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Abstract
In this dissertation we prove certain regularity properties of three unrelated families of operators arising from separate problems in harmonic analysis. The first result concerns the classical Bochner-Riesz operators $S\sp{\delta}$ on Euclidean spaces ${\rm\bf R}\sp{n}$ (as well as more general Riesz means on manifolds). By the work of C. Fefferman, P. Tomas, E. Stein, and M. Christ, one can obtain regularity results on these operators from $L\sp2$ restriction theory. We encompass these results by using the restriction theorem to prove an optimal weak-type estimate for the index $\delta={n-1\over2(n+1)},$ which is the sharpest possible result one can obtain from $L\sp2$ restriction theory alone.
The second result addresses the question of the pointwise convergence of various wavelet sampling methods. In applications one often samples a function $f=\sum\sb{j,k}a\sb{j,k}\psi\sb{j,k}$ by discarding all but the largest wavelet co-efficients, leaving a reconstructed function of the form $\sum\sb{\mid a\sb{j,k}\mid>\lambda}a\sb{j,k}\psi\sb{j,k}.$ We show that under general conditions the sampled function converges pointwise almost everywhere to the original function as $\lambda{\buildrel{.\enspace}\over{\rightarrow}}0.$ This is achieved by approximating the sampling operator by a linear sampling method whose convergence was established by Kelly, Kon, and Raphael.
Our final result extends recent work by M. Christ, J. L. Rubio de Francia, and A. Seeger, on weak (1,1) estimates for rough operators in Euclidean spaces, to more general homogeneous groups. This is still a work in progress by the author, and the results presented here are somewhat partial in nature. We show that a homogeneous singular integral convolution operator is of weak-type (1,1) if it is bounded on $L\sp2$ and the kernel is $L\log L$ on the sphere.
