Democracy of quasi-greedy bases in $p$-Banach spaces with applications to the efficiency of the TGA in the Hardy spaces $H_p({\mathbb{D}}^d)$

We use new methods, specific of non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0<p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p(\mathbb{D})$ for $0<p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb{D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [F. Albiac, J. L. Ansorena, and P. Wojtaszczyk, $\mathit{\text{Quasi-greedy}}\mathit{\text{bases}}\mathit{\text{in}}{\mathrm{\ell }}_p\mathit{\text{(}}0<p<1\mathit{\text{)}}\mathit{\text{are}}\mathit{\text{democratic}}$, J. Funct. Anal. $\mathbf{\text{280}}$ (2021), no. 7, 108871, 21]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.