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Table 1 Convergence ( 1.2 ) holds for \(\pmb{f\in H^{s}(\mathbb{R}^{n})}\)

From: Global \(L^{2}\) estimates for a class of maximal operators associated to general dispersive equations

Dim.

Range of s

Authors

n = 1

\(s\geq\frac{1}{4}\)

Carleson [1] in 1979

n ≥ 2

\(s>\frac{1}{2}\)

Sjölin [3] in 1987 and Vega [4] in 1988, independently

n = 2

for some \(s<\frac{1}{2}\)

Bourgain [5] in 1992

n = 2

s>κ with \(\frac{20}{41}<\kappa<\frac{41}{84}\)

Moyua, Vargas and Vega [6] in 1996

n = 2

\(s> \frac{15}{32}\)

Tao and Vargas [7] in 2000

n = 2

\(s> \frac{2}{5}\)

Tao [8] in 2003

n = 2

\(s>\frac{3}{8}\)

Lee [9] in 2006

n ≥ 3

\(s>\frac{1}{2}-\frac{1}{4n}\)

Bourgain [10] in 2013