0$, $1\leq \rho\leq\infty$, $$ \|u\|_{\tilde{L}_T^\rho(B_p^{\alpha,q} )} =\|u\|_{L^\rho_T(L^p) } +\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|Q_{s}(u)\|_{L^\rho_T(L^p) }^q\frac{ds}{s}\Big\}^{1/q}<\infty, $$ where for any $u\in L^\rho_T(L^p)(\mathbb{R}^n)$, $$ \|u\|_{L^\rho_T(L^p) }=\Big\{\int_0^T \|u\|_{L^p}^\rho dt\Big\}^{1/\rho}<\infty. $$ \end{definition} \begin{remark} \label{rmk2.3} \rm Note that for any $u\in L^\rho_T(W^{k,p})(\mathbb{R}^n)$, similar definition can be given in the following form $$ \|u\|_{L^\rho_T(W^{k,p}) }=\{\int_0^T \|u\|_{W^{k,p}}^\rho dt\}^{1/\rho}<\infty, $$ where $W^{k,p}(\mathbb{R}^n)$ is Sobolev space and $k\in \mathbb{Z}^+$. Here for $p,q,\rho=\infty$, definitions for the above spaces can be given conventionally. Moreover by the coordinate transform, we have $$ \|u\|_{B_p^{\alpha,q}}\sim\|u\|_{L^p}+\Big\{\int_0^2 s^{-\frac{1}{2}\alpha q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}<\infty. $$ \end{remark} Now we give the main theorem in the paper. \begin{theorem} \label{tm2.1} Suppose that $L$ satisfies assumptions (a) and (b). Let $\alpha\in(-\mu,\mu)$ $(0<\mu\leq 1)$ and $1< p,q,\rho<\infty$. Let $u_0\in B_p^{\alpha,q}$ and $f\in \tilde{L}_T^\rho(B_p^{\alpha-2+\frac{2}{\rho},q})\cap {L}_T^\infty(L^p)$. Then the initial-value problem of the parabolic equation has a solution $u \in \tilde{L}_T^\rho (B_p^{\alpha+\frac{2}{\rho},q}) \cap {L}_T^1(W^{2,p})$ and there exists a constant $C>0$ depending only on $n$ and such that $$ \|u\|_{\tilde{L}_T^\rho(B_p^{\alpha+\frac{2}{\rho_1},q})}\leq C\big((1+T^{1/\rho_1})\|u_0\|_{B_p^{\alpha,q}}+ (1+T^{1+\frac{1}{\rho_1}-\frac{1}{\rho}}) \|f\|_{\tilde{L}_T^\rho(B_p^{\alpha-2+\frac{2}{\rho},q})}\big), $$ where $\rho_1\in(\rho,+\infty)$ satisfying $|\alpha+\frac{2}{\rho_1}|\leq \mu$ and $|\alpha-2+\frac{2}{\rho}|\leq \mu$. \end{theorem} \section{The proof of main theorem} Before we prove the main theorem, we need the following lemmas. \begin{lemma} \label{lem3.1} Let $k_{s,t}(x,y)$ be the kernel of $P_t(Q_s)$. If $t\geq s$, there exists a constant $c>0$ such that \begin{equation} \label{eq} |k_{s,t}(x,y)|\leq c\big(\sqrt{s/t}\big)^\mu h_s(x,y). \end{equation} If $t\leq s$, there exists a constant $c>0$ such that \begin{equation} \label{eq3.2} |k_{s,t}(x,y)|\leq ch_t(x,y). \end{equation} \end{lemma} \begin{proof} The proof is similar to \cite[Lemma B.1]{FJ}, which uses the vanishing conditions of the kernel of $Q_s$ and H\"older continuity conditions. \end{proof} \begin{remark} \label{rmk3.1} \rm Here the kernels of $Q_t(Q_s)$ and $Q_t(\tilde{Q}_s)$ have better properties due to the vanishing moment conditions; that is, let $K_{s,t}(x,y)$ be the kernel of $Q_t(Q_s)$, then if $t\geq s$, there exists a constant $c>0$ such that \begin{equation} \label{eq3.3} |K_{s,t}(x,y)|\leq c\big(\sqrt{s/t}\big)^\mu h_s(x,y); \end{equation} Also if $t\leq s$, there exists a constant $c>0$ such that \begin{equation} \label{eq3.4} |K_{s,t}(x,y)|\leq c\big(\sqrt{t/s}\big)^\mu h_t(x,y). \end{equation} \end{remark} \begin{lemma} \label{lem3.2} Let $1< p$, $q<\infty$ and $\alpha\in (-\mu,\mu)$. For $u\in L^p(\mathbb{R}^n)$, $$ \|u\|_{B_p^{\alpha,q}}\sim \|u\|_{L^p}+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}<\infty. $$ \end{lemma} \begin{proof} Firstly we prove there exists a constant $c>0$ such that $$ \|u\|_{B_p^{\alpha,q}}\leq c(\|u\|_{L^p}+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q})<\infty. $$ Here we need to verify that \begin{equation} \label{eq3.5} \Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|Q_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q} \leq c\Big(\|u\|_{L^p}+\Big\{\int_0^1 s^{-\frac{1}{2}\alpha q}\|\tilde{Q}_{s}(u)\|_p^q\frac{ds}{s}\Big\}^{1/q}\Big). \end{equation} Note that the following identity holds in $L^p$ for $1

0)$ for $\omega\in A_2$, where the elliptic operator was defined by \begin{equation} \label{eq3.8} L_\omega=-\omega^{-1}\mathop{\rm div}A\nabla \end{equation} where $A=(a_{i,j})_{n\times n}$ was a matrix of complex-valued, measurable functions satisfying some degenerate elliptic conditions. They pointed that if $A$ is real and symmetric valued, the heat kernel $p_t(x,y)$ of the semigroup $e^{-tL_\omega}(t>0)$ also satisfies Gaussian upper bounds. We think that regularity results in adapted weighted Besov spaces for solutions of the corresponding parabolic equations can also be obtained by using similar methods to the ones described in this paper. \end{remark} \subsection*{Acknowledgements} The author would like thank anonymous the referee for his/her good suggestions on the improvement of this paper. \begin{thebibliography}{10} \bibitem{AT} P. Auscher and P. Tchamitchian; \textit{Square root problem for divergence operators and related topics}, Asterisque, \textbf{249}(1998). \bibitem{CR} D. Cruz-Uribe and C. Rios; \textit{Gaussian bounds for degenerate parabolic equations}, J. Funct. Anal., \textbf{225}(2008), 283-312. \bibitem{FJ} M. Frazier and B. Jawerth; \textit{ A discrete transform and decompositions of distributional spaces}, J. Funct. Anal., \textbf{93}(1990), 34-170. \bibitem{L} P. G. Lemmari$\acute{e}$-Rieusset; \textit{Recent developments in the Navier-Stokes problem}, volume 431 of Chapman $\And$ Hall/CRC Research Notes in Mathematics, Chapman $\And$ Hall/CRC, BocaRaton, FL, (2002). \bibitem{Mc} Alan. McIntosh; \textit{Operators which have an $H_\infty$-calculus}, Miniconference on operator theory and partial differential equations, Proc. Center. Math.Analysis., ANU, Canberra, \textbf{14} (1986),210-231. \bibitem{P} A. Pazy; \textit{Semigroups of Linear Operators and Applications to Partial Differential Equations}, Springer-Verlag, New York, (1983). \bibitem{Sh} Z. Shen; \textit{On fundamental solution of generalized Sch$ddot{o}$dinger operators}, J. Funct. Anal., \textbf{167}(1999), 521-564. \end{thebibliography} \end{document}