Fractional integrals on product manifolds
A mixed norm estimate is established for multi-parameter fractional integrals on product spaces. This result is then shown to imply boundedness of several particular kinds of fractional integrals, namely those of Nagel-Stein type on product manifolds, of Folland-Stein type on products of homogeneous groups and the discrete fractional integrals of Stein-Wainger type. Let \(\left( \mathcal{X}_{i},\mathcal{M}_{i},\mu _{i}\right) \), \( i=1,2\), be two measure spaces and define \(P=(p_{1},p_{2})\). Define the mixed norm space \[ \begin{multlined} L^{P}\left( \mathcal{X}_{1}\times \mathcal{X}_{2}\right) := \left\{f: \mathcal{X}_{1}\times \mathcal{X}_{2}\rightarrow \mathbb{R}^{1},f \text{ is } \mu _{1}\otimes \mu _{2}\text{-measurable},\right. \\ \left.\left\| f\right\| _{L^{P}\left( \mathcal{X}_{1}\times \mathcal{X}_{2}\right) }=\left\| \left\| f\right\| _{L^{p_{1}}\left( \mathcal{X}_{1},\mu _{1}\right) }\right\| _{L^{p_{2}}\left( \mathcal{X}_{2},\mu _{2}\right) }<\infty \right\},\end{multlined} \] For \(p_{2}=\infty \), \(1\leq r<\infty \), define the norm of \( L^{r,\infty }\left( \mathcal{X},\mu \right) \) to be \[ \left\| f\right\| _{L^{r,\infty }}:=\sup_{t>0}\left[ t^{r}\mu \left( \{x\in \mathcal{X}:\left| f(x)\right| >t\}\right) \right] ^{1/r}<\infty. \] Next define the two parameter integral operator \(T_{\alpha _{1},\alpha _{2}}\) by \[ T_{\alpha _{1},\alpha _{2}}f(x_{1},x_{2})=\int\nolimits_{\mathcal{X} _{2}}\int\nolimits_{\mathcal{X} _{1}}K(x_{1},y_{1};x_{2},y_{2})f(y_{1},y_{2})d\mu _{1}(y_{1})d\mu _{2}(y_{2}) , \] where \(K\) is a measurable function on \(\mathcal{X}_{1}\times \mathcal{X}_{2}\). (The indices \(\alpha _{i}\) refer to the conditions (a) - (d) given below). The authors' main result is the following theorem: Suppose \(0<\alpha _{1},\alpha _{2}<1\). If the kernel \(K\) satisfies the conditions: {\parindent=8mm \begin{itemize}\item[(a)]\(\sup_{y_{2}}\left\| \sup_{y_{1}}\left\| K\left( \cdot ,y_{1};\cdot ,y_{2}\right) \right\| _{L_{x_{1}}^{1/(1-\alpha _{1}),\infty }}\right\| _{L_{x_{2}}^{1/(1-\alpha _{2}),\infty }}<\infty \) \item[(b)]\(\sup_{y_{2}}\left\| \sup_{x_{1}}\left\| K\left( x_{1},\cdot ;\cdot ,y_{2}\right) \right\| _{L_{y_{1}}^{1/(1-\alpha _{1}),\infty }}\right\| _{L_{x_{2}}^{1/(1-\alpha _{2}),\infty }}<\infty \) \item[(c)]\(\sup_{x_{2}}\left\| \sup_{y_{1}}\left\| K\left( \cdot ,y_{1};x_{2},\cdot \right) \right\| _{L_{x_{1}}^{1/(1-\alpha _{1}),\infty }}\right\| _{L_{y_{2}}^{1/(1-\alpha _{2}),\infty }}<\infty \) \item[(d)]\(\sup_{x_{2}}\left\| \sup_{x_{1}}\left\| K\left( x_{1},\cdot ;x_{2},\cdot \right) \right\| _{L_{y_{1}}^{1/(1-\alpha _{1}),\infty }}\right\| _{L_{y_{2}}^{1/(1-\alpha _{2}),\infty }}<\infty \), \end{itemize}} then for all \(P=\left( p_{1},p_{2}\right) \) and \( Q=(q_{1},q_{2})\) with \(1<p_{i}<1/\alpha _{i}\) and \( 1/p_{i}-1/q_{i}=\alpha _{i}\) (\(i=1,2\)), we have \[ \left\| T_{\alpha _{1},\alpha _{2}}f\right\| _{L^{Q}\left( \mathcal{X} _{1}\times \mathcal{X}_{2}\right) }\leq C\left\| f\right\| _{L^{P}\left( \mathcal{X}_{1}\times \mathcal{X}_{2}\right) }, \] where \(C=C(P,\alpha _{1},\alpha _{2})\). The proof is established by first proving a similar result for a one parameter integral operator, and then applying this result to obtain the two parameter theorem. The one parameter result is proved by showing that the integral operator is both type \(L^{\infty },L^{\infty }\) and weak-type \( L^{1},L^{1}\) and using interpolation. The authors say their paper was motivated by \textit{A. Nagel} and \textit{E. Stein}, [Rev. Mat. Iberoam. 20, 531--561 (2004; Zbl 1057.42016)]. Other related work is contained in [\textit{G. Folland} and \textit{E. Stein}, Hardy spaces on homogeneous groups. Mathematical Notes, 28. Princeton, New Jersey: Princeton University Press; University of Tokyo Press. (1982; Zbl 0508.42025)], \textit{C. Muscalu, J. Pipher, T. Tao} and \textit{C. Thiele}, [Rev. Mat. Iberoam. 22, 963--976 (2006; Zbl 1114.42005)], and \textit{E. Stein} and \textit{S. Wainger} [J. Anal. Math. 80, 335--355 (2000; Zbl 0972.42010)].
scientific article; zbMATH DE number 5550527
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