ScalingStacks

Lemma 6.9 . [02HX]

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Lemma 6.9.

For every δ<1\delta<1 there exists a constant C>0C>0 independent of ϵ\epsilon such that

‖u​v‖Cδ−10,α≤C​ϵδ−1​‖u‖Cδ−10,α​‖v‖Cδ−10,α.\|u\,v\|_{C^{0,\alpha}_{\delta-1}}\leq C\epsilon^{\delta-1}\|u\|_{C^{0,\alpha}_{\delta-1}}\|v\|_{C^{0,\alpha}_{\delta-1}}.
Proof.

From the definition of the Cδ−10,αC^{0,\alpha}_{\delta-1}–norm it is immediate to check that

‖u​v‖Cδ−10,α≤C​‖ρϵδ−1‖C0​‖u‖Cδ−10,α​‖v‖Cδ−10,α.\|u\,v\|_{C^{0,\alpha}_{\delta-1}}\leq C\|\rho_{\epsilon}^{\delta-1}\|_{C^{0}}\|u\|_{C^{0,\alpha}_{\delta-1}}\|v\|_{C^{0,\alpha}_{\delta-1}}.

Since ρϵ≥c​ϵ\rho_{\epsilon}\geq c\,\epsilon and δ−1<0\delta-1<0 the result follows. ∎

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