\(L^p\)-Sobolev regularity for integral operators over certain hypersurfaces (Q2877799)

In this paper, the authors prove sharp \(L^p\)-regularity estimates for averaging operators with convolution kernel associated to hypersurfaces in \(\mathbb{R}^d\) (\(d\geq2\)) of the form \(y\rightarrow (y,\gamma(y))\), where \(y\in\mathbb{R}^{d-1}\) and \(\gamma(y):=\sum_{i=1}^{d-1}\pm|y_i|^{m_i}\) for \(2\leq m_1\leq\cdots\leq m_{d-1}<\infty\). Precisely, for any smooth function \(f\) on \(\mathbb{R}^d\), let the averaging operator \(\mathcal{A}\) be defined by NEWLINE\[NEWLINE \mathcal{A}f(x):=\int_{\mathbb{R}^{d-1}}f(x-(y,\gamma(y)))\chi(y)\,dy, NEWLINE\]NEWLINE where \(x\in\mathbb{R}^d\) and \(\chi\) is a smooth function with a compact support near the origin with \(\chi(0)\neq0\).NEWLINENEWLINEThen, for \(\alpha\in[0,\infty)\) and \(p\in(1,\infty)\), the \(L^p\)-Sobolev space \(L^p_{\alpha}(\mathbb{R}^d)\) is defined as the class of all suitable functions \(f\) such that NEWLINE\[NEWLINE \|f\|_{L^p_{\alpha}(\mathbb{R}^d)} :=\|[(1+|\cdot|^2)^{\alpha/2}\widehat{f}]^{\vee}\|_{L^p(\mathbb{R}^d)}<\infty. NEWLINE\]NEWLINE Moreover, let \(p\in[2,\infty)\), and \(\nu_k\) and \(\alpha(p)\) be defined by \(\nu_k:=\sum_{j=k}^{d-1}\frac1{m_j}\) for \(k\in\{1,\ldots,d-1\}\), \(\nu_d:=0\) and \(\alpha(p):=\min_{k\in\{1,\ldots,d\}}\{\nu_k+\frac{k-1}{p}\}\).NEWLINENEWLINESuppose that \(p\in[2,\infty)\) and \(2\leq m_1\leq\cdots\leq m_{d-1}<\infty\). The authors prove that the operator \(\mathcal{A}\) is bounded from \(L^p(\mathbb{R}^d)\) to \(L^p_{\alpha}(\mathbb{R}^d)\) if and only if either \(p=m_i\) and \(\alpha<\alpha(p)\), or \(p\neq m_i\) and \(\alpha\leq\alpha(p)\), where \(i\in\{1,\ldots,d-1\}\).