On Some Questions Related to the Maximal Operator on Variable L Spaces
نویسنده
چکیده
Let P(Rn) be the class of all exponents p for which the HardyLittlewood maximal operator M is bounded on Lp(·)(Rn). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupttype condition A under some restrictions on the behavior of p at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type ( p(·), p(·) ) property of M in terms of A for radially decreasing p. Finally, we construct an example showing that p ∈ P(Rn) does not imply p(·)− α ∈ P(Rn) for all α < p− − 1. Similarly, p ∈ P(Rn) does not imply αp(·) ∈ P(Rn) for all α > 1/p−.
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