Singular integral operators supported in higher dimensional surfaces (Q6095351)

In the paper, the \(L^p\) mapping properties of a class of operators, called by the author singular integral operators with kernels supported in surfaces in \(\mathbb R^n\times \mathbb R^n\), are studied. Given a suitable mapping \(\gamma: \mathbb R^n\times \mathbb R^n\to \mathbb R^n\), and \(\Omega\in L^1(S^{n-1}\times S^{n-1})\) satisfying \[ \int_{S^{n-1}}\Omega(u^\prime,\cdot)\,d\sigma(u^\prime)=\int_{S^{n-1}}\Omega(\cdot,v^\prime)\,d\sigma(v^\prime)=0, \] and \(\Omega(tx,sy)=\Omega(x,y)\) for any \(t,s>0\) and \(x,y\in \mathbb R^n\), the operator \(T_{\gamma,\Omega}\) is defined by \[ T_{\gamma,\Omega}f(x)=\mathrm{p.v.} \int_{\mathbb R^n\times \mathbb R^n}f\big(x-\gamma(u,v)\big)\frac{\Omega(u',v')}{|u|^n|v|^n}\,du\,dv; \] here \(y^\prime=y/|y|\) for \(y\neq0\). It was proved by \textit{A. Al-Salman} [Banach J. Math. Anal. 16, Paper No. 48 (2022; Zbl 1495.42004)] that if \(\gamma=\mathcal P\), where \(\mathcal P=(P_1,\ldots,P_n)\) is a polynomial mapping, then with the additional assumption \[ \Omega\in L(\log L)^2(S^{n-1}\times S^{n-1}), \tag{1} \] the operator \(T_{\mathcal P,\Omega}\) is bounded on \(L^p(\mathbb R^n)\) for \(p\in(1,\infty)\); moreover, for \(p=2\), the exponent 2 cannot be replaced by \(2-\varepsilon\) for any \(\varepsilon>0\). The present paper treats general mappings \(\gamma_{\varphi_1,\varphi_2}\) of the form \[ \gamma_{\varphi_1,\varphi_2}(u,v)=\varphi_1(|u|)u^\prime+\varphi_2(|v|)v^\prime, \] where \(\varphi_1,\varphi_2: (0,\infty)\to\mathbb R\) are suitable functions. More precisely, three types of such functions are considered: I) semi-convex functions; II) functions satisfying specific growth conditions; III) restrictions to \((0,\infty)\) of one variable real-valued polynomials vanishing at 0. The main result of the paper, Theorem 1.4, says that under the additional assumption (1), for \(\varphi_1\), \(\varphi_2\) from any of these three classes of functions, the operator \(T_{\gamma_{\varphi_1,\varphi_2},\Omega}\) is bounded on \(L^p(\mathbb R^n)\) for \(p\in(1,\infty)\).