Lp bounds for Stein's spherical maximal operators

2023/03/15 by Liu, Naijia, Shen, Minxing, Song, Liang +1
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.08655

Abstract

Let \frak M^α be the spherical maximal operators of complex order α on \mathbb Rⁿ. In this article we show that when n≥ 2, suppose ‖\frak M^α f ‖_L^p(\mathbb Rⁿ) ≤ C‖f ‖_L^p(\mathbb Rⁿ) holds for some α and p≥ 2, then we must have \rm Re α≥ max \1/p-(n-1)/2, -(n-1)/p \. When n=2, we prove that ‖\frak M^α f ‖_L^p(\mathbb R²) ≤ C‖f ‖_L^p(\mathbb R²) if \rm Re α>max\1/p-1/2, -1/p\, and hence the range of α is sharp in the sense the estimate fails for \rm Re α<max\1/p-1/2, -1/ p\.

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