Singular integrals and maximal functions associated with highly monotone curves

Abstract:

Let $\gamma :[ - 1,1] \to {{\mathbf {R}}^n}$ be an odd curve. Set \[ {H_\gamma }f(x) = {\text {PV}}\int {f(x - \gamma (t)) (dt/t)} \] and \[ {M_\gamma }f(x) = \sup {h^{ - 1}}\int _0^h {|f(x - \gamma (t))| dt} \] . We introduce a class of highly monotone curves in ${{\mathbf {R}}^n}$, $n \geqslant 2$, for which we prove that ${H_\gamma }$ and ${M_\gamma }$ are bounded operators on ${L^2}({{\mathbf {R}}^n})$. These results are known if $\gamma$ has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput’s estimate of trigonometric integrals.