Estimates for Hyperbolic Equations of Space Dimension 3

✍ Scribed by Mitsuru Sugimoto

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 465 KB
• Volume: 160
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3296

✦ Synopsis

We discuss the problem of boundedness from L p (R n ) to L p$ (R n ) (1Âp+1Âp$=1, 1 p 2) of operators of the type M=F &1 e i.(!) a(!) F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property of 7=. &1 (1), and its dependence has been exactly determined in the cases n=2, 1 p 2 and n 3, p=1, 2 (M. Sugimoto, Math. Z. 215 (1994), 519 531; 222 (1996), 521 531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest case n=3.

1998 Academic Press

Here P(D t , D x ) is a homogeneous constant coefficient partial differential operator of degree m in the time t and the space x # R n , which is strictly article no.