On the regularity of the one-sided Hardy-Littlewood maximal functions

Abstract

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A Note on the Regularity of the Two-Dimensional One-Sided Hardy-Littlewood Maximal Function

“…We remark that M + maps 1,∞ (R) into 1,∞ (R) boundedly by similar arguments to those in [1,Remark (iii)]. Recently, Liu and Mao [7] proved the following result.…”

“…Inequality (14) is not optimal, and it was asked in [29] whether the sharp constant for inequality (14) is in fact = 2; this question was resolved in the affirmative by Madrid in [31]. Later on, the above results were extended to a fractional case in [17,32,33], to a one-sided case in [7], and to a high dimensional case in [34]. For other interesting works we can consult [19,20,[35][36][37].…”

“…For other interesting works we can consult [19,20,[35][36][37]. In particular, Liu and Mao [7] investigated the regularity of the one-dimensional discrete one-sided Hardy-Littlewood maximal operator…”

“…In 1997, Kinnunen [1] first studied the Sobolev regularity of the usual centered Hardy-Littlewood maximal function M and showed that M is bounded on the firstorder Sobolev spaces 1, (R ) for all 1 < ≤ ∞ (the same conclusion also holds for the uncentered versionM by a simple modification of Kinnunen's arguments or [2,Theorem 1]). Later on, Kinnunen's result was extended to a local version in [3], to a fractional version in [4] and to a multisublinear version in [5,6] and to a one-sided version in [7]. Due to the lack of the sublinearity for the derivative of the maximal function, the continuity of M :…”

Operator Inequalities of Morrey Spaces Associated with Karamata Regular Variation

“…The reasons to study one-sided operators involve the requirements of ergodic theory [11]. The study of weighted theory for onesided operators was first introduced by Sawyer [12] and many authors thereafter ( [13][14][15][16][17][18][19]). The one-sided Hardy-Littlewood maximal operators [12] are defined by…”