Lemma 7 [057V]

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Lemma 7

Consider the energy $E_{t}(\varphi_{t})$ in the formalism for degenerating background metrics as in the set-up (1.7). Then we have for any $p,q>1$ and ${1\over p}+{1\over q}=1$ ,

\displaystyle E_{t}(\varphi_{t})\leq{c_{t}^{n}\over V_{t}}\,\|e^{nF_{t}}\|_{L^{q}}\,\|\varphi_{t}\|_{L^{p}}.

(5.1)