2021/04/08 by Moyan Qin, Qin, Moyan, Huoxiong Wu +3
Mathematics · #42B20 #42B25 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #msc:42B20 #msc:42B25
paper · pdf · doi:10.48550/arxiv.2104.03497
18 pages
arxiv created 2021/04/08 · openalex publication_date 2021/04/08 · arxiv updated 2021/04/09 · openalex created_date 2021/04/13 · openalex updated_date 2026/07/28
It is well known that the weak (1,1) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak Llog L type norm inequality. Let Φn(t)=t(1+(log+t)n-1) and the space LΦn(\mathbb Rn) be the set of all measurable functions on \mathbb Rn such that ‖f‖_LΦn(\mathbb Rn) :=‖Φn(|f|)‖_L1(\mathbb Rn)<∞. In this paper, we introduce a new weak norm space LΦn1,∞(\mathbb Rn), which is more larger than L1,∞(\mathbb Rn) space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that max\2n((n-1)!)-1,1\ is a lower bound for the best constant of the LΦn→ LΦn1,∞ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.