Wavelet Bases in Banach Function Spaces

We show that if the Hardy–Littlewood maximal operator is bounded on a separable Banach function space X(R) and on its associate space X^′(R) , then the space X(R) has an unconditional wavelet basis. This result extends previous results by Soardi (Proc Am Math Soc 125:3669–3673, 1997) for rearrangement-invariant Banach function spaces with nontrivial Boyd indices and by Fernandes et al. (Banach Center Publ 119:157–171, 2019) for reflexive Banach function spaces. We specify our result to the case of Lorentz spaces L^p,q(R, w) , 1 < p< ∞, 1 ≤ q< ∞ with Muckenhoupt weights w∈ A_p(R).