Weak type (1,1) estimates for a class of discrete rough maximal functions

Abstract

We prove weak type (1, 1) estimate for the maximal function associated with the sequence [m α ], 1 < α < 1 + 1 1000 . As a consequence, the sequence [m α ] is universally L 1 -good.

On convergence of oscillatory ergodic Hilbert transforms

“…We give sufficient conditions for maximal truncations of discrete singular integral operators to have sparse bounds. Our argument has as its antecedents the Fefferman [13] and Christ [3] T T * approach to proving weak-type (1, 1) bounds for rough singular integral operators on Euclidean space; this approach has already appeared in the discrete context in the work of LaVictoire [19] where ℓ 1 → ℓ 1,∞ endpoint estimates for certain (random) maximal functions are proven, and in Urban and Zienkiewicz [24] and Mirek [20] where endpoint estimates for (deterministic) maximal functions taken over "thin" subsets of the integers are established.…”

On variants of the Halton sequence

“….) where I. g(n) = n ω if ω > 1 and ω / ∈ N. In fact there is a interval ω ∈ [1, a] for very small a where ([n ω ]) n≥1 is L 1 good universal [27]. II.…”

Weak type (1, 1) inequalities for discrete rough maximal functions

“…Therefore, it will cause no confusion if we use the same letter M h f in the definitions (1.2) and (1.8). The proof of Theorem 1.7 (see Section 6) will be based on the concepts of [20]. In [20] the authors used a subtle version of Calderón-Zygmund decomposition, which was pioneered by Fefferman [9] and later on developed by Christ [8], to study maximal functions.…”

“…The proof of Theorem 1.7 (see Section 6) will be based on the concepts of [20]. In [20] the authors used a subtle version of Calderón-Zygmund decomposition, which was pioneered by Fefferman [9] and later on developed by Christ [8], to study maximal functions. Fefferman's ideas turned out to be applicable to the discrete settings as it was shown in [20], and recently also in [12] and [7].…”

“…In [20] the authors used a subtle version of Calderón-Zygmund decomposition, which was pioneered by Fefferman [9] and later on developed by Christ [8], to study maximal functions. Fefferman's ideas turned out to be applicable to the discrete settings as it was shown in [20], and recently also in [12] and [7]. Heuristically speaking, the weak type (1, 1) bound of M h f is obtained by considering the recalcitrant part of the Calderón-Zygmund decomposition in ℓ 2 (see Lemma 6.6 and Theorem 6.1 in Section 6), using the fact that K h,N * f, K h,N * g = K h,N * K h,N * f, g , and decomposing K h,N * K h,N (x) 1 into several manageable pieces (a delta mass at 0, a slowly varying function G N (x), and a small error term E N (x), see Section 5) obtained by special Van der Corput estimates, (we refer to Section 3).…”