Square-summability decomposition for V^2 frequency components

Establish the estimate $\\left(\\sum_k\\|T_kv\\|_{U^2}^2\\right)^{1/2}\\lesssim\\|v\\|_{V^2}$ for orthogonal spatial frequency projections $T_k$ and arbitrary $v\\in V^2$, thereby obtaining the $V^2$ analogue of the known $U^2$ square-summability estimate.

Background

The resolution-space analysis uses the atomic spaces $U^p$ and variation spaces $V^p$. Proposition \ref{Pref} provides square-summability under orthogonal decompositions for $U^p$ when $p\leq2$, and a different summation estimate for $V^p$ when $p\geq2$.

The estimate needed in the paper is the $V^2$ counterpart of the $U^2$ square-summability bound. The authors explicitly state that this estimate is not known, and therefore replace it with a weaker frequency-localized lemma involving a logarithmic loss.