Boundedness of Marcinkiewicz functions.

The ${\displaystyle L^p}$ boundedness(1 < p < ∞) of Littlewood-Paley's g-function, Lusin's S function, Littlewood-Paley's ${\displaystyle g{\cdot }_{\lambda }}$-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 ${\displaystyle {\mu }_S^{\varrho }}$ and ${\displaystyle {\mu }_{\lambda }^{\cdot ,\varrho }}$ to S and ${\displaystyle g{\cdot }_{\lambda }}$. The definition of ${\displaystyle {\mu }_S^{\varrho }\left(f\right)}$ is as follows: ${\displaystyle {\mu }_S^{\varrho }\left(f\right)\left(x\right)={\left({ʃ}_{|y-x|<t}{|\frac{1}{t^{\varrho }}{ʃ}_{\left|z\right|\le t}\Omega \frac{z}{{\left|z\right|}^{n-\varrho }}f(y-z)dz|}^2\frac{dydt}{t^{n+1}}\right)}^{\frac{1}{2}}}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere ${\displaystyle S^{n-1}}$, and ${\displaystyle {ʃ}_{S^{n-1}}\Omega (x\prime )d\sigma (x\prime )=0}$. We show that if σ = Reϱ > 0, then ${\displaystyle {\mu }_S^{\varrho }}$ is ${\displaystyle L^p}$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then ${\displaystyle L^p}$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for ${\displaystyle {\mu }_{\lambda }^{\cdot ,\varrho }}$. Their boundedness in the Campanato space ${\displaystyle {\epsilon }^{\alpha ,p}}$ is also considered.