Singular integrals associated to homogeneous mappings with rough kernels
Let \(n\geq 2\) and \(S^{n-1}\) be the unit sphere in \(\mathbb{R}^n\) equipped with the normalized Lebesgue measure \(d\sigma\). Suppose that \(\Omega\) is a homogeneous function of degree zero on \(\mathbb{R}^n\) that satisfies \(\Omega\in L(S^{n-1})\) and \[ \int_{S^{n-1}}\Omega(x)\,d\sigma= 0 \] For a suitable mapping \(\Psi: \mathbb{R}^n\to \mathbb{R}^m\), define the singular integral operator \(T\) and its truncated maximal operator \(T^*\) by \[ Tf(x)= \text{p.v.\;}\int_{\mathbb{R}^n} f(x- \Psi(y))\,\Omega(y) |y|^{-n} dy, \] and \[ Tf(x)= \sup_{\varepsilon> 0}\,\Biggl|\, \int_{|y|>\varepsilon} f(x- \Psi(y))\,\Omega(y)|y|^{-n} dy\Biggr |, \] respectively.The main result is the following theorem: Theorem. Suppose that \(\Psi\) is a homogeneous mapping of degree \(d= (d_1,\dots, d_m)\) with all \(d_j\neq 0\), \(j= 1,2,\dots, m\), and \(\Psi|_{S^{n-1}}\) is real analytic. If \(\Omega\) is in the block space \(B^{0,0}_q\) with \(q> 1\), then both \(T\) and \(T^*\) are bounded operators on \(L^p(\mathbb{R}^m)\), \(1< p<\infty\). Using this result, the authors establish an application on the oscillatory singular integrals.
scientific article; zbMATH DE number 2140982