Marcinkiewicz integrals with non-doubling measures

Let mu, be a positive Radon measure on R-d which may be non doubling. The only condition that mu must satisfy is mu(B(x, r)) <= Cr-n for all x is an element of R-d, r > 0 and some fixed constants C > 0 and n is an element of (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some Hormander-type condition, and assume that it is bounded on L-2(mu). We then establish its boundedness, respectively, from the Lebesgue space L-1(mu) to the weak Lebesgue space L-1,L-infinity (mu), from the Hardy space H-1 (mu) to L-1 (mu) and from the Lebesgue space L-infinity (mu) to the space RBLO(mu). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L-p (mu) with p is an element of (1, infinity). Moreover, we establish the boundedness of the commutator generated by the RBMO(mu) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger Hormander-type condition, respectively, from L-p (mu) with p is an element of (1, infinity) to itself, from the space L log L(mu) to L-1,L-infinity(mu) and from H-1(mu) to L-1,L-infinity (mu). Some of the results are also new even for the classical Marcinkiewicz integral.