The Critical Smoothness of Generalized Functions

For each integrability parameter $p \in (0,\infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$ (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter $p$. Our main result is a simple characterization of all the possible critical smoothness functions $p\mapsto s_f(p)$ when $f$ describes the space of generalized periodic functions. We moreover characterize the compressibility of generalized periodic functions in wavelet bases from the knowledge of their critical smoothness function.