A note on maximal operator on ℓ{pn} and Lp(x)(ℝ)

Abstract

We consider a discrete analogue of Hardy-Littlewood maximal operator on the generalized Lebesque space ℓ{pn} of sequences defined on ℤ. It is known a necessary and sufficient condition P which guarantees an existence of a real number p>1 such that the norms in the space ℓ{pn} and in the classical space ℓp are equivalent. Of course, this condition immediately implies the boundedness of maximal operator on ℓ{pn} and, moreover, lim⁡|n|→∞⁡pn=p. We construct two examples of sequences {pn} satisfying lim⁡|n|→∞⁡pn=p in this paper. In the first example the maximal operator is unbounded on ℓ{pn} and the sequence {pn} from the second example does not satisfy P but the maximal operator is bounded. Moreover, it is known a sufficient integral condition to a behavior of a function p(x) at infinity which guarantees the boundedness of the maximal operator on Lp(⋅)(ℝn). As a main result of this paper we construct a function p(x) which does not satisfy this integral condition nevertheless the maximal operator is bounded.