A remark on the maximal operator for radial measures

Abstract

Abstract. The purpose of this paper is to prove that there exist measures dμ(x) = γ(x)dx, with γ(x) = γ 0 (|x|) and γ 0 being a decreasing and positive function, such that the Hardy-Littlewood maximal operator, M μ , associated to the measure μ does not mapThis result answers an open question of P. Sjögren and F. Soria. Statement of resultsLet μ be a non-negative measure in R n , finite on compact sets. If f ∈ L 1 loc (μ), we define the maximal operatorwhere the supremum is taken over all balls containing the point x. If we consider only balls centered at x, we obtain that the operator-associated maps from L 1 (dμ) into L 1,∞ (dμ) are said to be of weak type (1,1). This can be proved using the Besicovitch covering lemma. If μ is a doubling measure, then M μ is of weak type (1,1). This can be proved using the Vitali covering lemma. For n ≥ 2, if. A. Vargas in [4] showed that M μ being of weak type (1,1) is equivalent to μ being a doubling measure away from the origin. In [1] it is shown that the operator associated with the measure dμ(x) = e −|x| 2 dx is bounded on L p μ , n ≥ 2, for 1 < p ≤ ∞. P. Sjögren and F. Soria

Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

“…In [50] the authors provide a sufficient condition on a radial measure µ so that M b,µ satisfies certain weak type inequalities close to L 1 (µ) which in turn imply that M b,µ ∶ L p (µ) → L p (µ). An example of a radial measure µ such that M b,µ is unbounded on all L p (µ)-spaces, p < +∞, can be found in [25]. Note that in the non-doubling case, the operators M Q,µ and M b,µ can behave quite differently, unlike the doubling case.…”