Nonisotropic operators arising in the method of rotations

This thesis is concerned with the mapping properties of the related objects, Mf(x,w) := suph_1J(01h) f(x-6w)dt, h>O Mrf(x) sup h1 fo,h) f(x F(t)) dt h>O and their associated singular integral operators, H and Hr respectively. Here, it := exp((log t)P) and P is a real d by d matrix whose eigenvalues have positive real part, and F R -+ R° parameterises a curve. For p in (1, max(2, (d + 1)/2)], we prove that M maps L1' to LP(L) for an optimal range of q (modulo an endpoint). For H, the same optimality is achieved for pin (1, 2]. If F(t) = (t, P(7(t))), where P is a real polynomial and 'y is a convex function, then we give sufficient conditions in order for Mr, and Hr to be bounded on L, for all p in (1, cc), with bounds independent of the coefficients of P. We also consider when these operators map L log L to weak L' locally. The same conclusions are shown to hold for the corresponding hypersurface in Rd (d > 2) under weaker hypotheses on F. We give sufficient conditions on a convex curve F in R° (d > 2) in order for NCI, and Hi-j to map L log L to weak L' locally. Finally, it is shown that if F is a piecewise linear version of a parabola then the best one can expect, in terms of Orlicz spaces locally near L', is that Mr maps L(log L)1/2 to L"°°.