(q, r) boundedness of general rough pseudo-differential operators

Abstract

In this paper, we consider the (q,r) boundedness of the pseudo-differential operators with the amplitude a\in L^p{S}_{\rho }^m(p\ge 1,m\in \mathbb{R},0\le \rho \le 1). When , 1⩽p, q\le \mathrm{\infty }, r\le p, \frac{1}{r}\le \frac{1}{p}+\frac{1}{q}, we prove that if

　　　　　　　　　　

then for any a\in L^p{S}_{\rho }^m, the pseudo-differential operator T_a is bounded from L^q to L^r. It is a generalization and improvement of the known theorems and in general the conditions on r, m are sharp.

Keywords

Pseudo-differential operator / rough Hrmander class / (q, r) boundedness

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