Endpoint Inequalities for Spherical Multilinear Convolutions

✍ Scribed by Jong-Guk Bak; Yong-Sun Shim

• Book ID: 102590488
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 376 KB
• Volume: 157
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3269

✦ Synopsis

Write _=(_ 1 , ..., _ n ) for an element of the sphere 7 n&1 and let d_ denote Lebesgue measure on 7 n&1 . For functions f 1 , ..., f n on R, define

Let R=R(n) denote the closed convex hull in R 2 of the points (0, 0), (1Ân, 1), ((n+1)Â(n+2), 1), ((n+1)Â(n+3), 2Â(n+3)), ((n&1)Â(n+1), 0). We show that if n 3, then the inequality

holds if and only if (1Âp, 1Âq) # R. Our results fill in the gap in the necessary and sufficient conditions when n 3 in Oberlin's previous work.

A negative result is given along with some positive results, when n=2, thus narrowing the gap in the necessary and sufficient conditions in this case. 1998 Academic Press

for, say, bounded Borel functions f 1 , ..., f n on R.

We would like to consider the problem of determining all pairs (1Âp, 1Âq) # [0, 1]_[0, 1] such that there is an inequality