On maximal functions for Mikhlin–Hörmander multipliers

Abstract

Given Mikhlin-Hörmander multipliers mi, i = 1, . . . , N , with uniform estimates we prove an optimal log(N + 1) bound in L p for the maximal function sup i |F −1 [mi f ]| and related bounds for maximal functions generated by dilations. These improve results in [7].

Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane

“…See also [12], where the same method had been first applied to the study of maximal one-dimensional multipliers, and [7]. We apply the same reduction, and then show that the resulting square function is bounded on all L p , 1 < p < ∞ uniformly in the cardinality of V (see (3.3) …”

“…We summarize the reduction to a square function in the following proposition; the proof is contained in the reference [12], see also [6].…”

Low regularity classes and entropy numbers

“…Instead of the Pichorides conjecture we shall then use the well known bounds for a martingale analogue, due to Chang, Wilson, and Wolff [2]. This philosophy also applies to the proof of Theorem 1.1; it has been used in other papers, among them [7], [8], [5] (see also references contained in these papers). …”

Maximal functions of multipliers on compact manifolds without boundary

“…[3,14,20]). Our study of Theorem 1.2 was motivated by the study of Grafakos-Honzik-Seeger [11] where the maximal function of multipliers was studied on the Euclidean space.…”

“…One part will be handled by modifying the argument of [11] and the other part will be estimated using the L p − L q bound results of the spectral projection operators.…”