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-Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces $\boldsymbol {H_{p}({\mathbb {D}}^{d})}$
We use new methods, specific for 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 [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.
