Averaging and orthogonal operators on variable exponent spaces

Given a measurable space ( Ω , μ ) and a sequence of disjoint measurable subsets A = ( A n ) n , the associated averaging projection P A and the orthogonal projection T A are considered. We study the boundedness of these operators on variable exponent spaces L p ( ⋅ ) ( Ω ) . These operators are unbounded in general. Sufficient conditions on the sequence A in order to achieve that P A or T A be bounded are given. Conditions which provide the boundedness of P A imply that T A 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 L p ( ⋅ ) ( Ω ) .