Let (X, d, µ) be an Ahlfors metric measure space. We give sufficient conditions on a closed set F ⊆ X and on a real number β in such a way that d(x, F ) β becomes a Muckenhoupt weight. We give also some illustrations to regularity of solutions of partial differential equations and regarding some classical fractals.2010 Mathematics Subject Classification. Primary 28A25; Secondary 28A78.
We study mixed weak type inequalities for the commutator [b, T ], where b is a BMO function and T is a Calderón-Zygmund operator. More precisely, we prove that for everyand v ∈ A∞(u). Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of L log L type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight u and a radial function v which is not even locally integrable.2010 Mathematics Subject Classification. 42B20, 42B25.
In this paper we prove mixed inequalities for the maximal operator Φ , for general Young functions Φ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given ≥ 1, if , are weights belonging to the 1-Muckenhoupt class and Φ is a Young function as above, then the inequality ({ ∈ ℝ ∶ Φ ()() () > }) ≤ ∫ ℝ Φ (| ()|) () () dx holds for every positive. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of Φ. Moreover, it is well-known that for the particular case Φ() = (1 + log +) with ∈ ℕ these maximal functions control, in some sense, certain operators in harmonic analysis.
In this note we combine the dyadic families introduced by M. Christ in (Colloq. Math. 60/61(2):601-628, 1990) and the discrete partitions introduced by J.M. Wu in (Proc. Am. Math. Soc. 126 (5): [1453][1454][1455][1456][1457][1458][1459] 1998) to get approximation of a compact space of homogeneous type by a uniform sequence of finite spaces of homogeneous type. The convergence holds in the sense of a metric built on the Hausdorff distance between compact sets and on the Kantorovich-Rubinshtein metric between measures.
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