Boundedness of Marcinkiewicz functions

The L-p boundedness (1 < p < infinity) of Littlewood-Paley's g-function, Lu sin's S function, Littlewood-Paley's g(lambda)*-functions, and the Marcinki ewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley's g-function. In this note, we tr eat counterparts mu(S)(rho) and mu(lambda)(*,rho) to S and g(lambda)*. [GRAPHICS] where Omega(x) is a homogeneous function of degree 0 and Lipschitz continuo us of order beta (0 < beta less than or equal to 1) on the unit sphere Sn-1 , and integral(Sn-1) Omega(x') d sigma(x') = 0. We show that if sigma = Re rho > 0, then mu(S)(rho) is L-p bounded for max(1, 2n/(n + 2 sigma) < p < i nfinity, and for 0 < rho less than or equal to n/2 and 1 less than or equal to p less than or equal to 2n/(n + 2 rho), L-p boundedness does not hold i n general, in contrast to the case of the S function. Similar results hold for mu(lambda)(*,rho). Their boundedness in the Campanato space E-alpha,E-p is also considered.