SOME ESTIMATES FOR RADIAL FOURIER MULTIPLIER OPERATORS WITH SLOWLY DECAYING KERNELS

We consider the restriction to radial functions of a class of radial F ourier multiplier operators containing the Bochner-Riesz multiplier op erator. The convolution kernel K(rc) of an operator in this class deca ys 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(gam ma>0) gamma(n)/K(gamma.) f(x)/ associated with such a kernel. The ma ximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, a nd strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim(gamma -->infinity) gamma(n)K(gamma.) f(x) = cf(x) a.e. for radial f epsilon L-p(IRn), a less than or equal to p < b.