Regularity of the Hardy-littlewood Maximal Operator on Block Decreasing Functions
نویسنده
چکیده
We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the l∞-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.
منابع مشابه
On the Regularity of Maximal Operators
We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W (R) × W (R) → W (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versi...
متن کاملBest Constants for Uncentered Maximal Functions
We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on Lp(R1). Consequently, we compute the operator norm of the “strong” maximal function on Lp(Rn), and we observe that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on Lp(Rn) grows exponentially as n → ∞. For a locally integrable function f on R, let (Mnf)(x) = sup B x 1...
متن کاملCórdoba–fefferman Collections in Harmonic Analysis
Córdoba–Fefferman collections are defined and used to characterize functions whose corresponding maximal functions are locally integrable. Córdoba–Fefferman collections are also used to show that, if Mx and My respectively denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in R, MHL denotes the standard Hardy–Littlewood maximal operator in R,...
متن کاملThe Best Constant for the Centered Maximal Operator on Radial Functions
We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.
متن کاملMathematische Zeitschrift Some remarks on the Hardy-Littlewood maximal function on variable L spaces
We show that any pointwise multiplier for BMO(R) generates a function p from the class P(Rn) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L space. In particular, this gives a positive answer to Diening’s conjecture saying that there are discontinuous functions which nevertheless belong to P(Rn). Mathematics Subject Classification (2000): 42B25
متن کامل