Rough singular integrals supported on submanifolds
The authors show that the rough singular integral operator \(\mathcal T_{\Omega,\mathcal P}\) along polynomials \(\mathcal P = (p_1 , \dots , p_d)\) is bounded on the Triebel-Lizorkin spaces \(\dot{F}_\alpha^{p,q}(\mathbb R^d)\) and Besov spaces \(\dot{B}_\alpha^{p,q}(\mathbb R^d)\), where \(\mathcal T_{\Omega,\mathcal P}\) is defined by \[ \mathcal T_{\Omega,\mathcal P}f(x)=\text{ p.v. }\int_{\mathbb R^n}\frac{b(|y|)\Omega(y)}{|y|^n} f(x-\mathcal P(y))dy,\quad x\in\mathbb R^d, \] with \(\Omega\in H^1 (S^{n-1})\), \(\Omega(\lambda y) =\Omega (y)\) for \(\lambda > 0\), and \(\int_{S^{n-1}}\Omega (y') d\sigma(y') = 0\). Moreover, \(\mathcal P = (p_1 , \dots , p_d)\) is a mapping from \(\mathbb R^n\) into \(\mathbb R^d\) with \(p_j\) being polynomials in \(y\in\mathbb R^n\). The results obtained in the paper under review are extensions of \textit{D. Fan} and \textit{Y. Pan}'s result [Am. J. Math. 119, No. 4, 799--839 (1997; Zbl 0899.42002)]. To get the above results, the authors establish an \(l^q\)-valued \(L^p (\mathbb R^d)\)-boundedness of the maximal operator \[ \mathcal M_{\mathcal P}f(x):=\sup_{r>0}\frac{1}{r^n}\int_{\{y\in\mathbb R^n:|y|\leq r\}}|f(x-\mathcal P(y))|dy,\quad x\in\mathbb R^d, \] along polynomials \(\mathcal P = (p_1 , \dots, p_d)\), which was defined by \textit{E. M. Stein} in 1986 [Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 196--221 (1987; Zbl 0718.42012)].
scientific article; zbMATH DE number 5712604
scientific article