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OleehyO
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latex-formulas

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\begin{align*}\tau_{-x_n}\mu^i_n=\tau_{-x_n}\mu_n-\sum_{0\le j\leq i}\tau_{x^j_n-x_n}\mu^j,\end{align*}
\begin{align*}r\alpha=\frac{r-1}{1+\frac 1N-\frac 1p}\geq p.\end{align*}
\begin{align*}m_v\coloneqq m-\sum_{0\leq i<k}m_i=\lim_{n\to\infty}\int_{B_{r_n}\setminus\cup_{0\leq i<k_n}B^i_n}u_n.\end{align*}
\begin{align*}H(m)=\theta\frac{H(m)}{m}+(m-\theta)\frac{H(m)}{m}\leq \theta\frac{H(\theta)}{\theta}+(m-\theta)\frac{H(m-\theta)}{m-\theta}.\end{align*}
\begin{align*}e(\partial B_{R_\ell}(x_0))=u(\partial B_{r_\ell}(x_0))=0,\\\shortintertext{and}\lim_{\ell\to\infty}\frac{e(B_{R_\ell}(x_0))}{u(B_{r_\ell}(x_0))}=\limsup_{R\to 0^+}\frac{e(B_R(x_0))}{u(B_R(x_0))}.\end{align*}
\begin{align*}\limsup_{R\to 0^+}\frac{e(B_R(x_0))}{u(B_R(x_0))} = \lim_{\ell\to\infty} \frac{e(B_{\rho_\ell}(x_0))}{u(B_{\rho_\ell}(x_0))},\end{align*}
\begin{align*}\lim_{n\to\infty}e_n(B_{R_\ell}(x_0))=e(B_{R_\ell}(x_0))\quad \lim_{n\to\infty}\int_{B_{r_\ell}(x_0)}u_n=u(B_{r_\ell}(x_0)).\end{align*}
\begin{align*}u(\{x_0\})=\lim_{\ell\to\infty}\int_{B_{r_\ell}(x_0)}u_{n_\ell}, e(\{x_0\}) = \lim_{\ell\to\infty} e_{n_\ell}(B_{R_\ell}(x_0)),\\\shortintertext{and}\limsup_{\ell\to\infty}\frac{e(B_{R_\ell}(x_0))}{u(B_{r_\ell}(x_0))}=\lim_{\ell\to\infty}\frac{e_{n_\ell}(B_{R_\ell}(x_0))}{\int_{B_{r_\ell}(x_0)}u_{n_\ell}}.\end{align*}
\begin{align*} &\lambda_1(D(T(a,b)))\geq\lambda_1(D(T(1,1))=8.2882,\\ &\lambda_2(D(T(a,b)))\geq\lambda_2(D(T(1,1))=-0.5578,\\ &\lambda_3(D(T(a,b)))\geq\lambda_3(D(T(1,1))=-0.7639,\\ &\lambda_4(D(T(a,b)))\geq\lambda_4(D(T(1,1))=-1.7304,\\ &\lambda_n(D(T(a,b)))\leq\lambda_5(D(T(1,1))=-5.2361.\end{align*}
\begin{align*}e(\{x_0\})\geq H(x_0,m_v) + \sum_{0\leq i<k}H(x_0,m_i) \geq H\Bigl(x_0,m_v+\sum_{0\leq i<k}m_i\Bigr)=H(x_0,u(\{x_0\})).\end{align*}
\begin{align*}H_f(m)=m^{1-\frac{p}{N}}H_f(1),0<H_f(1)<+\infty.\end{align*}
\begin{align*}\exists m_\ast\geq 0, \begin{cases}H[0,m_\ast],\\H(m_\ast,+\infty).\end{cases}\end{align*}
\begin{align*}\lambda = m^{\frac{s/p-1}{1+N-sN/p}}.\end{align*}
\begin{align*}\mathcal{F}(f)(\xi)= \int e^{-ix \cdot \xi} f(x) d x.\end{align*}
\begin{align*}\tilde{M}_{n}(t)=\sup_{s\in [0, t]}M_{n}(s)+ (1+t)^{n^3} \lesssim \big( \tilde{M}_{n}( t) \big)^{ 1- \epsilon}+ (1+t)^{n^3}, \Longrightarrow \tilde{M}_{n}(t)\lesssim (1+t)^{n^3}. \end{align*}
\begin{align*}\beta_t = \sup_{s\in[0, t]}\beta_t(s) \leq 1-2\epsilon, \alpha_t = \sup_{s\in[0, t]}\alpha_t(s) \leq \alpha^{\star}- \epsilon. \end{align*}
\begin{align*}\mathfrak{M}(\xi, v, V(s))= \frac{ \mathbf{e}_3 \langle V(s)\rangle }{|\xi|( |\xi| + \hat{V}(s )\cdot \xi)} - \frac{V_3(s) }{|\xi|( |\xi| + \hat{V}(s )\cdot \xi)^2} \big(\xi - \hat{V}(s) \hat{V}(s)\cdot \xi \big).\end{align*}
\begin{align*}\mathfrak{High}^{j }_{k, \tilde{k}}(t_1,t_2)= \int_{t_1}^{t_2} \langle V(s)\rangle^{-1} E(s, X(s))\cdot \widetilde{\mathfrak{High}}^{j }_{k, \tilde{k}}(s) d s, \end{align*}
\begin{align*}g_{0}=dr^2+ \sin^2 r \cdot h_{\mathbb{S}^{n-2}}+\cos^2 r \cdot ds^2\end{align*}
\begin{align*}&(-\lambda-2)^{2c-2}(\lambda^5+(4c-4)\lambda^4+(12c^2+36c+2)\lambda^3+(24c^2+88c+28)\lambda^2+(8c^2+72c+40)\lambda+16c+16)\\&=(-\lambda-2)^{2c-2}(\lambda^2+(2c+4)\lambda+4)(-\lambda^3+6c\lambda^2+(12c+6)\lambda+4c+4).\end{align*}
\begin{align*}g=\frac{dr^2}{\alpha(r)}+ \sin^2 r \cdot h_{\mathbb{S}^{n-2}}+e^{2\beta (s,r)} \cdot ds^2.\end{align*}
\begin{align*}\mathcal{R}=2K_{rs}+2(n-2)K_{ri}+2(n-2)K_{is}+(n-2)(n-3)K_{ij}\end{align*}
\begin{align*}K_{rs}=-\alpha (\beta_{rr}+(\beta_r)^2)\end{align*}
\begin{align*}K_{ij}=\frac1{\sin^2 r}\left( 1-\alpha \cos^2 r\right)\geq 0.\end{align*}
\begin{align*}\beta(s,r)=\varphi(r)\log\cos r + (1-\varphi(r))f(s,0),\end{align*}
\begin{align*}\beta_r=\varphi'(r)\big[\log\cos r - f(s,0)\big] -\varphi(r)\tan r\end{align*}
\begin{align*}\begin{aligned}\beta_{rr} &=\varphi''(r)\big[\log\cos r - f(s,0)\big] -2\varphi'(r)\tan r -\varphi(r)\sec^2 r\\&\leq |\varphi''(r)|\big[ -\log\cos R + \bar{f}\big] \end{aligned}\end{align*}
\begin{align*}\begin{aligned}d_h(a,b) &\leq d_h(a,\tilde x)+d_h(\tilde x,\tilde y)+d_h(\tilde y,b)\\&\leq 3\eta + \delta +3\eta\\&\leq 2\delta\end{aligned}\end{align*}
\begin{align*}g=\frac{dr^2}{\alpha(r)}+ r^2 \cdot h_{\mathbb{S}^{n-2}}+e^{2\beta (s,r)} \cdot ds^2.\end{align*}
\begin{align*}\beta(s,r)=(1-\varphi(r))f(s,0),\end{align*}
\begin{align*}F=\frac{1}{2}\begin{pmatrix}(\partial_{\xi_j}\partial_{x_k} q(x,\xi))_{j,k=1}^d&(\partial_{\xi_j}\partial_{\xi_k} q(x,\xi))_{j,k=1}^d \\-(\partial_{x_j}\partial_{x_k} q(x,\xi))_{j,k=1}^d&-(\partial_{x_j}\partial_{\xi_k} q(x,\xi))_{j,k=1}^d\end{pmatrix}.\end{align*}
\begin{align*}P(h) u_h= E(h) u_h \,\, M, \\\end{align*}
\begin{align*}\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=O(1).\end{align*}
\begin{align*}U_\Gamma=\{(x', x_n):\, x'\in \Gamma \,\, \,\, x_n\in (-c, c)\}\end{align*}
\begin{align*}\Lambda_{A} := \bigcup_{T}^{\infty} \Lambda_{A,T}\,, \qquad\Lambda_{A,T} := \bigcup_{t=-T}^{T} \phi_t(A)\end{align*}
\begin{align*}\Sigma_{T^*M}=\{(x,\xi) \in T^*M\, | \, |\xi |^2_g+V(x)=E\}.\end{align*}
\begin{align*}\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=o(1).\end{align*}
\begin{align*}\chi_\alpha(x_n)=\begin{cases}&0 |x_n|\geq 2\alpha \\&1 |x_n| \leq \alpha/2,\end{cases}\end{align*}
\begin{align*}\lim_{\alpha\to 0}\frac{i}h\int_{\Omega_\Gamma} [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h \overline{u_h} dx=o(1).\end{align*}
\begin{align*}&\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)} \\&= \left< Op_h \left(\{\sigma(-h^2\Delta+V(x)-E(h)), \sigma(\chi_\alpha(x_n) hD_n)\} \right)u_h, u_h \right>_{L^2(\Omega_\Gamma)}+ O(h),\end{align*}
\begin{align*}\{|(\xi', \xi_n)|_x^2+V(x)-E(h), \chi_\alpha(x_n)\xi_n\}= 2\chi_\alpha'(x_n) \xi_n^2-\chi_\alpha(x_n)\partial_{x_n} (R+V),\end{align*}
\begin{align*}a_{\ell} + a_k = b_{\ell} + b_k\end{align*}
\begin{align*}C_1(m,\lambda)=\prod_{k=0}^{m-1} (\lambda + 2k).\end{align*}
\begin{align*}|(\xi', \xi_n)|_x^2= \xi_n^2+R(x',x_n,\xi'),\end{align*}
\begin{align*}&\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)} \\=& \int_{\Sigma_{\Omega_\Gamma}} \left( 2\chi_\alpha'(x_n) \xi_n^2-\chi_\alpha(x_n)\partial_{x_n}(R+V) \right) \, d \mu+o(1) \\=& I_1-I_2+o(1)\end{align*}
\begin{align*}\int_{\Sigma_{\Omega_\Gamma} } \chi_\alpha(x_n) \big(1-\chi_\Gamma \big) \partial_{x_n}(R+V)\, d \mu =0\end{align*}
\begin{align*}\lim_{\alpha\to 0}\left< \frac{i}h [-h^2\Delta+V(x)-E(h), \chi_\alpha(x_n)hD_n] u_h,\, u_h \right>_{L^2(\Omega_\Gamma)}=o(1).\end{align*}
\begin{align*}\frac{1}{2T} \mu (\Lambda_{S^*_\Gamma M \backslash S^* \Gamma,\, T})=0.\end{align*}
\begin{align*}\|\lambda_j^{-1} \partial_\nu u_{\lambda_j} \|_{L^2(\Gamma)}=o(1).\end{align*}
\begin{align*}\lim_{h\to0}\lim_{\alpha\to 0} \frac{i}h\int_{\Omega_\Gamma} [-h^2\Delta-1, \chi_\alpha(x_n)hD_n] u_h \overline{u_h} dx=\int_{S^*_\Gamma M \backslash S^* \Gamma} |\xi_n| d\mu^\perp,\end{align*}
\begin{align*}\phi_h(x_1, x_2)=e^{\frac{i}h x_1},h^{-1}\in 2\pi \mathbb{Z}.\end{align*}
\begin{align*}\mu(x_1, x_2, \xi_1, \xi_2)= \delta_{(1,0)}(\xi_1, \xi_2) dx_1 dx_2.\end{align*}
\begin{align*}\frac{1}{2T} \mu (\Lambda_{S^*_\Gamma \mathbb{T}^2 \backslash S^*\Gamma })=0.\end{align*}
\begin{align*}A=\frac{1}{2}\mathrm{Tr}(Q \nabla_x^2) - \frac{1}{2} Rx\cdot x + Bx\cdot \nabla_x\end{align*}
\begin{align*}\|h \partial_{x_2} \phi_h(x_1, x_2) \|_{L^2(\Gamma)}=0\end{align*}
\begin{align*}C_{\gamma_1}\cap C_{\gamma_2}=\{\gamma_1\}\times C_J=\{\gamma_2\}\times C_J, \end{align*}
\begin{align*}\{H(e_1)\mid e_1\in E^-(v_1)\}=\{H(e_2)\mid e_2\in E^-(v_2)\}.\end{align*}
\begin{align*}\bar{\phi}_{C_1,C_2}:=\phi_{C'_{n-1},C'_n}\circ\cdots\circ\phi_{C'_1,C'_2}\circ\phi_{C'_0,C'_1}\end{align*}
\begin{align*}t(v\Uparrow u)=\cup\{t(u')\mid u'\in{\downarrow_1}(v): q(u)\prec q(u')\}\cup\{i\}.\end{align*}
\begin{align*}{\downarrow_1}(v\Uparrow u)=\{u'\in{\downarrow_1}(v)\mid q(u)\prec q(u')\}\cup\{u\}.\end{align*}
\begin{align*}\{[v']\mid v'\in{\downarrow_1}(v_1)\}=\{[v']\mid v'\in{\downarrow_1}(v_2)\}.\end{align*}
\begin{align*}w&:=v_1\Uparrow u=v_2\Uparrow u,\\w'&:=v'_1\Uparrow u'=v'_2\Uparrow u',\\y_1&:=v_1\Uparrow x_1=v'_1\Uparrow x_1,\\y_2&:=v_2\Uparrow x_2=v'_2\Uparrow x_2.\end{align*}
\begin{align*}\bar{\psi}_{C_1,C_2}:=\phi_{C_1,C_2}.\end{align*}
\begin{align*}A=\Delta_{x^{(2)}} - \frac{1}{4}|x^{(2)}|^2 - x^{(2)}\cdot\nabla_{x^{(1)}} + \nabla_{x^{(1)}}V(x^{(1)})\cdot\nabla_{x^{(2)}},\end{align*}
\begin{align*}\bar{\psi}_{C_1,C'_1}=\phi^{[y_1]}_{C_1,C'_1}\quad\quad\bar{\psi}_{C_2,C'_2}=\phi^{[y_1]}_{C_2,C'_2}.\end{align*}
\begin{align*}\lambda\cdot\bar{\psi}=\bar{\psi}\end{align*}
\begin{align*}\phi^{[\lambda_k\lambda'' v]}:=(\lambda_k\lambda'')\cdot\phi.\end{align*}
\begin{align*}\phi^{[\lambda\lambda_k\lambda'' v]}&=(\lambda\lambda_k\lambda'')\cdot\phi\\&=\lambda\cdot((\lambda_k\lambda'')\cdot\phi)\\&=\lambda\cdot\phi^{[\lambda_k\lambda''v]}.\end{align*}
\begin{align*}\phi_{\lambda C_1,\lambda C_2}=\lambda\circ\phi_{C_1,C_2}\circ\lambda^{-1}\end{align*}
\begin{align*}t(v)=t'(h_k v)\end{align*}
\begin{align*}\widetilde{M}_{n}(t)=\sup_{s\in [0, t]}M_{n}(s)+ (1+t)^{n^3} \lesssim \big( \widetilde{M}_{n}( t) \big)^{ 1- \epsilon}+ (1+t)^{n^3}, \Longrightarrow \widetilde{M}_{n}(t)\lesssim (1+t)^{n^3}. \end{align*}
\begin{align*}\int_{t_1}^{t_2} \tilde{V}(s)\cdot K(s,X(s), V(s)) d s, \tilde{V}(s):=V(s)/|V(s)|. \end{align*}
\begin{align*}\mathcal{F}(f)(\xi)= \int e^{-ix \cdot \xi} f(x) d x.\end{align*}
\begin{align*}K_g^2:= |E\cdot \omega|^2 + |B\cdot \omega |^2 +|E-\omega \times B|^2 +|B+\omega \times E|^2. \end{align*}
\begin{align*}(Q^{1/2},BQ^{1/2},\dots,B^{d-1}Q^{1/2})=(Q^{1/2},BQ^{1/2},Q^{1/2},\dots,BQ^{1/2})\end{align*}
\begin{align*}c_p:= (\hat{v}_2\delta_{3p}-\hat{v}_3\delta_{2p}) - \frac{\hat{v}_2\omega_3-\hat{v}_3\omega_2}{1+\hat{v}\cdot \omega}\hat{v}_p.\end{align*}
\begin{align*}\big((\hat{v}+\omega) \times B \big)_1= (\hat{v}+\omega)_2 B_3 - (\hat{v}+\omega)_3 B_2, \big((\hat{v}+\omega) \times B \big)_2=- (\hat{v}+\omega)_1 B_3 +(\hat{v}+\omega)_3 B_1\end{align*}
\begin{align*}u = u \cdot \omega \omega +\sum_{i=1,2,3} (u\cdot (\omega\times \mathbf{e}_i))(\omega\times \mathbf{e}_i)\Longrightarrow B_3= B\cdot \omega(\omega\cdot \mathbf{e}_3) -\sum_{i=1,2,3} (B\cdot(\omega \times \mathbf{e}_i))( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i.\end{align*}
\begin{align*} \big((\hat{v}+\omega) \times B \big)_a= (-1)^{a-1}\big[(\hat{v}+\omega)_{3-a} (\omega\cdot \mathbf{e}_3) (B\cdot \omega ) -\big(\sum_{i=1,2,3} (\hat{v}+\omega)_{3-a}( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i))+ (\hat{v}+\omega)_3 B_2\big)\big].\end{align*}
\begin{align*}EB^2(t,s,x ,\omega, v)= \big( \sum_{i=1,2,3} - (\hat{v}+\omega)_2( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i))- (\hat{v}+\omega)_3 B_2, \sum_{i=1,2,3} (\hat{v}+\omega)_1( \omega \times \mathbf{e}_3)\cdot \mathbf{e}_i (B\cdot(\omega \times \mathbf{e}_i)) \end{align*}
\begin{align*}\big| det\big(\begin{bmatrix}\sin \theta \cos\phi - \sin \tilde{\theta} \cos\tilde{\phi} & \sin \theta \sin\phi - \sin \tilde{\theta}\sin \tilde{\phi} & \cos \theta - \cos \tilde{\theta}\\ -s \sin \theta \sin\phi & s \sin \theta \cos\phi & 0\\ s\cos\theta\cos\phi & s\cos\theta \sin \phi & -s\sin\theta\\ \end{bmatrix}\big)\big|\end{align*}
\begin{align*}= s^2\sin \theta\big[ 1-\cos(\theta- \tilde{\theta})+\sin \theta\sin \tilde{\theta}(1-\cos(\phi- \tilde{\phi}) )\big]\gtrsim s^2\sin \theta | 1-\cos(\theta- \tilde{\theta})|. \end{align*}
\begin{align*}\beta_t = \sup_{s\in[0, t]}\beta_t(s) \leq 1-2\epsilon, \alpha_t = \sup_{s\in[0, t]}\alpha_t(s) \leq (2/3+\iota)- \epsilon. \end{align*}
\begin{align*}c^{m, E }_{j,l,r;err}(t-s,v,\omega):= \big(\frac{\hat{v} +\omega }{(1+\hat{v}\cdot \omega)^2}\big(1 -|\hat{v}^2| \big) + \omega - \frac{(\omega +\hat{v} )\hat{v}\cdot \omega}{1+\hat{v}\cdot \omega}\big) \varphi_{l; r}(\tilde{v}+\omega ) \varphi_j(v) \varphi_{m;-10 M_t }( t-s )\end{align*}
\begin{align*}\sum_{i=0,1}\big| T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x, \zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta)\big|+ \big|\widetilde{T}_{k,j;n}^{bil;\mu,0}( \mathfrak{m}, E)(t,x, \zeta )+ \hat{\zeta}\times \widetilde{T}_{k,j;n}^{bil;\mu,0 }(\mathfrak{m}, B )(t,x, \zeta)\big| \end{align*}
\begin{align*}\bigcap_{j = 0}^{d-1} \ker(RB^j)=\ker(R)\cap\ker(RB).\end{align*}
\begin{align*}\sum_{i=3,4}\big\| T_{k,j;n,l,r}^{\mu,m,i}(\mathfrak{m},E)(t,x, \zeta) + \hat{\zeta}\times T_{k,j;n,l,r}^{\mu,m, i}(\mathfrak{m},B)(t,x, \zeta) \big\|_{L^\infty_x}\lesssim \| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty} \big[ 2^{(1- 19.5\epsilon)M_{t^{\star}}} \end{align*}
\begin{align*} \lesssim 2^{ 4\epsilon M_t}\| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty} \big( \min\{2^{n+4j/3}, 2^{k+ n }\} \mathbf{1}_{l> -j} + \min\{ 1+ 2^{n+j}, 2^{k/2+n}\}\mathbf{1}_{l=-j} \big)\end{align*}
\begin{align*} \lesssim \| \mathfrak{m}(\cdot, \zeta)\|_{\mathcal{S}^\infty} 2^{m/4+ k/2 + n+\max\{-j, p, n+\epsilon M_t\}/2-p/2 + \max\{l,p\} + 8\epsilon M_t } \big(2^{\min\{n,p\}+ j}\mathbf{1}_{l=-j, l> n+\epsilon M_t} \end{align*}
\begin{align*} \sum_{i=3,4}\big| T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x, \zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta)\big| \end{align*}
\begin{align*}\sum_{i=3,4}\big|\mathbf{P}\big( T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x,\zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta)\big) \big|+ 2^{(\gamma_1-\gamma_2)M_{t^{\star}}} \big| T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x,\zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta) \big|\end{align*}
\begin{align*}\Phi_2(\xi, \eta, \zeta):= \hat{\zeta} \cdot(\xi+ \eta ) + + \mu |\xi| + \mu_1 |\eta|.\end{align*}
\begin{align*}\Phi_3(\xi, \eta, \sigma ,\zeta):= \hat{\zeta} \cdot(\xi+ \eta +\sigma ) + \mu_2 |\sigma| + \mu |\xi| + \mu_1 |\eta|.\end{align*}
\begin{align*}\Phi_4(\xi, \eta, \sigma, \kappa,\zeta):= \hat{\zeta} \cdot(\xi+ \eta +\sigma+\kappa) + \mu_3 |\kappa| + \mu_2 |\sigma| + \mu |\xi| + \mu_1 |\eta|.\end{align*}
\begin{align*}\Phi_5(\xi, \eta, \sigma, \kappa,\chi, \zeta):= \hat{\zeta} \cdot(\xi+ \eta +\sigma+\kappa+\chi) +\mu_4 |\chi|+ \mu_3 |\kappa| + \mu_2 |\sigma| + \mu_1 |\eta| + \mu |\xi|.\end{align*}
\begin{align*}\sum_{i=0,1,2} |\nabla_v \mathcal{K}^{\mu,i}_{k,j; n,l}(y, v, V(s))| + \sum_{a=3,4} 2^{-\max\{n, (\gamma_1-\gamma_2) M_{t^\star}\} -\epsilon M_t} |\nabla_v \mathcal{K}^{\mu,a }_{k,j ; n,l}(y, v, V(s))|\end{align*}
\begin{align*}\sigma(H)=\sigma(H_1) + \sigma(H_2) + \sigma(H_3),\end{align*}
\begin{align*} a= a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2, c=\cos\theta\cos \phi \mathbf{e}_1+ \cos\theta \sin \phi \mathbf{e}_2+\sin\theta \mathbf{e}_3,(a-c)\times b = -(a_2-\cos\theta \sin \phi) \mathbf{e}_3 + \sin\theta \mathbf{e}_2 \end{align*}
\begin{align*} \forall i,i_1\in\{0,1,2,3,4\}, \mu_1\in \{+, -\}, |HypEll_{k_1,j_1;n_1}^{i,i_1, \mu_1}(t_1,t_2)| \leq \sum_{a=1,2,3 } |{}^1_a LastEll_{k_1,j_1;n_1}^{i,i_1, \mu_1}(t_1,t_2)|, \end{align*}
\begin{align*}\sum_{i=0, 1,2,3,4}\big|{}_0^{1}Err^i_{k,j;n}(t_1, t_2)\big|\lesssim \mathcal{M}(C) 2^{ (\alpha^\star + 3\iota + 140\epsilon) M_t- (k+2n)/2} \lesssim 2^{ M_t/2} \mathcal{M}(C). \end{align*}