Maximal functions associated with Fourier multipliers of Mikhlin-H�rmander type

Abstract

We show that maximal operators formed by dilations of MikhlinHörmander multipliers are typically not bounded on L p (R d ). We also give rather weak conditions in terms of the decay of such multipliers under which L p boundedness of the maximal operators holds.

Maximal functions of multipliers on compact manifolds without boundary

“…where the operator Q is defined in (8). For a smooth function ψ ∈ C ∞ (1/2, 1), abusing a notation a bit, we set ψ j (·) := ψ(·/2 j ) for j ∈ N and…”

A Cotlar type maximal function associated with Fourier multipliers

“…Let χ ∈ C ∞ c (R) be supported in (1/2, 2) such that ∞ j=−∞ χ(2 j t) = 1 and let φ = χ(|.|). For a given Schwartz function f , let Sf := F −1 [mf ], and for n ∈ Z let S n be defined by (2) S n f (ξ) := j≤n φ(2 −j ξ)m(ξ)f (ξ).…”

Multilinear and multiparameter spectral multipliers on stratified groups

“…Then Mihlin and Hömander prove Mihlin multiplier theorem and Hörmander multiplier theorem. After that, plenty of authors have been focusing on investigating the boundedness properties of Fourier multipliers, for instance, Besov, Christ et al, Lu et al, Tomita, Yabuta, Grafakos et al, Chen et al, Noi, Yang et al, and Zhao et al…”

“…Then Mihlin 2 and Hömander 3 prove Mihlin multiplier theorem and Hörmander multiplier theorem. After that, plenty of authors have been focusing on investigating the boundedness properties of Fourier multipliers, for instance, Besov, 4 Christ et al, 5 Lu et al, 6 Tomita, 7 Yabuta, 8 Grafakos et al, [9][10][11] Chen et al, 12 Noi, 13 Yang et al, 14 and Zhao et al 15,16 Since Fourier multipliers are special cases of spectral multipliers, it is very natural to consider whether there exist some similar properties for other kinds of spectral multipliers. After the forerunner work made by Wendel,17 there appear a number of studies that focus on investigating multiplier theorems for spectral multipliers on some kinds of nilpotent Lie groups, for example, Alexopoulos, 18 Chen et al, 19 Gong and Yan, 20 Christ, 21 Christ and Müller, 22 Duong, 23 Kolomoitsev, 24 Lin, 25 Martini, 26 Mauceri and Meda, 27 Michele and Mauceri, 28 and Pini.…”