref{lem:2.9}, Corollary \ref{co:1} and Remark \ref{remark1}. (iii) For a initial data $ a\in\dot{B}^{\gamma_1, \gamma_2}_{p,q}$, the index $\gamma_1$ represents the regularity of $a$. In Theorem \ref{mthmain}, taking $\dot{B}^{\gamma_1, \gamma_2}_{p,q}=\dot{B}^{-1+w, w}_{p,q}$ with $w>0$, $p>2$ and $q>2$ yields $\dot{B}^{-1+w, w}_{p,q}\subset BMO^{-1}(\mathbb{R}^{n})$. Compared with the ones in $BMO^{-1}(\mathbb{R}^{n})$, the elements of $\dot{B}^{-1+w, w}_{p,q}$ have higher regularity. Furthermore, Theorem \ref{mthmain} implies that the regularity of our solutions becomes higher along with the growth of $\gamma_1$. (iv) Interestingly, Federbush \cite{Feder} employed the divergence-free wavelets to study the classical Navier-Stokes equations, while the wavelets used in this paper are classical Meyer wavelets. In addition, when constructing a contraction mapping, Federbush's method was based on the estimates of ``long wavelength residues". Nevertheless, our wavelet approach based on Lemmas \ref{le6}-\ref{le7} and the Cauchy-Schwarz inequality is to convert the bilinear estimate of $B(u,v)$ into various efficient computations involved in the wavelet coefficients of $u$ and $v$. \end{remark} The remaining of this article is organized as follows. In Section \ref{sec2}, we list some preliminary knowledge on wavelets and give the wavelet characterization of the Besov-Q spaces. In Sections \ref{sec3}-\ref{sec4} we define the initial data spaces and the corresponding solution spaces. Section \ref{sec5} carries out a necessary analysis of some non-linear terms and a prior estimates. In Section \ref{sec6}, we verify Theorem \ref{mthmain} via Lemmas \ref{le6}-\ref{le7} which will be demonstrated in Sections \ref{sec7}-\ref{sec8} respectively. \subsection*{Notation} ${\mathsf U}\approx{\mathsf V}$ indicates that there is a constant $c>0$ such that $c^{-1}{\mathsf V}\le{\mathsf U}\le c{\mathsf V}$ whose right inequality is also written as ${\mathsf U}\lesssim{\mathsf V}$. Similarly, one writes ${\mathsf V}\gtrsim{\mathsf U}$ for ${\mathsf V}\ge c{\mathsf U}$. \section{Preliminaries}\label{sec2} First of all, we would like to say that we will always utilize tensorial orthogonal wavelets which may be regular Daubechies wavelets (only used for characterizing Besov and Besov-Q spaces) and classical Meyer wavelets, but also to recall that the regular Daubechies wavelets are such Daubechines wavelets that are smooth enough and have more sufficient vanishing moments than the relative spaces do; see Lemma \ref{le9} and the part before Lemma \ref{lem:c}. Next, we present some preliminaries on Meyer wavelets $\Phi^{\epsilon}(x)$ in detail and refer the reader to \cite{Me}, \cite{Woj} and \cite{Yang1} for further information. Let $\Psi^{0}$ be an even function in $ C^{\infty}_0 ([-\frac{4\pi}{3}, \frac{4\pi}{3}])$ with \begin{gather*} 0\leq\Psi^{0}(\xi)\leq 1; \\ \Psi^{0}(\xi)=1\quad \text{for }|\xi|\leq \frac{2\pi}{3}. \end{gather*} If $$ \Omega(\xi)= \sqrt{(\Psi^{0}(\frac{\xi}{2}))^{2}-(\Psi^{0}(\xi))^{2}}, $$ then $\Omega$ is an even function in $ C^{\infty}_0([-\frac{8\pi}{3}, \frac{8\pi}{3}])$. Clearly, \begin{gather*} \Omega(\xi)=0\quad \text{for }|\xi|\leq \frac{2\pi}{3};\\ \Omega^{2}(\xi)+\Omega^{2}(2\xi)=1=\Omega^{2}(\xi)+\Omega^{2}(2\pi-\xi)\quad \text{for }\xi\in [\frac{2\pi}{3},\frac{4\pi}{3}]. \end{gather*} Let $\Psi^{1}(\xi)= \Omega(\xi) e^{-\frac{i\xi}{2}}$. For any $\epsilon= (\epsilon_1,\dots, \epsilon_{n}) \in \{0,1\}^{n}$, define $\Phi^{\epsilon}(x)$ via the Fourier transform $\hat{\Phi}^{\epsilon}(\xi)= \prod^{n}_{i=1} \Psi^{\epsilon_{i}}(\xi_{i})$. For $j\in \mathbb{Z}$ and $k\in\mathbb{Z}^{n}$, set $\Phi^{\epsilon}_{j,k}(x)= 2^{\frac{nj}{2}} \Phi^{\epsilon} (2^{j}x-k)$. Furthermore, we put \begin{gather*} E_{n}=\{0,1\}^{n}\backslash\{0\}; \\ F_{n}=\{(\epsilon,k):\epsilon\in E_{n}, k\in\mathbb{Z}^{n}\};\\ \Lambda_{n}=\{(\epsilon,j,k), \epsilon\in E_{n}, j\in\mathbb{Z}, k\in \mathbb{Z}^{n}\}, \end{gather*} and for any $\epsilon\in \{0,1\}^{n}, k\in \mathbb{Z}^{n}$ and a function $f$ on $\mathbb R^n$, we write $f^{\epsilon}_{j,k}= \langle f, \Phi^{\epsilon}_{j,k}\rangle .$ The following result is well-known. \begin{lemma}\label{le1} The Meyer wavelets $\{\Phi^{\epsilon}_{j,k}\}_{(\epsilon,j,k)\in \Lambda_{n}}$ form an orthogonal basis in the space $L^{2}(\mathbb{R}^{n})$. Consequently, for any $f\in L^{2}(\mathbb{R}^{n})$, the following wavelet decomposition holds in the $L^2$ convergence sense: $$ f=\sum_{(\epsilon,j,k)\in\Lambda_{n}}f^{\epsilon}_{j,k}\Phi^{\epsilon}_{j,k}. $$ \end{lemma} Moreover, for $j\in\mathbb{Z}$, let $$ P_jf= \sum_{k\in \mathbb{Z}^{n}} f^{0}_{j,k}\Phi^{0}_{j,k}, \quad Q_jf= \sum_{(\epsilon,k)\in F_{n}} f^{\epsilon}_{j,k} \Phi^{\epsilon}_{j,k}. $$ For the above Meyer wavelets, by Lemma \ref{le1}, the product of any two functions $u$ and $v$ can be decomposed as \begin{equation}\label{eq:decompose} \begin{aligned} uv&= \sum_{j\in \mathbb{Z}} P_{j-3}u Q_jv + \sum_{j\in \mathbb{Z}} Q_ju Q_jv + \sum_{0\frac{1}{2}$ and $\gamma_1-\gamma_2 = 1-2\beta$, each $\dot{B}^{\gamma_1,\gamma_2}_{p,q}$ is a critical space; i.e., $$ \|\lambda^{\gamma_2-\gamma_1} f(\lambda \cdot)\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}} \approx \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}\quad\forall \lambda>0. $$ To better understand why the Besov-Q spaces are larger than many spaces cited in the introduction, we should observe the basic fact below. \begin{lemma}\label{lem:2.9} Given $10, 1\leq q_1\leq \infty$, one has $\dot{B}^{\gamma_1, \gamma_2+w}_{p,q} \subset \dot{B}^{\gamma_1-w,\gamma_2} _{\frac{p}{p_1}, \frac{q}{q_1}}$. \end{itemize} \end{lemma} For $0\leq \alpha-\beta+1$ and $\alpha+\beta-1\leq n/2$, we say that $f$ belongs to the Q-type space $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ provided $$ \sup_{Q} r^{2(\alpha+\beta-1)-n} \int_{Q}\int_{Q} \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2(\alpha-\beta+1)}} dxdy<\infty, $$ where the supremum is taken over all cubes with sidelength $r$. This definition was used in \cite{LZ} to extend the results in \cite{X} which initiated a PDE-analysis of the original Q-spaces introduced in \cite{EJPX} (cf. \cite{DX, DX1,PY,WX,Yang1} for more information). The following is a direct consequence of Lemmas \ref{lem:c} and \ref{lem:2.9}. \begin{corollary}\label{co:1} \begin{itemize} \item[(i)] If $0\leq \alpha-\beta+1< 1, \alpha+\beta-1\leq \frac{n}{2}$, then $Q_{\alpha}^{\beta}(\mathbb{R}^{n})= \dot{B}^{\alpha-\beta+1, \alpha+\beta-1}_{2,2}$. \item[(ii)] If $p=\frac{n}{\gamma_2}$, then $\dot{B}^{\gamma_1, \gamma_2}_{p,q}=\dot{B}^{\gamma_1,q}_{p}(\mathbb{R}^{n})$. \item[(iii)] Given $w=0,v=1$ or $w>0$, $1\leq v\leq \infty$. If $p=n/(\gamma_2+w)$, then $$ \dot{B}^{\gamma_1, q}_{p}(\mathbb{R}^{n}) \subset \dot{B}^{\gamma_1-w,\gamma_2} _{\frac{n}{u(w+\gamma_2)}, \frac{q}{v}}. $$ \end{itemize} \end{corollary} \begin{remark}\label{remark1} \rm Wu \cite{W2} obtained the well-posedness of \eqref{eqn:ns} with an initial data in the critical Besov space $\dot{B}^{1+\frac{n}{p}-2\beta,q}_{p}(\mathbb{R}^{n})$. Given $10$, $$ \dot{B}^{1+\frac{n}{p}-2\beta,q}_{p}(\mathbb{R}^{n})\subset \dot{B}^{1+\frac{n}{p}-2\beta,\frac{n}{p}}_{p_0,q_0}. $$ \end{remark} \section{Besov-Q spaces via semigroups}\label{sec4} To establish a semigroup characterization of the Besov-Q spaces, recall the following semigroup characterization of $Q^{\beta}_{\alpha}(\mathbb{R}^{n})$, see \cite{LZ}: Given $\max\{\alpha,1/2\}<\beta<1$ and $\alpha+\beta-1\geq0$. $f\in Q^{\beta}_{\alpha}(\mathbb{R}^{n})$ if and only if $$ \sup_{x\in\mathbb R^n,\, r\in (0,\infty)} r^{2\alpha-n+2\beta-2}\int^{r^{2\beta}}_0 \int_{|y-x|0$, let $\hat{K}^{\beta}_{t} (\xi) = e^{-t|\xi|^{2\beta}}$. We have \begin{equation*} f(t,x)= e^{-t(-\Delta)^{\beta}} f(x) = K_{t}^{\beta}*f(x). \end{equation*} For the Meyer wavelets $\{\Phi^{\epsilon}_{j,k}\}_{(\epsilon,j,k)\in\Lambda_{n}}$, let $a^{\epsilon}_{j,k}(t) = \langle f(t,\cdot), \Phi^{\epsilon}_{j,k}\rangle$ and $a^{\epsilon}_{j,k} = \langle f, \Phi^{\epsilon}_{j,k}\rangle$. By Lemma \ref{le1} we obtain $$ f(x)= \sum_{\epsilon,j,k} a^{\epsilon}_{j,k} \Phi^{\epsilon}_{j,k}(x)\quad \text{and}\quad f(t,x)= \sum_{\epsilon, j, k} a^{\epsilon}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ If $f(t,x)=K^{\beta}_{t}*f(x)$, then \begin{equation}\label{eq3} \begin{aligned} a^{\epsilon}_{j,k}(t) &= \sum_{\epsilon',|j-j'|\leq 1, k'} a^{\epsilon'}_{j',k'} \langle K^{\beta}_{t} \Phi^{\epsilon'}_{j',k'}, \Phi^{\epsilon}_{j,k}\rangle\\ &= \sum_{\epsilon',|j-j'|\leq 1, k'} a^{\epsilon'}_{j',k'} \int e^{-t 2^{2j\beta} |\xi|^{2\beta}} \hat{\Phi}^{\epsilon'} (2^{j-j'}\xi) \hat{\Phi}^{\epsilon} (\xi) e^{-i(2^{j-j'}k'-k)\xi}\, d\xi. \end{aligned} \end{equation} \begin{lemma}\label{le4} Let $\{\Phi^{\epsilon}_{j,k}\}_{(\epsilon,j,k)\in\Lambda_{n}}$ be Meyer wavelets. For $\beta>0$ there exist a large constant $N_{\beta}>0$ and a small constant $\tilde c>0$ such that if $N>N_{\beta}$ then \begin{equation}\label{eq3.5} |a^{\epsilon}_{j,k}(t)|\lesssim e^{-\tilde c t 2^{2j\beta}} \sum_{\epsilon',|j-j'|\leq 1, k'} |a^{\epsilon'}_{j',k'}| (1+|2^{j-j'}k'-k|)^{-N}\quad\forall\, t 2^{2\beta j} \geq 1 \end{equation} and \begin{equation}\label{eq3.6} |a^{\epsilon}_{j,k}(t)|\lesssim \sum_{|j-j'|\leq 1} \sum_{\epsilon',k'} |a^{\epsilon'}_{j',k'}| (1+|2^{j-j'}k'-k|)^{-N}\quad\forall\, 0< t 2^{2\beta j} \leq 1. \end{equation} \end{lemma} \begin{proof} Formally, we can write \begin{align*} a^{\epsilon}_{j,k}(t) &=\sum_{\epsilon',|j-j'|\leq 1, k'}a^{\epsilon'}_{j',k'}\langle K^{\beta}_{t}\ast\Phi^{\epsilon'}_{j',k'}, \Phi^{\epsilon}_{j,k}\rangle\\ &=\sum_{\epsilon',|j-j'|\leq 1, k'}a^{\epsilon'}_{j',k'}\langle e^{-t(-\Delta)^{\beta}}\Phi^{\epsilon'}_{j',k'}, \Phi^{\epsilon}_{j,k}\rangle\\ &=\sum_{\epsilon',|j-j'|\leq 1, k'}a^{\epsilon'}_{j',k'}\int e^{-t2^{2j\beta}|\xi|^{2\beta}}\widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi)\widehat{\Phi^{\epsilon}}(\xi)e^{-i(2^{j-j'}k'-k)\xi}d\xi. \end{align*} We divide the rest of the argument into two cases. \smallskip \noindent\textbf{Case 1:} $|2^{j-j'}k'-k|\le 2$. Notice that $\widehat{\Phi^{\epsilon}}$ is supported on a ring. By a direct computation, we obtain $$ |a^{\epsilon}_{j,k}(t)| \lesssim \sum_{\epsilon',|j-j'|\leq 1,k'}|a^{\epsilon'}_{j',k'}|e^{-t2^{2j\beta}} \lesssim \sum_{\epsilon',|j-j'|\leq 1, k'}\frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k'-k|)^{N}}e^{-t2^{2j\beta}}. $$ \noindent\textbf{Case 2:} $|2^{j-j'}k'-k|\geq 2$. Denote by $l_{i_0}$ the largest component of $2^{j-j'}k'-k$. Then $(1+|l_{i_0}|)^{N}\sim (1+|2^{j-j'}k'-k|)^{N}$. We have \begin{align*} &a^{\epsilon}_{j,k}(t)\\ &=\sum_{\epsilon',|j-j'|\leq 1, k'}\frac{a^{\epsilon'}_{j',k'}}{(l_{i_0})^{N}} \int e^{-t2^{2j\beta}|\xi|^{2\beta}}\widehat{\Phi^{\epsilon'}} (2^{j-j'}\xi)\widehat{\Phi^{\epsilon}}(\xi) [(\frac{-1}{i}\partial_{\xi_{i_0}})^{N}e^{-i(2^{j-j'}k'-k)\xi}]d\xi. \end{align*} By an integration-by-parts, we can obtain that if $C^l_N$ is the binomial coefficient indexed by $N$ and $l$ then \begin{align*} |a^{\epsilon}_{j,k}(t)| &=\Big|\sum_{\epsilon',|j-j'|\leq 1, k'}(-1)^{N}\frac{a^{\epsilon'}_{j',k'}}{(l_{i_0})^{N}} \int \sum^{N}_{l=0}C^{l}_{N}\partial_{\xi_{i_0}}^{l} (e^{-t2^{2j\beta}|\xi|^{2\beta}})\\ &\quad\times \partial_{\xi_{i_0}}^{N-l}(\widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi) \widehat{\Phi^{\epsilon}}(\xi))e^{-i(2^{j-j'}k'-k)\xi}d\xi\Big|\\ &\lesssim \sum_{\epsilon',|j-j'|\leq 1, k'}\frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k'-k|)^{N}}\Big| \int \sum^{N}_{l=0}C^{l}_{N}(-t2^{2j\beta}|\xi|^{2\beta})^{l}|\xi|^{2\beta-2} \xi_{i_0}\\ &\quad \times e^{-t2^{2j\beta}|\xi|^{2\beta}} \partial_{\xi_{i_0}}^{N-l}(\widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi) \widehat{\Phi^{\epsilon}}(\xi))e^{-i(2^{j-j'}k'-k)\xi}d\xi\Big|. \end{align*} If $t2^{2j\beta}\ge1$, there exists a constant $c$ such that $(t2^{2j\beta})^{l}e^{-t2^{2j\beta}}\lesssim e^{-ct2^{2j\beta}}$. Since $\Phi^{\epsilon'}$ is defined on a ring, we obtain \begin{align*} |a^{\epsilon}_{j,k}(t)| &\lesssim \sum_{\epsilon',|j-j'|\leq 1, k'}\frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k' -k|)^{N}}(t2^{2j\beta})^{l}e^{-t2^{2j\beta}}\\ &\lesssim \sum_{\epsilon',|j-j'|\leq 1,k'}e^{-ct2^{2j\beta}} \frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k'-k|)^{N}}. \end{align*} If $00$, $10$ and $\gamma\in\mathbb{R}$. We say that \begin{gather*} a(\cdot,\cdot)\in \mathbb{B}^{\gamma}_{\tau,\infty}\quad \text{if } \sup_{t2^{2j\beta}\geq 1} (t2^{2j\beta})^{\tau} 2^{\frac{nj}{2}} 2^{j\gamma}|a^{\epsilon}_{j,k}(t)| +\sup_{0p$ and $m',\tau>0$. \begin{itemize} \item[(i)] If $m>0$, then $\mathbb{B}^{\gamma_1,\gamma_2}_{p, q, m,m'} \subset\mathbb{B}^{\gamma_1-\gamma_2}_{\frac{m}{p},\infty}$. \item[(ii)] If $-2\beta\tau<\gamma<0<\beta$, then $\mathbb{B}^{\gamma}_{\tau,\infty}\subset \mathbb{B}^{\gamma}_{0,\infty}$. \end{itemize} \end{lemma} For convenience, for any dyadic cube $Q_{j_0, k_0}$, we always use $\widetilde{Q}_{j_0,k_0}$ to denote the dyadic cube containing $Q_{j_0, k_0}$ with side length $2^{8-j_0}$. Given $(\epsilon,j,k)\in \Lambda_{n}$. If $\epsilon\in E_{n}$ and $Q_{j,k}\subset Q_{j_0, k_0}$, we write $(\epsilon,k)\in S^{j}_{j_0,k_0}$. For any $w\in \mathbb{Z}^{n}$, denote $\widetilde{Q}^w_{j_0,k_0}= 2^{8-j_0}w + \widetilde{Q}_{j_0, k_0}$. Denote $(\epsilon,k)\in S^{w,j}_{j_0,k_0}$ whenever $Q_{j,k}\subset \widetilde{Q}^w_{j_0,k_0}$. Furthermore, we frequently utilize the so-called $\alpha$-triangle inequality: $$ (a+b)^{\alpha}\leq a^{\alpha}+b^{\alpha}\quad\forall (\alpha,a,b)\in (0,1]\times(0,\infty)\times(0,\infty). $$ Now we characterize the Besov-Q spaces by using a semigroup operator. \begin{theorem}\label{th1} Given $10$, $11, m>0$, by \eqref{eq3.5}, there exists a constant $N$ large enough such that $$ |a^{\epsilon}_{j,k}(t)| \lesssim e^{-\tilde c t 2^{2j\beta}} \sum_{\epsilon',|j-j'|\leq 1, k'}\frac{ |a^{\epsilon'}_{j',k'}|}{ (1+|2^{j-j'}k'-k|)^{N}} \lesssim 2^{\frac{-nj}{2}} 2^{j(\gamma_2-\gamma_1)}e^{-\tilde c t 2^{2j\beta}}. $$ Choosing a sufficiently large $N'$ (depending on $N$) in the last estimate we have \begin{align*} I^{\gamma_1,\gamma_2}_{p,q, Q_{r},m}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq\max\{-\log_2r, -\frac{\log_2t}{2\beta}\}} 2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big[\sum_{(\epsilon,k)\in S^{j}_{r}}e^{-\tilde{c}t2^{2j\beta}}\Big(\sum_{\epsilon',|j-j'|\leq1, k'}\frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k'-k|)^{N}}\Big)^p(t2^{2j\beta})^{m}\Big]^{q/p}, \end{align*} where $p>1$ has been used. In the sequel, denote by $S^{w,j'}_r$ the set of all indexes $(\epsilon',k')$ ensuring $Q_{j',k'}\subset Q_r^w$, and divide the proof into two cases. \textit{Case 1.1:} $q\le p$. Because $|j-j'|\leq1$ and $j>-\log_2r$, one gets $2^{-(j'+1)n}\leq r^{n}$. This implies that $(2^{nj'}|Q|)^{-N'}\lesssim1$. Hence \begin{align*} &I^{\gamma_1,\gamma_2}_{p,q,Q_{r}, m}(t)\\ &\lesssim \sum_{w\in\mathbb{Z}^{n}}\frac{|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}} {(1+|w|)^{{Nq}/{p}}}\sum_{j'\geq\max\{-\log_2r,-\frac{\log_2t}{2\beta}\}} 2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})} [\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|a^{\epsilon'}_{j',k'}|^p]^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \textit{Case 1.2:} $q>p$. Applying H\"{o}lder's inequality to $w\in\mathbb{Z}^{n}$, we similarly have \begin{align*} &I^{\gamma_1,\gamma_2}_{p,q,Q_{r},m}(t)\\ &\lesssim \sum_{w\in\mathbb{Z}^{n}}\Big\{|Q_{r}|^{\frac{q\gamma_2}{p}-\frac{q}{p}} \sum_{j'\geq\max\{-\log_2r,-\frac{\log_2t}{2\beta}\}}2^{qj' (\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|a^{\epsilon'}_{j',k'}|^p\Big]^{q/p}\Big\}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \noindent\textbf{Situation 2:} $K^{\beta}_{t}\ast f\in \mathbb{B}^{\gamma_1,\gamma_2, II}_{p,q}$. For $t2^{2\beta j}\leq1$ and $m'>0$, by \eqref{eq3.6}, there exists a natural number $N$ large enough such that $N>2n$ and $$ |a^{\epsilon}_{j,k}(t)|\lesssim \sum_{\epsilon', |j-j'|\leq1, k'}|a^{\epsilon'}_{j',k'}|(1+|2^{j-j'}k'-k|)^{-N}. $$ Consequently, \begin{align*} II^{\gamma_1,\gamma_2}_{p,q,Q_{r}}(t) &\lesssim |Q|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j\leq-\frac{\log_2t}{2\beta}}2^{jq(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big[\sum_{(\epsilon,k)\in S^{j}_{r}}\Big(\sum_{\epsilon',|j'-j|\leq1, k'}\frac{|a^{\epsilon'}_{j',k'}|}{(1+|2^{j-j'}k'-k|)^{N}}\Big)^p\Big]^{q/p}. \end{align*} \textit{Case 2.1:} $q\le p$. Notice that $(2^{nj'}|Q|)^{-N}\lesssim1$. We have \begin{align*} &II^{\gamma_1,\gamma_2}_{p,q,Q_{r}}(t)\\ &\lesssim \sum_{|w|\in\mathbb{Z}^{n}} \frac{|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}}{(1+|w|)^{qN/p}} \sum_{-\log_2r-1\leq j'\leq-\frac{\log_2t}{2\beta}-1}2^{j'q(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big(\sum_{(\epsilon', k')\in S^{w,j'}_{r}}|a^{\epsilon'}_{j',k'}|^p\Big)^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \textit{Case 2.2:} $q>p$. By H\"{o}lder's inequality, in a similar manner, we can obtain \begin{align*} II^{\gamma_1,\gamma_2}_{p,q,Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r-1\leq j'\leq-\frac{\log_2t}{2\beta}-1}2^{j'q(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big[\sum_{(\epsilon',k')\in S^{w,j'}_r} |a^{\epsilon'}_{j',k'}|^p(1+|2^{j-j'}k'-k|)^{-N'}\Big]^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \noindent\textbf{Situation 3:} $K^{\beta}_{t}\ast f\in \mathbb{B}^{\gamma_1,\gamma_2,III}_{p,q,m}$. For this case we have $2^{-2j\beta} 2^{n}}](1+|w|)^{-N} \sum_{(\epsilon',k')\in S^{w,j'}_{r}}e^{-ct2^{2j'\beta}}|a^{\epsilon'}_{j',k'}|^p\frac{dt}{t}\Big\}^{q/p}\\ &=: M_1+M_2. \end{align*} By the definition of $\dot{B}^{\gamma_1,\gamma_2}_{p,q}$, it is easy to see that $M_1\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}$. For the term $M_2$, we divide the estimate into two cases. \textit{Case 3.1:} $q\le p$. For this case, $j'\geq-\log_2r-3$ implies $(2^{nj'}r^{n})^{N'}\lesssim 1$ and \begin{align*} M_2&\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}} \sum_{j'\geq-\log_2r-3}2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})} \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-\frac{qN'}{p}}\\ &\quad\times \Big[\int^{r^{2\beta}}_{2^{-2j\beta}}(t2^{2j'\beta})^{m}\sum_{(\epsilon',k')\in S^{w,j'}_{r}}e^{-cpt2^{2j'\beta}}|a^{\epsilon}_{j',k'}|^p(2^{nj'}|Q|)^{-N'}\frac{dt}{t}\Big]^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \textit{Case 3.2:} $q>p$. For this case, by H\"older's inequality and $j\sim j'$ we obtain \begin{align*} &\Big[\int^{r^{2\beta}}_{2^{-2j\beta}}(t2^{2j\beta})^{m}\sum_{|w|>2^{n}} \sum_{(\epsilon',k')\in S^{w,j'}_{r}}e^{-cpt2^{2j\beta}}(1+|w|)^{-N} |a^{\epsilon'}_{j',k'}|^p(2^{nj'}|Q|)^{-N'}\frac{dt}{t}\Big]^{q/p}\\ &\lesssim\sum_{|w|>2^{n}}(1+|w|)^{-\frac{qN'}{p}} \Big(\int^{r^{2\beta}}_{2^{-2j\beta}}(t2^{2j'\beta})^{m} \sum_{(\epsilon',k')\in S^{w,j'}_{r}}e^{-cpt2^{2j'\beta}}|a^{\epsilon'}_{j',k'}|^p\frac{dt}{t}\Big)^{q/p}. \end{align*} The rest of the argument is similar to that of Case 3.1, and so omitted. \smallskip \noindent\textbf{Situation 4:} $K^{\beta}_{t}\ast f\in \mathbb{B}^{\gamma_1,\gamma_2, IV}_{p,q,m}$. Because $|j-j'|\leq 1$ and $0 2^{n}}\frac{1}{(1+|w|)^{N}}\sum_{(\epsilon',k')\in S^{w,j'}_{r}}\frac{|a^{\epsilon'}_{j',k'}|^p}{(2^{nj'}|Q|)^{N'}}\Big]^{q/p}. \end{align*} \textit{Case 4.1:} $q\le p$. For this case, by the $\alpha$-triangle inequality we have \begin{align*} IV^{\gamma_1,\gamma_2}_{p,q,Q_{r},m'} &\lesssim \sum_{w\in\mathbb{Z}^{n}}\frac{|Q|^{\frac{q\gamma_2}{n} -\frac{q}{p}}}{(1+|w|)^{\frac{qN}{p}}} \sum_{j'\geq-\log_2r-1} 2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big(\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|a^{\epsilon'}_{j',k'}|^p\Big)^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} \textit{Case 4.2:} $q>p$. Using H\"older's inequality we have \begin{align*} IV^{\gamma_1,\gamma_2}_{p,q,Q_{r},m'} &\lesssim \sum_{w\in\mathbb{Z}^{n}} \frac{|Q|^{\frac{q\gamma_2}{n}-\frac{q}{p}}}{(1+|w|)^{N}} \sum_{j'\geq-\log_2r-1} 2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big(\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|a^{\epsilon'}_{j',k'}|^p\Big)^{q/p}\\ &\lesssim \|f\|_{\dot{B}^{\gamma_1,\gamma_2}_{p,q}}. \end{align*} This completes the proof. \end{proof} We close this section by showing the following continuity of the Riesz transforms acting on the Besov-Q spaces; see also \cite{Al} and \cite{Yang1} for some related results. \begin{theorem}\label{th4} For $1p$, and $m'>0$, the Riesz transforms $R_1,R_2,\dots ,R_n$ are continuous on $\mathbb{B}^{\gamma_1,\gamma_2}_{p, q, m,m'}$. \end{theorem} \begin{proof} For convenience, we choose the classical Meyer wavelet basis $\{\Phi^{\epsilon}_{j,k}\}_{(\epsilon,j,k)\in\Lambda_{n}}$. For any $g(\cdot,\cdot)\in\mathbb{B}^{\gamma_1,\gamma_2}_{p,q,m,m'}$ and $l=1,2,\dots ,n$, we need to prove $(R_lg)(\cdot,\cdot)\in \mathbb{B}^{\gamma_1,\gamma_2}_{p,q,m,m'}$. Write $g(t,x)=\sum_{(\epsilon,j,k)\in\Lambda_{n}}g^{\epsilon}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)$. Then $$ R_lg(t,x)=\sum_{(\epsilon,j,k)\in\Lambda_{n}}g^{\epsilon}_{j,k}(t) R_l\Phi^{\epsilon}_{j,k}(x) =:\sum_{(\epsilon,j,k)\in\Lambda_{n}}b^{\epsilon}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x), $$ where $b^{\epsilon}_{j,k}(t)$ is defined by $$ b^{\epsilon}_{j,k}(t) =\sum_{(\epsilon',j',k')\in\Lambda_{n}}g^{\epsilon'}_{j',k'}(t)\big\langle R_l\Phi^{\epsilon'}_{j',k'}, \Phi^{\epsilon}_{j,k}\big\rangle=: \sum_{|j-j'|\leq1}\sum_{\epsilon',k'}a^{\epsilon,\epsilon'}_{j,k,j',k'}g^{\epsilon'}_{j',k'}(t). $$ Because $R_l$ is a Calder\'on-Zygmund operator, by \eqref{eq7} we obtain $$ |a^{\epsilon,\epsilon'}_{j,k,j',k'}|\lesssim2^{-|j-j'|(\frac{n}{2}+N_0)} \Big(\frac{2^{-j}+2^{-j'}}{2^{-j}+2^{-j'}+|2^{-j}k-2^{-j'}k'|}\Big)^{n+N_0}. $$ The rest of the proof is similar to Theorem \ref{th1}. We omit the details. \end{proof} \section{Nonlinear terms and their a priori estimates}\label{sec5} \subsection{Decompositions of non-linear terms} For the rest of this article, let $$ u(t,x)=\sum_{(\epsilon,j,k)\in \Lambda_n} u^{\epsilon}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x);\quad v(t,x)=\sum_{(\epsilon,j,k)\in \Lambda_n} v^{\epsilon}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ For $l=1,\dots, n$, we will derive some inequalities about $$ B_l(u,v)(t,x)=\int^{t}_0 e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l}(uv) ds. $$ Here, it is worth pointing out that \eqref{eq:decompose} gives $$ e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l}(uv)(s,t,x)=\sum_{j'\in\mathbb{Z}}\sum_{i=1}^{4}I^{i,l}_{j'}(s,t,x), $$ where \begin{gather*} \begin{aligned} &I^{1,l}_{j'}(u,v)(s,t,x)\\ &= \sum_{\epsilon',k'}\sum_{k''} u^{\epsilon'}_{j',k'}(s) v^{0}_{j'-3,k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{0}_{j'-3,k''}(x)), \end{aligned}\\ \begin{aligned} &I^{2,l}_{j'}(u,v)(s,t,x)\\ &=\sum_{\epsilon',k'}\sum_{\epsilon'',k''} u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j',k''}(x)), \end{aligned}\\ \begin{aligned} &I^{3,l}_{j'}(u,v)(s,t,x)\\ &= \sum_{0<|j'- j''|\leq 3}\sum_{\epsilon',k'} \sum_{\epsilon'',k''} u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j'',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j'',k''}(x)), \end{aligned}\\ \begin{aligned} &I^{4,l}_{j'}(u,v)(s,t,x)\\ &= \sum_{\epsilon',k'} \sum_{k''} v^{\epsilon'}_{j',k'}(s) u^{0}_{j'-3,k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{0}_{j'-3,k''}(x)). \end{aligned} \end{gather*} Hence $$ B_l(u,v)(t,x) =:\int^{t}_0\sum_{j'\in\mathbb{Z}}\sum_{i=1}^{4}I^{i,l}_{j'}(s,t,x)ds =: \sum_{i=1}^{4}\int^{t}_0I^{i}_l(s,t,x)ds. $$ Therefore, we can write \begin{equation}\label{eq:de} B_l(u,v)(t,x) = \sum^{4}_{i=1} I^{i}_l(u,v)(t,x), \end{equation} where $$ I^{i}_l(u,v)(t,x)=\int^{t}_0I^{i}_l(s,t,x)ds. $$ To estimate the bilinear term $B(u,v)$ in some suitable function spaces on $\mathbb R^{1+n}_+$, we are required to decompose the terms $I^{i}_l(u,v)(t,x)$, $i=1,2,\dots, 4$, respectively. \subsection*{Decomposition of $I^{1}_l(u,v)(t,x)$} The term $I^{1}_l(u,v)(t,x)$ is decomposed according to two cases. Case $[I^{1}_l]_1$: $t\geq 2^{-2j\beta}$. For this case, we write $I^1_l(u,v)(t,x)$ as the sum of the following three terms: \begin{align*} I^{1}_l(u,v)(t,x) &= \sum_{\epsilon',j',k'} \sum_{k''} \Big(\int^{2^{-1-2j'\beta}}_0+\int^{t/2}_{2^{-1-2j'\beta}} +\int_{t/2}^{t}\Big)\\ &\quad\times \Big\{u^{\epsilon'}_{j',k'}(s)v^{0}_{j'-3,k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{0}_{j'-3,k''}(x))\Big\}ds\\ &=: I^{1,1}_l(u,v)(t,x)+I^{1,2}_l(u,v)(t,x)+I^{1,3}_l(u,v)(t,x). \end{align*} For $i=1,2,3$, denote $$ I^{1,i}_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} a^{\epsilon,i}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ Case $[I^{1}_l]_2$: $t<2^{-2j\beta}$. For this case , we denote $a^{\epsilon,4}_{j,k}(t)=a^{\epsilon}_{j,k}(t)$ and then have $$ I^1_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} a^{\epsilon,4}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ \subsection*{Decomposition of $I^{2}_l(u,v)(t,x)$} The decomposition of $I^{2}_l(u,v)(t,x)$ is made according to two cases. Case $[I^{2}_l]_1$: $t\geq 2^{-2j\beta}$. Naturally, $I^2_l(u,v)(t,x)$ can be divided into the following three terms: \begin{align*} I^{2}_l(u,v)(t,x) &= \sum_{j'} \sum_{\epsilon',k'} \sum_{\epsilon'',k''} \Big(\int^{2^{-1-2j'\beta}}_0+\int^{t/2}_{2^{-1-2j'\beta}} +\int_{t/2}^{t}\Big)\\ &\quad\times \Big\{u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j',k''}(x))\Big\}ds\\ &=: I^{2,1}_l(u,v)(t,x)+I^{2,2}_l(u,v)(t,x)+I^{2,3}_l(u,v)(t,x). \end{align*} Case $[I^{2}_l]_2$: $t< 2^{-2j\beta}$. This $I^{2}_l(u,v)(t,x)$ can be decomposed into the sum of $II^{4}(u,v)(t,x)$ and $II^{5}(u,v)(t,x)$, where \begin{align*} I^{2}_l(u,v)(t,x) &= \sum_{j'} \sum_{\epsilon',k'} \sum_{\epsilon'',k''} \Big(\int^{2^{-2j'\beta}}_0+\int^{t}_{2^{-2j'\beta}}\Big)\\ &\quad\times \Big\{u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j',k''}(x))\Big\}ds\\ &=: I^{2,4}_l(u,v)(t,x)+I^{2,5}_l(u,v)(t,x). \end{align*} For $i=1,2,3,4,5$, set $$ I^{2,i}_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} b^{\epsilon,i}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ \subsection*{Decompositions of $I^{3}_l(u,v)(t,x)$} Similarly, we have the following two cases: Case $[I^{3}_l]_1$: $t\geq 2^{-2j\beta}$. This $I^3_l(u,v)(t,x)$ can be divided into the following three terms: \begin{align*} I^{3}_l(u,v)(t,x) &= \sum_{0<|j'- j''|\leq 3} \sum_{\epsilon',k'}\sum_{\epsilon'',k''}\Big(\int^{2^{-1-2j'\beta}}_0 +\int^{t/2}_{2^{-1-2j'\beta}}+\int_{t/2}^{t}\Big)\\ &\quad\times \Big\{u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j'',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j'',k''}(x))\Big\}ds\\ &=: I^{3,1}_l(u,v)(t,x)+I^{3,2}_l(u,v)(t,x) +I^{3,3}_l(u,v)(t,x). \end{align*} Case $[I^{3}_l]_2$: $t <2^{-2j\beta}$. This $I^{3}_l(u,v)(t,x)$ can be decomposed into the sum of $II^{4}(u,v)(t,x)$ and $II^{5}(u,v)(t,x)$, where \begin{align*} I^{3}_l(u,v)(t,x) &= \sum_{0<|j'- j''|\leq 3} \sum_{\epsilon',k'} \sum_{\epsilon'',k''} \Big(\int^{2^{-2j'\beta}}_0+\int^{t}_{2^{-2j'\beta}}\Big)\\ &\quad\times \Big\{u^{\epsilon'}_{j',k'}(s) v^{\epsilon''}_{j'',k''}(s) e^{-(t-s)(-\Delta)^{\beta}} \frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{\epsilon''}_{j'',k''}(x))\Big\}ds\\ &=: I^{3,4}_l(u,v)(t,x)+I^{3,5}_l(u,v)(t,x). \end{align*} For $i=1,2,3,4,5$, denote $$ I^{3,i}_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} b^{\epsilon,i}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ \subsection*{Decomposition of $I^{4,l}_j(u,v)(t,x)$} It is easy to see that the terms $I^{1, l}_j(u,v)(t,x)$ and $I^{4,l}_j(u,v)(t,x)$ are symmetric associated with $u(t,x)$ and $v(t,x)$. Hence for $I^{4}_l(u,v)$ we have a similar decomposition. Case $[I^{4}_l]_1$: $t\geq 2^{-2j\beta}$. For this case, we write $I^{4}_l(u,v)(t,x)$ as the sum of the following three terms: \begin{align*} I^{4}_l(u,v)(t,x) &=\sum_{\epsilon',j',k'} \sum_{k''} \Big(\int^{2^{-1-2j'\beta}}_0+\int^{t/2}_{2^{-1-2j'\beta}} +\int_{t/2}^{t}\Big)\\ &\quad\times \Big\{ v^{\epsilon'}_{j',k'}(s) u^{0}_{j'-3,k''}(s) e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l} (\Phi^{\epsilon'}_{j',k'}(x)\Phi^{0}_{j'-3,k''}(x))\Big\}ds\\ &=: I^{4,1}_l(u,v)(t,x)+I^{4,2}_l(u,v)(t,x)+I^{4,3}_l(u,v)(t,x). \end{align*} For $i=1,2,3$, denote $$ I^{4,i}_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} a^{\epsilon,i}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ Case $[I^{4}_l]_2$: $t<2^{-2j\beta}$. For this case, we denote $a^{\epsilon,4}_{j,k}(t)=a^{\epsilon}_{j,k}(t)$ and then have $$ I^1_l(u,v)(t,x) = \sum_{(\epsilon,j,k)\in \Lambda_n} a^{\epsilon,4}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x). $$ \subsection{Induced a priori estimates} In the sequel we are about to dominate the above-defined $a^{\epsilon,i}_{j,k}$, $b^{\epsilon,i}_{j,k}$ by $u^{\epsilon'}_{j',k'}$ and $v^{\epsilon''}_{j',k''}$. \begin{lemma}\label{le6} There is a constant $\tilde{c}>0$ such that: \begin{itemize} \item[(i)] For $i=1, 2$, \begin{align*} |a^{\epsilon,i}_{j,k}(t)| &\lesssim 2^{\frac{nj}{2}+j} \sum_{|j-j'|\leq 2} \sum_{\epsilon',k',k''} \int_{I_{i}}\frac{|u^{\epsilon'}_{j',k'}(s)|}{(1+ |2^{j-j'}k'-k|)^{N}} \frac{| v^{0}_{j'-3,k''}(s)|}{(1+|2^{j-j'+3}k''-k'|)^{N}}\\ &\quad\times e^{-\tilde c t2^{2j\beta}}ds, \end{align*} where $I_1=[0, 2^{-1-2j'\beta}]$ and $I_2=[2^{-1-2j'\beta}, \frac{t}{2}]$. \item[(ii)] For $i=3, 4$, \begin{align*} |a^{\epsilon,i}_{j,k}(t)| &\lesssim 2^{\frac{nj}{2}+j} \sum_{|j-j'|\leq 2} \sum_{\epsilon',k',k''} \int_{I_{i}}\frac{|u^{\epsilon'}_{j',k'}(s)|}{(1+ |2^{j-j'}k'-k|)^{N}} \frac{| v^{0}_{j'-3,k''}(s)|}{(1+|2^{j-j'+3}k''-k'|)^{N}}\\ &\quad\times e^{-\tilde c (t-s)2^{2j\beta}}ds, \end{align*} where $I_{3}=[t/2, t]$ and $I_{4}=[0, t]$. \end{itemize} \end{lemma} \begin{proof} \begin{align*} a^{\epsilon,1}_{j,k}(t) &=\big\langle I^{1,1}_l(u,v), \Phi^{\epsilon}_{j,k}\big\rangle\\ &=\sum_{\epsilon',k',k''}~\sum_{|j-j'|\leq2}\int^{2^{-1-2j'\beta}}_0 \Big\{u^{\epsilon'}_{j',k'}(s)v^{0}_{j'-3,k''}(s)\\ &\quad\times \Big\langle e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l}(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''}), \Phi^{\epsilon}_{j,k}\Big\rangle \Big\}ds. \end{align*} The Fourier transform gives \begin{align*} &\Big\langle e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l}(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''}),\ \Phi^{\epsilon}_{j,k}\Big\rangle\\ &=\int e^{-(t-s)|\xi|^{2\beta}}\xi_l\widehat{(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''})}(\xi) 2^{-jn/2}e^{-i2^{-j}k\xi}\widehat{\Phi^{\epsilon}}(2^{-j}\xi)d\xi\\ &=\int e^{-(t-s)|\xi|^{2\beta}}\xi_le^{-i2^{-j'}k'\xi} \Big[\int e^{ik'\eta}\widehat{\Phi^{\epsilon'}}(2^{-j'}\xi-\eta) e^{-8ik''\eta}\widehat{\Phi^{0}}(8\eta)d\eta\Big]\\ &\quad\times 2^{-jn/2}e^{-i2^{-j}k\xi}\widehat{\Phi^{\epsilon}}(2^{-j}\xi)d\xi. \end{align*} Because $00$ such that $$ |\partial_{\xi_{i_0}}^{l}(e^{-t2^{2j\beta}|\xi|^{2\beta}}\xi_l)| \lesssim e^{-ct2^{2j\beta}}. $$ Consequently, we have a constant $\tilde{c}>0$ such that \begin{align*} &\Big|\Big\langle e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l}(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''}),\ \Phi^{\epsilon}_{j,k}\Big\rangle\Big|\\ &\lesssim\frac{2^{jn/2+j}}{(1+|2^{j-j'}k'-k|)^{N}}\Big|\int e^{-i(k-2^{j-j'}k')\xi}\sum^{N}_{l=0}C^{l}_{N}\partial_{\xi_{i_0}}^{l}(e^{-t2^{2j\beta}|\xi|^{2\beta}}\xi_l)\\ &\quad\times\partial_{\xi_{i_0}}^{N-l}\Big(\int \widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi-\eta)\widehat{\Phi^{0}}(8\eta) e^{-i(k-8k'')\eta}d\eta\Big) \widehat{\Phi^{\epsilon}}(\xi)d\xi\Big|\\ &\lesssim e^{-\tilde c t2^{2j\beta}} 2^{jn/2+j} (1+ |2^{j-j'}k'-k|)^{-N} (1+|2^{j-j'+3}k''-k'|)^{-N}. \end{align*} \textit{Case 4:} $|k'-8k''|\geq 2$. In a similar manner to treat Case 3, we denote by $k_{i_0}$ the largest component of $2^{j-j'}k'-k$, and then obtain \begin{align*} &\Big|\Big\langle e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l}(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''}), \Phi^{\epsilon}_{j,k}\Big\rangle\Big|\\ &\lesssim\frac{2^{jn/2+j}}{(1+|2^{j-j'}k'-k|)^{N}}\int \sum^{N}_{l=0}C^{l}_{N}\Big| \partial_{\xi_{i_0}}^{l}(e^{-t2^{2j\beta}|\xi|^{2\beta}}\xi_l)\Big|\\ &\quad\times\Big|\int \partial_{\xi_{i_0}}^{N-l} \Big(\widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi-\eta)\Big)\widehat{\Phi^{0}}(8\eta) e^{-i(k-8k'')\eta}d\eta\Big| |\widehat{\Phi^{\epsilon}}(\xi)|d\xi. \end{align*} As in Case 2, upon choosing $l_{j_0}$ as the largest component of $k'-8k’’$, applying an integration-by-parts, and utilizing the fact that $\widehat{\Phi^{\epsilon}}$ is supported on a ring, we can get a constant $\tilde{c}>0$ such that \begin{align*} &\Big|\Big\langle e^{-(t-s)(-\Delta)^{\beta}}\frac{\partial}{\partial x_l}(\Phi^{\epsilon'}_{j',k'}\Phi^{0}_{j'-3,k''}),\ \Phi^{\epsilon}_{j,k}\Big\rangle\Big|\\ &\lesssim\frac{2^{jn/2+j}}{(1+|2^{j-j'}k'-k|)^{N}(1+|2^{j-j'+3}k''-k'|)^{N}} \int \sum^{N}_{l=0}C^{l}_{N}\Big|\partial_{\xi_{i_0}}^{l}(e^{-t2^{2j \beta}|\xi|^{2\beta}}\xi_l)\Big|\\ &\quad\times\Big|\int \partial_{\xi_{i_0}}^{N-l} \Big(\widehat{\Phi^{\epsilon'}}(2^{j-j'}\xi-\eta)\Big)\widehat{\Phi^{0}}(8\eta) (\frac{1}{i}\partial_{\eta_{i_0}})^{N}\Big(e^{-i(k'-8k'')\eta}\Big)d\eta\Big| |\widehat{\Phi^{\epsilon}}(\xi)|d\xi\\ &\lesssim e^{-\tilde c t2^{2j\beta}} 2^{jn/2+j} (1+ |2^{j-j'}k'-k|)^{-N} (1+|2^{j-j'+3}k''-k'|)^{-N}. \end{align*} This completes the estimate of $a^{\epsilon,1}_{j,k}(t)$ The estimate of $a^{\epsilon,i}_{j,k}(t)$, for $i=1,2,3,4$, can be obtained similarly. \end{proof} Using the same method, we can obtain the following estimates for $b^{\epsilon,i}_{j,k}(t)$, $i=1,2,3,4,5$. \begin{lemma}\label{le7} There is a constant $\tilde{c}>0$ such that: \begin{itemize} \item[(i)] For $i=1, 2$, \begin{align*} &|b^{\epsilon, i}_{j,k}(t)|\\ &\lesssim 2^{\frac{nj}{2}+j} \sum_{j\leq j'+ 2} \sum_{\epsilon',k',\epsilon'',k''} \int_{I_{i}}\frac{|u^{\epsilon'}_{j',k'}(s)|}{(1+ |2^{j-j'}k'-k|)^{N}} \frac{| v^{\epsilon''}_{j',k''}(s)|}{(1+|2^{j-j'}k''-k'|)^{N}} e^{-\tilde c t2^{2j\beta}}ds, \end{align*} where $I_1=[0, 2^{-1-2j'\beta}]$ and $I_2=[2^{-1-2j'\beta}, \frac{t}{2}]$. \item[(ii)] For $i=3, 4, 5$, \begin{align*} |b^{\epsilon,i}_{j,k}(t)| &\lesssim 2^{\frac{nj}{2}+j} \sum_{j\leq j'+ 2} \sum_{\epsilon',k',\epsilon'',k''} \int^{t}_{t/2}\frac{|u^{\epsilon'}_{j',k'}(s)|} {(1+ |2^{j-j'}k'-k|)^{N}}\frac{| v^{\epsilon''}_{j',k''}(s)|} {(1+|2^{j-j'}k''-k'|)^{N}} \\ &\quad\times e^{-\tilde c (t-s)2^{2j\beta}}ds, \end{align*} where $I_{3}=[t/2, t]$, $I_{4}=[0, 2^{-2j'\beta}]$ and $I_{5}=[2^{-2j'\beta}, t]$. \end{itemize} \end{lemma} Let $Q_{j,k}$ and $Q_{j',k'}$ be two dyadic cubes, and for $w\in\mathbb{Z}^{n}$ denote by $Q^{w}_{j,k}$ the dyadic cube $\widetilde{Q}_{j,k}+2^{8-j}w$, where $\widetilde{Q}_{j,k}$ denotes the dyadic cube containing $Q_{j,k}$ with side length $2^{8-j}$. The forthcoming lemmas can be deduced from the Cauchy-Schwartz inequality. \begin{lemma}\label{inequality1} {\rm (i)} For $j, j'\in \mathbb{Z}$ and $w,k,k'\in \mathbb{Z}^{n}$, if $Q_{j',k'}\subset Q_{j,k}^{w}$, then \begin{equation}\label{eqn:est1} (1+ |2^{j-j'}k'-k|)^{-N} \lesssim(1+|w|)^{-N}. \end{equation} {\rm (ii)} Let $0 2^{n}$. If $Q_{j',k'}\subset Q_{j,k}^{w}$ and $Q_{j'',k''}\subset Q_{j,k}^{w'}$, then \begin{equation}\label{eqn:est2} (1+|2^{j'-j''}k''-k'|)^{-N}\lesssim 2^{N(j-j')} (1+|w-w'|)^{-N}. \end{equation} \end{lemma} \begin{lemma}\label{inequality3} Let $Q_{j,k}$ be a dyadic cube with radius $2^{-j}$. For $w\in\mathbb{Z}^{n}$, set $Q^{w}_{j,k}$ be the dyadic cube $2^{8-j}w+\widetilde{Q}_{j,k}$. Then \begin{equation}\label{eqn:est3} \begin{aligned} &\sum_{\epsilon',k'}\sum_{\epsilon'',k''}|u^{\epsilon'}_{j',k'}(s)| |v^{\epsilon''}_{j',k''}(s)|^{p-1} (1+|2^{j-j'}k'-k|)^{-8N}(1+|k'-k''|)^{-8N}\\ &\lesssim \sum_{w\in\mathbb{Z}^{n}} \sum_{w'\in\mathbb{Z}^{n}}(1+|w|)^{-N}(1+|w'|)^{-N}\\ &\quad \times\Big(\sum_{(\epsilon',k')\in S^{w,j'}_{ j,k}}|u^{\epsilon'}_{j',k'}(s)|^p\Big)^{1/p} \Big(\sum_{(\epsilon'',k'')\in S^{w',j'}_{j,k}}|v^{\epsilon''}_{j',k''}(s)|^p \Big)^{1/p'}. \end{aligned} \end{equation} \end{lemma} \begin{lemma}\label{inequality4} Let $Q_{j,k}$ be a dyadic cube with radius $2^{-j}$. For $w\in\mathbb{Z}^{n}$, denote by $Q^{w}_{j,k}$ the dyadic cube $2^{8-j}w+\widetilde{Q}_{j,k}$. If $\delta>0$ is small enough, then \begin{equation}\label{eqn:est5} \begin{aligned} &\sum_{(\epsilon,k)\in S^{j}_{r}} \Big\{ \sum_{j\leq j'+5} \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N} \Big(\sum_{(\epsilon',k')\in S^{w,j'}_{j,k}} |a^{\epsilon}_{j',k'}|^p\Big)^{1/p}\Big\} ^p\\ &\lesssim \sum_{j\leq j'+5} 2^{\delta (j'-j)} \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N} \sum_{(\epsilon',k')\in S^{w,j}_{r}} |a^{\epsilon}_{j',k'}|^p. \end{aligned} \end{equation} \end{lemma} Here, it is worth mentioning that the proof of Lemma \ref{inequality4} needs also the following fact: for fixed $j$, the number of $Q_{j',k'}$ which are contained in the dyadic cube $ Q_{j,k}^{w}=2^{8-j}w+\widetilde{Q}_{j,k}$ equals $2^{n(8+j'-j)}$. On the other hand, for any dyadic cube $Q_{r}$ with radius $r$, the number of $Q_{j,k}\subset Q_{r}$ equals to $(2^{j}r)^{n}$. Then the number of $Q_{j',k'}$ which are contained in the dyadic cube $Q^{w}_{r}$ equals $(2^{8+j'}r)^{n}$. In the proof of the main lemmas in Sections \ref{sec7} and \ref{sec8}, we will use this fact again. \begin{lemma} \label{lem5.6} Let $Q_{j,k}$ be a dyadic cube with radius $2^{-j}$. For $w\in\mathbb{Z}^{n}$, denote by $Q^{w}_{j,k}$ the dyadic cube $2^{8-j}w+\widetilde{Q}_{j,k}$. If $j2m'\beta$, $j\sim j'$ and H\"older's inequality, we apply \eqref{eqn:est1} to obtain \begin{align*} I^{m,1}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq\max\{-\log_2r, -\frac{\log_2t}{2\beta}\}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \\ &\quad\times \Big[\sum_{(\epsilon,k)\in S^{j}_{r}}e^{-cpt2^{2j\beta}} \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N} \\ &\quad\times \sum_{(\epsilon',k')\in S^{w,j'}_{j,k}}\int^{2^{-1-2j'\beta}}_0 |u^{\epsilon'}_{j',k'}(s)|^p(s2^{2j'\beta})^{m'}\frac{ds}{s}(t2^{2j\beta})^{m} \Big]^{q/p}. \end{align*} If $q\leq p$, by the $\alpha$-triangle inequality we obtain \begin{align*} I^{m,1}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq\max\{-\log_2r, -\frac{\log_2t}{2\beta}\}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})}\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-\frac{qN}{p}}\\ &\quad\times \Big[\sum_{(\epsilon',k')\in S^{w,j'}_{r}}\int^{2^{-1-2j'\beta}}_0|u^{\epsilon'}_{j',k'}(s)|^p (s2^{2j'\beta})^{m'}\frac{ds}{s}\Big]^{q/p}\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2, IV}_{p,q,m'}}\lesssim1. \end{align*} If $q>p$, H\"older's inequality implies \begin{align*} I^{m,1}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq\max\{-\log_2r, -\frac{\log_2t}{2\beta}\}}2^{qj(\gamma_1+\frac{n}{2} -\frac{n}{p})}\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}\\ &\quad\times \Big[\sum_{(\epsilon',k')\in S^{w,j'}_{r}}\int^{2^{-1-2j'\beta}}_0|u^{\epsilon'}_{j',k'}(s) |^p(s2^{2j'\beta})^{m'}\frac{ds}{s}\Big]^{q/p}\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2,IV}_{p,q,m'}} \lesssim 1. \end{align*} In a similar manner, we can obtain the following two assertions. \textit{Case 7.2:} $(t,x)\mapsto\sum_{(\epsilon,j,k)}a^{\epsilon,2}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,I}_{p,q,m}$; \textit{Case 7.3:} $(t,x)\mapsto \sum_{(\epsilon,j,k)}a^{\epsilon,3}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,I}_{p,q,m}$. \subsection{Setting (ii)} For $i=1, 2, 3$, define $$ III^{m,i}_{a, Q_{r}} =|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j \geq-\log_2r}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[\int^{r^{2\beta}}_{2^{-2j\beta}}\sum_{(\epsilon,k)\in S^{j}_{r}}|a^{\epsilon,i}_{j,k}(t)|^p(t2^{2j\beta})^{m}\frac{dt}{t}\Big]^{q/p}. $$ We consider the following three cases: \textit{Case 7.4:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in\Lambda_{n}}a^{\epsilon,1}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2, III}_{p,q,m}$; \textit{Case 7.5:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in\Lambda_{n}}a^{\epsilon,2}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2, III}_{p,q,m}$; \textit{Case 7.6:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in\Lambda_{n}}a^{\epsilon,3}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,III}_{p,q,m}$. It is sufficient to check Case 7.4 since Cases 7.5 and 7.6 can be dealt with similarly. In fact, for $v\in\mathbb{B}^{\gamma_1,\gamma_2}_{p,q,m,m'}$ we have, by (i) of Lemma \ref{le6}, $$ |a^{\epsilon,1}_{j,k}(t)|\lesssim 2^{j}\sum_{|j-j'|\leq2} \sum_{\epsilon',k'}\int^{2^{-1-2j'\beta}}_0 |u^{\epsilon'}_{j',k'}(s)| e^{-\tilde{c}t2^{2j\beta}}(1+|2^{j-j'}k'-k|)^{-N}s^{\frac{1}{2\beta}}\frac{ds}{s}, $$ whence obtaining \begin{align*} III^{m,1}_{a, Q_{r}} &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r} 2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[\int^{r^{2\beta}}_{2^{-2j\beta}}2^{pj}e^{-\tilde{c}pt2^{2j\beta}}\\ &\quad\times \sum_{(\epsilon,k)\in S^{j}_{r}} \Big(\sum_{|j-j'|\leq2}\sum_{\epsilon',k'}\int^{2^{-1-2j'\beta}}_0 |u^{\epsilon'}_{j',k'}(s)| (1+|2^{j-j'}k'-k|)^{-N}s^{\frac{1}{2\beta}}\frac{ds}{s}\Big)^p\\ &\quad\times (t2^{2j\beta})^{m}\frac{dt}{t}\Big]^{q/p}. \end{align*} Applying H\"older's inequality to $k'$ and $s$ respectively, along with \eqref{eqn:est1} and $|j-j'|\leq2$, we find \begin{align*} &III^{m,1}_{a, Q_{r}}\\ &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r} 2^{qj(\gamma_1+\frac{n}{2}-\frac{q}{p})} \Big[\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}\int^{r^{2\beta}}_{2^{-2j\beta}}2^{jp} e^{-\tilde{c}pt2^{2j\beta}}\\ &\quad\times \sum_{(\epsilon,k)\in S^{j}_{r}}2^{-jp}\Big(\sum_{(\epsilon',k')\in S^{w,j'}_{ j,k}}\int^{2^{-1-2j'\beta}}_0|u^{\epsilon'}_{j',k'}(s)|^p(s2^{2j\beta})^{m'} \frac{ds}{s}\Big)(t2^{2j\beta})^{m}\frac{dt}{t}\Big]^{q/p}. \end{align*} If $q\leq p$, by changing variables we obtain \begin{align*} &III^{m,1}_{a, Q_{r}}\\ &\lesssim \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-\frac{qN}{p}} |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r} 2^{qj(\gamma_1+\frac{n}{2}-\frac{q}{p})}\\ &\quad\times \Big[\int^{r^{2\beta}}_{2^{-2j\beta}} e^{-\tilde{c}t2^{2j\beta}}(t2^{2j\beta})^{m}\Big(\sum_{(\epsilon',k')\in S^{w,j'}_{ r}}\int^{2^{-1-2j'\beta}}_0|u^{\epsilon'}_{j',k'}(s)|^p(s2^{2j\beta})^{m'} \frac{ds}{s}\Big)\frac{dt}{t}\Big]^{q/p}\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2, IV}_{p,q,m'}}. \end{align*} If $q>p$, via applying H\"older's inequality for $w$ and $\frac{q}{p}>1$, we similarly have \begin{align*} &III^{m,1}_{a, Q_{r}}\\ &\lesssim \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N} |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r} 2^{qj(\gamma_1+\frac{n}{2}-\frac{q}{p})}\\ &\Big[\int^{r^{2\beta}}_{2^{-2j\beta}}e^{-\tilde{c}t 2^{2j\beta}}(t2^{2j\beta})^{m}\Big(\sum_{(\epsilon',k')\in S^{w,j'}_{ r}}\int^{2^{-1-2j'\beta}}_0|u^{\epsilon'}_{j',k'}(s)|^p (s2^{2j\beta})^{m'}\frac{ds}{s}\Big)\frac{dt}{t}\Big]^{q/p}\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2, IV}_{p,q,m'}}. \end{align*} \subsection{Setting (iii)} The argument for that the function $$ (t,x)\mapsto\sum_{(\epsilon,j,k)\in\Lambda_{n}}a^{\epsilon,4}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)\quad \text{is in }\mathbb{B}^{\gamma_1,\gamma_2,II}_{p,q} \cap \mathbb{B}^{\gamma_1,\gamma_2,IV}_{p,q, m'} $$ is divided into two cases. \textit{Case 7.7:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in\Lambda_{n}}a^{\epsilon,4}_{j,k}(t) \Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,II}_{p,q}$. Let $$ II^{4}_{a, Q_{r}}(t) = |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j<-\frac{\log_2t}{2\beta}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[\sum_{(\epsilon,k)\in S^{j}_{r}}|a^{\epsilon,4}_{j,k}(t)|^p\Big]^{q/p}. $$ Then, by (ii) of Lemma \ref{le6} we have $$ |a^{\epsilon,4}_{j,k}(t)| \lesssim 2^{j}\sum_{|j-j'|\leq2}\sum_{\epsilon',k'}\int^{t}_0 |u^{\epsilon'}_{j',k'}(s)| e^{-\tilde{c}(t-s)2^{2j\beta}} (1+|2^{j-j'}k'-k|)^{-N}s^{\frac{1}{2\beta}-1}ds, $$ whence via H\"older's inequality and \eqref{eqn:est1}, we obtain \begin{align*} II^{4}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j<-\frac{\log_2t}{2\beta}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[2^{pj}\sum_{|j-j'|\leq2}\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}\\ & \quad\times \sum_{(\epsilon',k')\in S^{w,j'}_{r}}t^{p-1-\mu p}\int^{t}_0|u^{\epsilon'}_{j',k'}(s)|^ps^{(\frac{1}{2\beta}-1+\mu)p}ds\Big]^{q/p} \end{align*} If $q>p$, by H\"older's inequality we have \begin{align*} II^{4}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j<-\frac{\log_2t}{2\beta}}2^{qj(\gamma_1+\frac{n}{2} -\frac{n}{p})}\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}\\ &\quad\times 2^{qj}\sum_{|j-j'|\leq2}t^{\frac{(p-1-p\mu)q}{p}} \Big[\int^{t}_0\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|u^{\epsilon'}_{j',k'}(s)|^ps^{(\frac{1}{2\beta}-1+\mu)p}ds\Big]^{q/p}. \end{align*} Because $2^{2j\beta}t\leq1$, one has $2^{qj}\leq t^{-\frac{q}{2\beta}}$. This in turn gives \begin{align*} II^{4}_{a, Q_{r}}(t) &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j<-\frac{\log_2t}{2\beta}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}2^{qj}\\ &\quad\times \sum_{|j-j'|\leq2}t^{\frac{q}{2\beta}}t^{-p(\frac{1}{2\beta}-1+\mu)-1} \Big[\int^{t}_0(\sum_{\epsilon',k'}|u^{\epsilon'}_{j',k'}(s)|^p)^{q/p} s^{(\frac{1}{2\beta}-1+\mu)p}ds\Big]\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2,II}_{p,q}}. \end{align*} If $q\leq p$, we have \begin{align*} &II^{4}_{a, Q_{r}}(t)\\ &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{-\log_2r\leq j<-\frac{\log_2t}{2\beta}}2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big[\sum_{w\in\mathbb{Z}^{n}}(1+|w|)^{-N}\sum_{|j-j'|\leq2}2^{jp} \sum_{(\epsilon',k')\in S^{w,j'}_{r}}\Big(\int^{t}_0|u^{\epsilon'}_{j',k'}(s) |s^{\frac{1}{2\beta}-1}ds\Big)^p\Big]^{q/p}. \end{align*} Because $t\leq2^{-2j\beta}$ and $0p$, by H\"older's inequality, we have \begin{align*} IV^{4,m'}_{a,Q_{r}} &\lesssim \sum_{w\in\mathbb{Z}^{n}}|Q_{r}|^{\frac{q\gamma_2}{n} -\frac{q}{p}}(1+|w|)^{-N} \sum_{j\geq-\log_2r_{w}}2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big[\int^{2^{-2(j'-2)\beta}}_0\sum_{(\epsilon',k')\in S^{w,j'}_{r}}|u^{\epsilon'}_{j',k'}(s)|^p(s2^{2j'\beta}) ^{\frac{p}{2\beta}+1-p+p\mu}\frac{ds}{s}\Big]^{q/p}\\ &\lesssim \|u\|_{\mathbb{B}^{\gamma_1,\gamma_2,III}_{p,q,m}} +\|u\|_{\mathbb{B}^{\gamma_1,\gamma_2,IV}_{p,q,m'}}. \end{align*} \section{Proof of Lemma \ref{lem54}}\label{sec8} \subsection{Setting (i)} For $i=1,2,3$, define \begin{align*} &I^{m,i}_{b, Q_{r}}(t)\\ &=|Q_{r}|^{\frac{q\gamma_2}{n} -\frac{q}{p}}\sum_{j\geq\max\{-\log_2r,-\frac{\log_2t}{2\beta}\}} 2^{qj(\gamma_1+\frac{n}{2}-\frac{n}{p})} \Big[\sum_{(\epsilon,k)\in S^{j}_{r}}|b^{\epsilon,i}_{j,k}(t)|^p(t2^{2j\beta})^{m} \Big]^{q/p}. \end{align*} We divide the proof into three cases: \textit{Case 8.1:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in \Lambda_{n}}b^{\epsilon,1}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,I}_{p,q,m}$; \textit{Case 8.2:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in \Lambda_{n}}b^{\epsilon,2}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,I}_{p,q,m}$; \textit{Case 8.3:} $(t,x)\mapsto\sum_{(\epsilon,j,k)\in \Lambda_{n}}b^{\epsilon,3}_{j,k}(t)\Phi^{\epsilon}_{j,k}(x)$ is in $\mathbb{B}^{\gamma_1,\gamma_2,I}_{p,q,m}$. But, we demonstrate only Case 8.1 and omit the proofs of Cases 8.2 and 8.3 due to their similarity. Assume first $10$ take $0<\deltap$, by H\"older's inequality we obtain \begin{align*} I^{m,1}_{b, Q_{r}}(t) &\lesssim\sum_{w\in\mathbb{Z}^{n}}\frac{|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}}{(1+|w|)^{N}} \sum_{j\geq\max\{-\log_2r,-\frac{\log_2t}{2\beta}\}}\Big[\sum_{j\leq j'+2}2^{(j'-j)[\delta-(p\gamma_2+2-2\beta)]}\Big]^{\frac{q-p}{p}}\\ &\quad\times\Big\{\sum_{j\leq j'+2}2^{(j'-j)[\delta-(p\gamma_2+2-2\beta)]}2^{qj'(\gamma_1+\frac{n}{2}-\frac{n}{p})}\\ &\quad\times \Big(\int^{2^{-1-2j'\beta}}_0\sum_{(\epsilon',k')\in S^{w,j'}_{ r}}|u^{\epsilon'}_{j',k'}(s)|^p(s2^{2j'\beta})^{m'}\frac{ds}{s}\Big)^{q/p}\Big\}\\ &\lesssim\|u\|_{\mathbb{B}^{\gamma_1,\gamma_2, IV}_{p,q,m'}}. \end{align*} Then assume $2p$, by H\"older's inequality we obtain \begin{align*} I^{m,1}_{b, Q_{r}}(t) &\lesssim \sum_{w\in\mathbb{Z}^{n}}\frac{|Q_{r}|^{\frac{q\gamma_2}{n} -\frac{q}{p}}}{(1+|w|)^{N}} \sum_{j\geq\max\{-\log_2r,-\frac{\log_2t}{2\beta}\}} \Big(\sum_{jp$, by H\"older's inequality we obtain \begin{align*} III^{m,1}_{b,Q_{r}} &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r} \Big(\sum_{j\leq j'+2}2^{p(j-j')(p\gamma_2+2-2\beta-\delta)}\Big)^{\frac{q-p}{p}}\\ &\quad\times \Big\{\sum_{j\leq j'+2}2^{q(j-j')(p\gamma_2+2-2\beta-\delta)}\Big[\int^{2^{-1-2j'\beta}}_0 \sum_{(\epsilon',k')\in S^{w,j'}_{r}}|u^{\epsilon'}_{j',k'}(s)|^p\\ &\quad\times (s2^{2j'\beta})^{m'}\frac{ds}{s}\Big]^{q/p}\Big\}. \end{align*} Upon taking $0<\delta0$, \begin{align*} &II^{4}_{b,Q_{r}}(t)\\ &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}} \sum_{-\log_2rp$, by H\"older's inequality we obtain \begin{align*} II^{4}_{b, Q_{r}}(t) &\lesssim \sum_{w\in\mathbb{Z}^{n}}\frac{|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}}{(1+|w|)^{N}} \sum_{j\geq-\log_2r}\Big(\sum_{j0$, \begin{align*} IV^{m',4}_{b,Q_{r}} &\lesssim |Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}\sum_{j\geq-\log_2r}2^{qj(\gamma_1+1+(p-1)\gamma_2)}2^{-\frac{q\delta j}{p}}\\ &\quad\times \Big\{\int^{2^{-2j\beta}}_0\sum_{(\epsilon,k)\in S^{j}_{r}} \sum_{j0$ and taking $0<\delta0. $$ and then changed the order of $j$ and $j'$. If $q>p$, by H\"older's inequality we have \begin{align*} IV^{m',4}_{b,Q_{r}} &\lesssim \sum_{w\in\mathbb{Z}^{n}}|Q_{r}|^{\frac{q\gamma_2}{n}-\frac{q}{p}}(1+|w|)^{-N} \sum_{j\geq-\log_2r} \Big[\sum_{j