Inclusions and noninclusion of spaces of convolution operators

Abstract:

Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type $(p,q)$ by $L_p^q(G)$. It is shown that, if $|1/q - 1/2| < |1/p - 1/2|$, then $L_q^q(G)$ is not included in $L_p^p(G)$. This result follows from estimates on the norms, in these spaces, of Rudin-Shapiro measures. The same method leads to a simple example of a convolution operator that is of strong type (q, q) for all q in the interval $(p,p’)$ but is not of restricted weak type (p, p) or of restricted weak type $(p’,p’)$. Other statements about noninclusion among the spaces $L_p^q(G)$ also follow from various assumptions about G. For instance, if G is unimodular, but not compact, $1 \leqslant p,q,r,s \leqslant \infty$, with $p \leqslant q$, and $\min (s,r’) < \min (q,p’)$, then $L_p^q(G)$ is not included in $L_r^s(G)$. Using Zafran’s multilinear interpolation theorem for the real method, it is shown that, if $1 < p < 2$, then there exists a convolution operator on G that is of weak type (p, p) but not of strong type (p, p); it is not known whether such operators exist when $p > 2$, but it is shown that if $p \ne 1,2,\infty$, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.