Democracy of quasi-greedy bases in -Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces

Abstract

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a -Banach space for 0 < p < p are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy Hp(D) space for 0 < p < 1 are democratic while, in contrast, no quasi-greedy basis of Hp(Dd) for d > 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.