Remarks on basic calculus in hyperboloidal foliation

TL;DR

This paper provides a systematic overview of fundamental calculus techniques within the hyperboloidal foliation framework, emphasizing estimates derived from commutator analysis to facilitate advanced mathematical investigations.

Contribution

It introduces a comprehensive summary of calculus methods in hyperboloidal foliation, focusing on commutator-based estimates, which are crucial for geometric analysis.

Findings

Developed estimates based on commutators in hyperboloidal foliation

Summarized key calculus techniques for hyperboloidal analysis

Enhanced understanding of geometric PDE methods

Abstract

In this article we will give a systematic summary on the basic calculus in the hyperboloidal foliation context, especially the estimates based on commutators.

Equations

$${\mathcal{H}}_s^{\ast }={\mathcal{H}}_s\cap \mathcal{K}.$$

$$\mathcal{K}=\bigcup _{1<s<\infty }{\mathcal{H}}_s^{\ast }.$$

$${\mathcal{K}}_{[s_0,s_1]}:=\{(t,x)\in \mathcal{K}|s_0^2+|x{|}^2\le t^2\le s_1^2+|x{|}^2\}$$

$${\mathcal{K}}_{[s_0,\infty )}:=\{(t,x)\in \mathcal{K}|s_0^2+|x{|}^2\le t^2\}.$$

$${\underline{\partial }}_a:=\frac{x^a}{t}{\partial }_t+{\partial }_a$$

$${\underline{\partial }}_0={\partial }_t.$$

$$\big{(}\underline{\Phi}_{\alpha}^{\beta}\big{)}=\big{(}{\underline{\Phi}_{\alpha}}^{\beta}\big{)}=\left(\begin{array}[]{cc}1&0\\ x/t&\text{Id}\end{array}\right),\qquad\qquad\big{(}\underline{\Psi}_{\alpha}^{\beta}\big{)}=\big{(}{\underline{\Psi}_{\alpha}}^{\beta}\big{)}=\left(\begin{array}[]{cc}1&0\\ -x/t&\text{Id}\end{array}\right).$$

$$s:=\sqrt{t^2-x^2},{\bar{x}}^a=x^a.$$

$${\partial }_{{\bar{x}}^a}:={\bar{\partial }}_a={\underline{\partial }}_a,{\partial }_s:={\bar{\partial }}_0=(s/t){\partial }_t.$$

$$\big{(}\overline{\Phi}_{\alpha}^{\beta}\big{)}=\big{(}{\overline{\Phi}_{\alpha}}^{\beta}\big{)}=\left(\begin{array}[]{cc}s/t&0\\ x/t&\text{Id}\end{array}\right),\qquad\qquad\big{(}\overline{\Psi}_{\alpha}^{\beta}\big{)}=\big{(}{\overline{\Psi}_{\alpha}}^{\beta}\big{)}=\left(\begin{array}[]{cc}t/s&0\\ -x/s&\text{Id}\end{array}\right).$$

$$T=T^{\alpha \beta }{\partial }_{\alpha }\otimes {\partial }_{\beta }={\underline{T}}^{\alpha \beta }{\underline{\partial }}_{\alpha }\otimes {\underline{\partial }}_{\beta }={\overline{T}}^{\alpha \beta }{\bar{\partial }}_{\alpha }\otimes {\bar{\partial }}_{\beta }$$

$${\underline{T}}^{\alpha \beta }={\underline{\Psi }}_{{\alpha }^{\prime }}^{\alpha }{\underline{\Psi }}_{{\beta }^{\prime }}^{\beta }{T}^{{\alpha }^{\prime }{\beta }^{\prime }},{\overline{T}}^{\alpha \beta }={\overline{\Psi }}_{{\alpha }^{\prime }}^{\alpha }{\overline{\Psi }}_{{\beta }^{\prime }}^{\beta }{T}^{{\alpha }^{\prime }{\beta }^{\prime }}.$$

$$\left(\underline{m}^{\alpha\beta}\right)_{0\leq\alpha,\beta\leq n}=\left(\begin{array}[]{cc}(s/t)^{2}&x/t\\ x/t&\text{Id}\end{array}\right),\quad\quad\left(\overline{m}^{\alpha\beta}\right)_{0\leq\alpha,\beta\leq n}=\left(\begin{array}[]{cc}1&x/s\\ x/s&\text{Id}\end{array}\right).$$

$$\mathcal{Z}=\mathcal{L}\cup \mathcal{P}\cup \mathcal{A}\cup \mathcal{H}$$

$$Z_i=\{\begin{array}{rlrl}& L_i,& & i=1,2,\cdots ,n,\\ & {\partial }_{i-(n+1)},& & i=n+1,\cdots ,2n+1,\\ & (s/t){\partial }_{i-2(n+1)},& & i=2n+2,\cdots ,3n+2,\\ & {\underline{\partial }}_{i-(3n+2)},& & i=3n+3,\cdots ,4n+2.\end{array}$$

$$Z^I:=Z_{i_1}{Z}_{i_2}\cdots Z_{i_N}$$

$${\partial }^I{L}^{J}u:={\partial }^I\left(L^{J}u\right).$$

$$\mathcal{G}(I):=\{(k,i_k)|k=1,2,\cdots N\}$$

$$\mathcal{A}=\left\{(k,i_k)|k\in A\text{ finite subset of }\mathbb{Z},i_k\in \{1,2,\cdots 4n+2\}\right\},$$

$$A=\text{D}(\mathcal{A}).$$

$${\mathcal{I}}_i\cap {\mathcal{I}}_j=\varnothing ,\mathcal{I}=\bigcup _{k=1}^m{\mathcal{I}}_k.$$

$${\mathcal{I}}_j\prec {\mathcal{I}}_k,\text{if}\min D({\mathcal{I}}_j)<\min D({\mathcal{I}}_k).$$

$${\mathcal{I}}_j\prec {\mathcal{I}}_k,\text{if}j<k.$$

$$Z_{j_l}=\{\begin{array}{rlrl}& Z_{i_k},& & \text{ if }l=k\in \text{D}(\mathcal{A}),\\ & 1,& & \text{ otherwise}.\end{array}$$

$$Z^I(u_1\cdot u_2)={\partial }_{\alpha }^2(u_1\cdot u_2)={\partial }_{\alpha }^2{u}_1\cdot u_2+u_1\cdot {\partial }_{\alpha }^2{u}_2+{\partial }_{\alpha }{u}_1\cdot {\partial }_{\alpha }{u}_2+{\partial }_{\alpha }{u}_1\cdot {\partial }_{\alpha }{u}_2.$$

$$Z^I(u_1\cdot u_2)={\partial }_{\alpha }^2(u_1\cdot u_2)={\partial }_{\alpha }^2{u}_1\cdot u_2+u_1\cdot {\partial }_{\alpha }^2{u}_2+{\partial }_{\alpha }{u}_1$$