On singular integral operators with rough kernel along surface
Let \(\Omega\) be a homogeneous function of degree 0 on \(\mathbb{R}^n\), with \(\Omega \in L^1(S^{n-1})\) and \( \int \Omega (y') d\sigma (y')=0. \) Suppose that \(\Phi\) is a nonnegative (or nonpositive) and monotonic \(C^1\) function on \((0, \infty)\) such that \( \varphi (t) := \frac{\Phi(t)}{t \Phi'(t)} \) is bounded. We say that \(b \in \varDelta_{\gamma}\) if \[ \sup_{R>0} \int_0^R | b(t) |^{\gamma} dt < \infty. \] We define the singular integral operator \(T_{\Omega, \Psi, b}\) by \[ T_{\Omega, \Phi, b}(f)(x) = \text{p.v.} \int_{\mathbb{R}^n} b( | y | ) \frac{\Omega (y')} {| y |^{n}} f(x - \Phi ( | y |)dy. \] The authors prove the following: If \(\Omega \in H^1(S^{n-1})\) (Hardy space on the unit sphere) and \(b \in \varDelta_{\gamma}\) for some \(\gamma >1\), then \(T_{\Omega, \Phi, b}\) is a bounded operator on \(L^p( \mathbb{R}^n)\) for \( | 1/p -1/2 | < \min (1/2, 1/\gamma')\). They can weaken the assumptions on \(\Phi\) in the paper by \textit{D. Fan} and \textit{Y. Pan} [Proc. Am. Math. Soc. 125, No. 12, 3695--3703 (1997; Zbl 0887.42009)]. For the proof they use the formula \[ \text{p.v.} \int_{\mathbb{R}^n} b( | y | ) \frac{\Omega (y')} {| y |^{n}} f(x - \Phi ( | y |)dy = \text{p.v.} \int_{\mathbb{R}^n} b( \Phi^{-1}(| y | ) ) \varphi( \Phi^{-1}(| y | ) ) \frac{\Omega (y')} {| y |^{n}} f(x - y)dy \] and transfer the effect of \(\Phi\) to the \(b\) part. They also consider weighted estimates and commutator operators.
scientific article; zbMATH DE number 5808431
scientific article