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We use new methods, specific of non-locally convex quasi-Banach spaces, toinvestigate when the quasi-greedy bases of a $p$-Banach space for $0<p<1$ aredemocratic. The novel techniques we obtain permit to show in particular thatall quasi-greedy bases of the Hardy space $H_p(\mathbb{D})$ for $0<p<1$ aredemocratic 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, \textit{Quasi-greedy bases in $\ell_p$ ($0<p<1$)are democratic}, J. Funct. Anal. \textbf{280} (2021), no. 7, 108871, 21].Applications of our results to other spaces of interest both in functionalanalysis and approximation theory are also provided.

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