A class of singular integrals (Q1192127)

There has been work on singular integral operators whose kernels are obtained by multiplying a standard Calderón-Zygmund kernel, homogeneous of degree -n, by a radial function. For such a kernel \(K\) (to be described below), the authors are interested in finding a class of kernels \(M\) such that the maximal operator \[ Tf(x) = \sup_{K \in M} | p.v. \int K(y) f(x - y) dy| \] is bounded in \(L^ p\) for some \(p\). The Calderón-Zygmund kernel is \ \(\Omega(x)/| x| ^ n\), where \(\Omega\) is homogeneous of degree 0. For the kernel \( K(x) = h(| x| )\Omega(x)/| x| ^ n\), where \(M = \{h| \int_ 0^{\infty} | h(r)| dr/r \leq 1 \}\), \( T: L^ p(R^ n) \rightarrow L^ p(R^ n)\), for \(n \geq 2\) and \(n/(n-1) < p < \infty\). If \(M = \{h| \;\| h\| _{\infty} \leq 1 \}\), \(T\) is unbounded on all \(L^ p\). One of the authors, in joint work [\textit{L.-K. Chen} and \textit{H. Lin}, Ill. J. Math. 34, No. 1, 120-126 (1990; Zbl 0682.42012)], has shown that if \[ M = \{h| \left(\int_ 0^{\infty} | h(r)| ^ s dr/r \right)^{1/s} \leq 1 \}, \] then \(T\) maps \(L^ p\) into \(L^ p\) for \(n \geq 2\) and \(sn/(sn-1) < p < \infty\). In this paper the authors let \(\{h_ j\}\) be any countable subset of \(L^ 2(S^{n-1})\) with \(\int_{S^{n-1}} h_ j d\sigma = 0\) such that \[ \sum_ j \| h_ j\| _ 2^ 2 \leq\infty. \] The authors take \[ M = \{ K| K(r\xi) =r^{-n} \sum a_ j(r) h_ j(\xi), \int_ 0^{\infty} \sum_ j| a_ j(r)| ^ 2 dr/r \leq 1\}. \] Their main result is that \(T\) as defined above maps \(L^ p\) into \(L^ p\) for \(n \geq 2\) and \(2n/(2n-1) < p < \infty\) and that the range of \(p\) is best possible.