\(L^p\) bounds for the parabolic Littlewood-Paley operator associated to surfaces of revolution (Q2902038)

Let \(\alpha_1, \dots, \alpha_n\) be fixed real numbers, \(\alpha_i \geq 1\), and set \(\alpha = \alpha_1 + \dots + \alpha_n\). For a fixed \(x\in\mathbb{R}^n\), the function \(F(x,\rho) = \sum_{i=1}^{n}\frac{x_i^2}{\rho^{2\alpha_i}}\) is a strictly decreasing function of \(\rho >0\) and thus there exists a unique \(\rho(x)\) such that \(F(x,\rho)=1\). This \(\rho\) came up in the work of \textit{E. B. Fabes} and \textit{N. M. Rivière} [Stud. Math. 27, 19--38 (1966; Zbl 0161.32403)], who showed that \(\rho\) is a metric on \(\mathbb{R}^n\). The space \((\mathbb{R}^n, \rho)\) is called the mixed homogeneity space related to \(\{ \alpha_i \}_{i=1}^{n}\). For \(\lambda > 0\), let \(A_{\lambda} = \mathrm{diag}[{\lambda}^{\alpha_{1}},\dots ,{\lambda}^{\alpha_{n}}]\) and let \(\Omega(x)\) be a real valued, measurable function that is homogeneous of degree zero with respect to \(A_{\lambda}\). Moreover, it is required that, \( \int_{S^{n-1}}\Omega(x')J(x')d\sigma(x') = 0 ,\) where \(d\sigma\) is the element of area of \(S^{n-1}\) and \(J(x')\) arises from the Jacobian when one switches to spherical coordinates in \(\mathbb{R}^n\).NEWLINENEWLINEThe paper studies \(L^p\) boundedness of the parabolic Littlewood-Paley operator associated to a surface of revolution, which is given by NEWLINE\[NEWLINE \mu_{\Phi, \Omega}(f)(x) = \left(\int_{0}^{\infty}|F_{\Phi,t}(x)|^2 \frac{dt}{t^3} \right)^{1/2} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE F_{\Phi,t}(x) = \int_{\rho(y)\leq t}\frac{\Omega(y)}{\rho(y)^{\alpha - 1}}f(x-\Phi(y))dy .NEWLINE\]NEWLINE This operator was first introduced and studied by \textit{Q. Y. Xue}, \textit{Y. Ding} and \textit{K. Yabuta} [Acta Math. Sin., Engl. Ser. 24, No. 12, 2049--2060 (2008; Zbl 1151.42007)] in the case where \(\Phi\) is the identity mapping. Their motivation came from the previously mentioned paper by E. Fabes and N. Rivière but they were also motivated by an improvement of that paper by \textit{A. Nagel}, \textit{N. M. Rivière} and \textit{S. Wainger} [Am. J. Math. 98, 395--403 (1976; Zbl 0334.44012)].NEWLINENEWLINEThe main result of this work establishes boundedness of \(\mu_{\Phi,\Omega}\) on \(L^{p}(\mathbb{R}^{n+1})\) for \(\frac{2+2\beta}{1+2\beta}<p<2+2\beta\), where \(\Phi(y) = (y,\phi(\rho(y)))\), \(\phi\) is a polynomial of degree \(m\) for some \(m\in\mathbb{N}\), \(\frac{d^{\alpha_i}\phi(t)}{dt}|_{t=0}=0\) and \(\alpha_i\) is a positive integer, less than \(m\), for all \(1\leq i \leq n\). In addition it is required that \(\Omega\) is integrable over \(S^{n-1}\) and for some \(\beta > 0\) NEWLINE\[NEWLINE \sup_{\xi\in S^{n-1}}\int_{S^{n-1}}|\Omega(\theta)|(\ln\frac{1}{|\theta \cdot \xi|})^{1+\beta}d\theta < \infty . NEWLINE\]NEWLINE This last condition was introduced by \textit{L. Grafakos} and \textit{A. Stefanov} [Indiana Univ. Math. J. 47, No. 2, 455--469 (1998; Zbl 0913.42014)] in the context of convolution type Calderón-Zygmund singular integral operators with rough kernels. It is important to note that the theory of parabolic Littlewood-Paley operators is motivated by and intimately tied to the theory of Calderón-Zygmund singular integral operators.NEWLINENEWLINEThe proof of the main theorem is in the spirit of the work by \textit{Q. Y. Xue}, \textit{Y. Ding} and \textit{K. Yabuta} [loc. cit.] and the work by \textit{Y. Ding} and \textit{Y. Pan} [Proc. Edinb. Math. Soc., II. Ser. 46, No. 3, 669--677 (2003; Zbl 1039.42014)].