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Subrearrangement-invariant function spaces

Rearrangement-invariance in function spaces can be viewed as a kind of generalization of 1-symmetry for Schauder bases. We define subrearrangement-invariance in function spaces as an analogous generalization of 1-subsymmetry. It is then shown that every rearrangement-invariant function space is also subrearrangement-invariant. Examples are given to demonstrate that not every function space on $(0,\infty)$ admits an equivalent subrearrangement-invariant norm, and that not every subrearrangement-invariant function space on $(0,\infty)$ admits an equivalent rearrangement-invariant norm. The latter involves constructing a new family of function spaces inspired by D.J.H.\ Garling, and we further study them by showing that they are Banach spaces containing copies of $\ell_p$.