Boundedness of Marcinkiewicz functions.

The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley's g-function, Lusin's S function, Littlewood-Paley's $g*_λ$-functions, and the Marcinkiewicz 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 treat counterparts $μ_{S}^{ϱ}$ and $μ_{λ}^{*,ϱ}$ to S and $g*_λ$. The definition of $μ_{S}^{ϱ}(f)$ is as follows: $μ_{S}^{ϱ}(f)(x) = (ʃ_{|y-x| < t}| 1/t^{ϱ} ʃ_{|z|≤ t} Ω(z)/(|z|^{n-ϱ}) f(y-z) dz|^2 (dydt)/(t^{n+1}) )^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 0, then $μ_{S}^{ϱ}$ is $L^p$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then $L^p$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for $μ_{λ}^{*,ϱ}$. Their boundedness in the Campanato space $ε^{α,p}$ is also considered