×

Boundedness of Marcinkiewicz functions. (English) Zbl 0930.42009

Summary: The \(L^p\) boundedness \((1<p<\infty)\) of Littlewood-Paley’s \(g\)-function, Lusin’s \(S\) function, Littlewood-Paley’s \(g^*_\lambda\)-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 \(\mu^\varrho_S\) and \(\mu^{*,\varrho}_\lambda\) to \(S\) and \(g^*_\lambda\). The definition of \(\mu^\varrho_S(f)\) is as follows: \[ \mu^\varrho_S(f)(x)= \Biggl(\int_{|y-x|< t} \Biggl|{1\over t^\varrho} \int_{|z|\leq t} {\Omega(z)\over|z|^{n-\varrho}} f(y- z) dz\Biggr|^2{dy dt\over t^{n+1}}\Biggr)^{1/2}, \] where \(\Omega(x)\) is a homogeneous function of degree \(0\) and Lipschitz continuous of order \(\beta\) \((0<\beta\leq 1)\) on the unit sphere \(S^{n-1}\), and \(\int_{S^{n-1}} \Omega(x') d\sigma(x')= 0\). We show that if \(\sigma= \text{Re }\varrho> 0\), then \(\mu^\varrho_S\) is \(L^p\) bounded for \(\max(1,2n/(n+ 2\sigma))< p<\infty\), and for \(0<\varrho\leq n/2\) and \(1\leq p\leq 2n/(n+ 2\varrho)\), \(L^p\) boundedness does not hold in general, in contrast to the case of the \(S\) function. Similar results hold for \(\mu^{*,\varrho}_\lambda\). Their boundedness in the Campanato space \({\mathcal E}^{\alpha,p}\) is also considered.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: EuDML