| R. Fefferman has shown that if a Calderon-Zygmund kernel k(x) = (OMEGA)(x)/(VBAR)x(VBAR)('n), where the L('(INFIN)) modulus of continuity of (OMEGA) satisfies a Lipschitz condition, is multiplied by a bounded radial function b((VBAR)x(VBAR)), then the function H(x) = b((VBAR)x(VBAR))k(x) is the kernel of a bounded operator on L('P)(R('n)) for 1 < P < (INFIN) and n (GREATERTHEQ) 2.; We improve this result by proving that (OMEGA) (epsilon) L('q)(S('n-1)), for some 1 < q (LESSTHEQ) (INFIN), is sufficient for the above result. Also, using Spherical Harmonics, we derive some other conditions under which R. Fefferman's result holds. In Chapter Two, we discuss necessary and sufficient conditions on b((VBAR)x(VBAR)) so that this map can be extended to a bounded operator from L('(INFIN)) to BMO. We also furnish an example of a "bad" b((VBAR)x(VBAR)) for which the operator can not be extended to L('(INFIN)). In Chapter Three, the boundedness of this operator from the weighted Hardy Space H(,w)('1) to L(,w)('1) is discussed and necessary and sufficient conditions are derived. Finally in Chapter Four, weighted weak type (1, 1) results are considered. |
