Convolution operators of weak type (2,2) which are not of strong type (2,2)

Abstract

It is well known that if G G is a locally compact and amenable group then the Banach spaces of operators of weak type ( 2 , 2 ) (2,2) and of strong type ( 2 , 2 ) (2,2) commuting with the right translations on G G are the same. In contrast we show that if G G is a nonabelian free group then there exists an operator of weak type ( 2 , 2 ) (2,2) commuting with the right translations on G G which is not of strong type ( 2 , 2 ) (2,2) .