Wavelet bases in rearrangement invariant function spaces

Abstract

We point out that the well known characterization of LP spaces (1 0 (3) la(f)IIx 0, (4) Ilr(f)Ix? < BIlflIx for all f E X. Conversely, let X' denote the associate Banach function space of X. Then X' is a r.i. Banach function space whose Boyd indices are a' = 1 d and d' = 1a, so that (3) holds with X' in place of X. Let f and g be bounded functions supported in sets of finite measure. Then f and g belong to L2 (R n), so that they have the expansion (5) f = (A) g = bb(A)OA. By the orthonormality of the system {? } we get j f-dx = j 5 c(A) Ab(A)A O dx. in in ,