Abstract. Let $P(s,t)$ denote a real-valued polynomial of real variables $s$ and $t$. For $f \in{\cal S}$ (i.e., a Schwartz class function), define the operator $T$ by (1) $ Tf(x,y) = \lim_{\epsilon,\eta\to 0}\int_{\epsilon\le |s| \le1} \int_{\eta\le |t|\le1 }f (x-s, y-P(s,t))_{\strut }^{\strut } {ds dt\over st}. $ We determine a necessary and sufficient condition on $P(s,t)$ so that the operator $T$ is bounded on $L^p({\msbm R}^2)$ for $1 < p < \infty $.

AMS Subject Classification (1991): 42B20

Keyword(s): Newton diagram, multiplier, van der Corput's lemma

Received January 20, 2010, and in revised form September 9, 2010. (Registered under 4/2010.)