2020/05/03 by Loukas Grafakos, Bae Jun Park, Grafakos, Loukas +1
Mathematics · #42B15 #42B25 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2005.01213
openalex publication_date 2020/05/03 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In this article, we provide a multilinear version of the H "ormander\nmultiplier theorem with a Lorentz-Sobolev space condition. The work is\nmotivated by the recent result of the first author and Slav 'ikov 'a where an\nanalogous version of classical H "ormander multiplier theorem was obtained;\nthis version is sharp in many ways and reduces the number of indices that\nappear in the statement of the theorem. As a natural extension of the linear\ncase, in this work, we prove that if mn/2<s<mn, then
big
Vert\nT
sigma(f1,
dots,fm)
big
Vert_Lp((
mathbbR)n)
lesssim\n
sup_k
in
mathbbZ
big
Vert\n
sigma(2k
;
vec
cdot
;)
widehat
Psi(m)
big
Vert_Lsmn/s,1(
mathbbRmn)
Vert\nf1
Vert_Lp1((
mathbbR)n)
cdots
Vert\nfm
Vert_Lpm((
mathbbR)n)\n for certain p,p1,\…,pm with 1/p=1/p1+\…+1/pm. We also show that\nthe above estimate is sharp, in the sense that the Lorentz-Sobolev space\nLsmn/s,1 cannot be replaced by Lsr,q for r<mn/s, 0<q\≤\n\∞, or by Lsmn/s,q for q>1.\n