Estimates for the kernel and continuity properties of pseudo-differential operators
This work studies the continuity of pseudo-differential operators in H6rmander's class .~emn (cf. [11]) in several function spaces, including L p spaces, Hardy spaces, weak L 1 and BMO. The basic assumption throughout the paper is that 0< Q<= 1, 0<_-6<I, in particular, all our results are valid for 6=>Q. The stress is on sharp conditions over the order and type of the operators. Our point of view is that in many spaces continuity should follow from the functional calculus and simple computations, once L 2 estimates and suitable estimates for the kernel are known. Thus, we prove three different types of estimates for kernels of pseudo-differential operators: pointwise, integral and "dyadic integral" in w 1, w 2 and w 5 respectively; the first two types extend [14, p. 1053] and [1, p. 75], the last one may be new. Then we combine these estimates with the L2-continuity results proved in [13] to obtain (L 1, weak L~), (L ~, BMO), (L p, L q) and (H p, L p) continuity conditions that extend or improve results due to C. Fefferman [9], L. H6rmander [11] and J. Alvarez and M. Milman [1] (most results are classic for 6< Q or 6_<-0). We also prove a pointwise estimate for the sharp maximal function (Lf) ~ in terms of the generalized Hardy--Littlewood maximal function Mpf for some pseudo-differential operators L extending [2, p. 424]. It is well-known that these pointwise estimates give weighted L p estimates for L. When 0= 1, pseudo-differential operators of non-positive order are associated to standard kernels, i.e., they are generalized Calderdn--Zygmund operators. However, when ~<1, in order to obtain the best continuity properties, one is led to consider kernels that blow up at the diagonal faster than standard kernels. It is then natural to ask to what extent properties valid for operators associated to stan-