Boyd indices were originally introduced for rearrangement-invariant Banach function spaces (see for example [D. W. Boyd, Can. J. Math. 21, 1245–1254 (1969; Zbl 0184.34802)]). They play an important role in order to study interpolation of operators and also to characterize the boundedness of some important operators that appear in harmonic analysis.
In this paper, a generalization of Boyd indices to general Banach function spaces \(\mathbb{X}\) is introduced and they are used to study some functional properties of this kind of spaces. In particular, ithe author proves a characterization in terms of generalized Boyd indices of some norm inequalities involving the action of the dilation operator acting on euclidean balls that, at the same time, guarantee that the Schwartz class of functions \({\mathcal S}({\mathbb R}^N)\) is a dense subset in \(\mathbb{X}\).
As an application, it is studied whether \(\mathbb{X}^p\), \(1<p<\infty\) satisfies the Littlewood-Paley characterization or not. Also, the validity of Fefferman-Stein type vector-valued inequalities on the norm \(\| \| \cdot\|_{\ell^q}\|_{\mathbb{X}^p}\), \(1<p,q<+\infty\). These norm inequalities are the main tool to give atomic and molecular decompositions of the Triebel-Lizorkin type spaces associated with \(\mathbb{X}\).