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I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$:

$\left(\int\left|\left|\psi(g)\right|+\delta\sum_{j=1}^{k}\left(\left|g_{j}\right|+G_{j}\right)\right|^{2}\right)^{1/2}\geq\delta\sum_{j=1}^{k}\left(\int\left|G_{j}\right|^{2}\right)^{1/2}$

$\psi$ maps continuously from $\mathbb{R}_k$ to a real number. $G_j$ are envelopes for the function classes in which $g_j$ live, though I don't think this is important.

The expression looks a bit like the opposite of the triangle inequality for L2 norms.

The expression clearly holds for k=1, but an attempt at an induction didn't get me anywhere.

EDIT: observe also that the The CR inequality gets us nowhere, only making LHS bigger:

$\left(\int\left||\psi(g)|+\delta\sum_{j=1}^{k}\left(\left|g_{j}\right|+G_{j}\right)\right|^{2}\right)\leq\epsilon^{2}k\left(\delta\sum_{j=1}^{k}\int\left|\left(\left|g_{j}\right|+G_{j}\right)\right|^{2}dQ+\int\left|\psi(g)\right|^{2}\right)$

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