Averaging and orthogonal operators on variable exponent spaces Lp(⋅)(Ω)Lp(⋅)(Ω)

Given a measurable space (Ω,μ)(Ω,μ) and a sequence of disjoint measurable subsets A=(An)nA=(An)n, the associated averaging projection PAPA and the orthogonal projection TATA are considered. We study the boundedness of these operators on variable exponent spaces Lp(⋅)(Ω)Lp(⋅)(Ω). These operators are unbounded in general. Sufficient conditions on the sequence A   in order to achieve that PAPA or TATA be bounded are given. Conditions which provide the boundedness of PAPA imply that TATA is also bounded. The converse is not true. Some applications are given. In particular, we obtain a sufficient condition for the boundedness of the Hardy–Littlewood maximal operator on spaces Lp(⋅)(Ω)Lp(⋅)(Ω).