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 ${\cal H}$ by $$(1)\qquad {\cal H} f(x) = \lim_{\epsilon,\eta\to 0}\int_{\epsilon\le |s| \le1} \int_{\eta\le |t|\le1 }f (x-P(s,t)) {ds dt\over st}. $$ We determine a necessary and sufficient condition on $P(s,t)$ so that the operator ${\cal H}$ is bounded on $L^p({\msbm R})$ for $1 < p < \infty $.

AMS Subject Classification (1991): 42B20

Keyword(s): Calderón--Zygmund kernel, multiplier, van der Corput's lemma

Received August 26, 2008, and in revised form October 20, 2008. (Registered under 6072/2009.)