Singular integrals and maximal functions associated with highly monotone curves

Abstract

Let γ : [ − 1 , 1 ] → R n \gamma :[ - 1,1] \to {{\mathbf {R}}^n} be an odd curve. Set \[ H γ f ( x ) = PV ∫ f ( x − γ ( t ) ) ( d t / t ) {H_\gamma }f(x) = {\text {PV}}\int {f(x - \gamma (t))\,(dt/t)} \] and \[ M γ f ( x ) = sup h − 1 ∫ 0 h | f ( x − γ ( t ) ) | d t {M_\gamma }f(x) = \sup {h^{ - 1}}\int _0^h {|f(x - \gamma (t))|\,dt} \] . We introduce a class of highly monotone curves in R n {{\mathbf {R}}^n} , n ⩾ 2 n \geqslant 2 , for which we prove that H γ {H_\gamma } and M γ {M_\gamma } are bounded operators on L 2 ( R n ) {L^2}({{\mathbf {R}}^n}) . These results are known if γ \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.