A note on a Marcinkiewicz integral operator (Q2747567)

Let \(\Omega\) be a homogeneous function of degree \(0\) satisfying NEWLINE\[NEWLINE\int_{S^{n-1}}\Omega(x') d\sigma(x')=0.NEWLINE\]NEWLINE Define NEWLINE\[NEWLINEF_{P,t}(x)=\int_{|y|<2^t}\frac {\Omega(y)}{|y|^{n-1}}f(x-P(|y|)y') dyNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\mu_P(f)(x)=\left(\int^\infty_{-\infty}|F_{P,t}(x)|^2 \frac {dt}{2^{2t}}\right)^{1/2},NEWLINE\]NEWLINE where \(P(t)=P_N(t)\) is a real polynomial on \(\mathbb R\) of degree \(N\) and satisfies \(P(0)=0\). If \(\alpha>0\) and \(\Omega\) satisfies NEWLINE\[NEWLINE\sup_{\xi\in S^{n-1}}\int_{S^{n-1}}|\Omega(y')|\left(\ln\frac 1{|\langle y',\xi\rangle|}\right)^{1+\alpha} d\sigma(y')<\infty,\tag \(*\) NEWLINE\]NEWLINE the authors prove that \(\mu_P\) is bounded in \(L^p(\mathbb R^n)\) for \(p\in \left(\frac {2+2\alpha}{1+2\alpha}, 2+2\alpha\right)\) and the bound of \(\mu_P\) is independent of the coefficients of \(P\). An analog of this result on the \(n\)-torus is also given. NEWLINENEWLINENEWLINEThe condition (\(\ast\)) was introduced by \textit{L. Grafakos} and \textit{A. Stefanov} [Indiana Univ. Math. J. 47, No. 2, 455-469 (1998; Zbl 0913.42014)]. They showed that there are functions \(\Omega\) that satisfy condition (\(\ast\)) for all \(\alpha>0\) but are not contained in the space \(L\log^+L\), and that every function \(\Omega\) that lies in \(L^q(S^{n-1})\) for any \(q>1\) satisfies condition (\(\ast\)) for all \(\alpha>0\).