@inbook{10.2307/j.ctt7zthns.13, URL = {http://www.jstor.org/stable/j.ctt7zthns.13}, abstract = {In this section we assume a space-time slab$\bigcup\nolimits_{[0,{{t}_{\star}}]}{{{\Sigma }_{t}}}$, foliated by a maximal time foliationtand an optical functionu, and we introduce our basic norms involving the curvature tensor R, second fundamental formk, and lapse functionϕ, as well as the optical quantitiesχ, ζ, ω.GivenS-tangent tensorfieldVwe first define${{\left| V \right|}_{p,S}}(t,u)={{\left( \int_{{{S}_{t,u}}}{{{\left| V \right|}^{p}}d{{\mu }_{\gamma }}} \right)}^{1/p}}\quad \text{if}\ 1\le p<\infty \caption {(10.1.1\text {a})}$$= \underset{{{S}_{t,u}}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty .$We also introduce the following norms defined in the interior and exterior regions$\Sigma _{t}^{i},\,\Sigma _{t}^{e}$of each slice:$\begin{matrix} {{\left\| V \right\|}_{p,i}}={{\left( \int_{\Sigma _{t}^{i}}{{{\left| V \right|}^{p}}} \right)}^{1/p}}\ \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{i}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty \\ {{\left\| V \right\|}_{p,e}}(t)={{\left( \int_{\Sigma _{t}^{e}}{{{\left| V \right|}^{p}}} \right)}^{1/p}}\quad \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{e}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty \\ {{\left\| \left| V \right| \right\|}_{p,e}}(t)=\underset{u\ge {{u}_{0}}(t)}{\mathop{\sup }}\,{{\left| V \right|}_{p,S}}(t,u)\quad \text{if}\ 1\le p<\infty \\ =\underset{\Sigma _{t}^{e}}{\mathop{\sup }}\,\left| V \right|\quad \text{if}\ p=\infty , \\ \end{matrix}\caption {(10.1.1\text {b})}$where we recall that$\Sigma _{t}^{i}=I$consists of points for which$r\le \frac{{{r}_{0}}(t)}{2}$while$\Sigma _{t}^{e}=E$consists of those for which$r\ge \frac{{{r}_{0}}(t)}{2}$, withr0(t) the}, bookauthor = {Demetrios Christodoulou and Sergiu Klainerman}, booktitle = {The Global Nonlinear Stability of the Minkowski Space (PMS-41)}, pages = {284--310}, publisher = {Princeton University Press}, title = {Third Version of the Main Theorem}, year = {1993} }