Singular integrals and maximal functions associated with highly monotone curves

W. C. Nestlerode
1981 Transactions of the American Mathematical Society  
Let y: [-1, 1] -»R" be an odd curve. Set HJ(x) = PV j fix -y(r)) (dt/t) and MJ(x) = sup h-lJo"\f(x -y(t))\ dt. We introduce a class of highly monotone curves in R", n > 2, for which we prove that Hy and My are bounded operators on L2(R"). These results are known if y 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.
doi:10.1090/s0002-9947-1981-0626482-0 fatcat:comv735lkbggng45lwwhkpwlae