Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels

We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L^p -boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup_{r>0 r}ⁿ∣K(r) * f(x)∣ associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim_γ→ γⁿ K(γ·) * f(x) = cf(x) a. e. for radial f ? L^P(?ⁿ), a ≥ p > b.