\(L^p(\mathbb{R}^d)\) boundedness for a class of nonstandard singular integral operators (Q6632426)

Let \(A\) be a function on \(\mathbb{R}^d\) such that \(\nabla A \in \operatorname{BMO}\left(\mathbb{R}^d\right)\) (\(d\ge 2\) ), and \(\Omega\) be homogeneous of degree zero, integrable on \(S^{d-1},\) and satisfy the vanishing moment condition that for all \(1 \leq n \leq d,\)\N\begin{align*}\N&\int_{S^{d-1}} \Omega\left(x^{\prime}\right) x_n^{\prime} d x=0.\N\end{align*}\NDefine the operator \(T_{\Omega, A}\) by \N\[\NT_{\Omega, A} f(x)=\text { p.v. } \int_{\mathbb{R}^d} \frac{\Omega(x-y)}{|x-y|^{d+1}}(A(x)-A(y)-\nabla A(y)(x-y)) f(y) d y.\N\]\NIn [ibid. 30, No. 3, Paper No. 32, 44 p. (2024; Zbl 1543.42020)], \textit{G. Hu} et al. showed that if \(\Omega \in L(\log L)^2(S^{d-1})\), then \(T_{\Omega, A}\) is bounded on \(L^p(\mathbb{R}^d)\) for all \(p \in(1, \infty)\). On the other hand, for \(\Omega \in L^1(S^{d-1})\) and \(\beta \in[1, \infty)\), we say that \(\Omega \in G S_\beta(S^{d-1})\) if\N\begin{align*}\N&\sup _{\zeta \in S^{d-1}} \int_{S^{d-1}}|\Omega(\theta)| \log ^\beta\left(\frac{1}{|\zeta \cdot \theta|}\right) d \theta<\infty.\N\end{align*}\NIn [Acta Math. Sin., Engl. Ser. 19, No. 2, 397--404 (2003; Zbl 1026.42017)], \textit{G. E. Hu} showed that if \(\Omega\in G S_\beta(S^{d-1})\) for some \(\beta>3\), then \(T_{\Omega, A}\) is bounded on \(L^2(\mathbb{R}^d)\). In the paper under review, the authors follow the line of research on the \(L^p\)-boundedness of \(T_{\Omega, A}\), and show that if \(\Omega\in G S_\beta(S^{d-1})\) for some \(\beta>2\), then \(T_{\Omega, A}\) is bounded on \(L^p(\mathbb{R}^d)\) for \(p\in (1+1/(\beta-1), \beta)\).