\(L^1\) estimates for maximal functions and Riesz transform on real rank 1 semisimple Lie groups
Let \(G\) be a real rank one semisimple Lie group and \(K\) a maximal compact subgroup of \(G\). Let \(f\in L^1(G//K)\) and denote by \(F_f\) the Abel transform of \(f\) and let \(F^1_f(x)= e^{\rho (x)}F(x)\), where \(\rho\) is the half sum of the positive roots of the Lie algebra of \(G\). For a real function \(F\) one introduces two Fourier multipliers \({\mathcal C}_+'\) and \({\mathcal Q}_s (s\geq 0)\): \[ \begin{matrix} ({\mathcal C}^1_+F)^\sim (\lambda)= c(-\lambda- i\rho)^{-1} F^\sim (\lambda), \\ ({\mathcal Q}_s F)^\sim (\lambda)= \bigl(1+| \lambda |^{-1}\bigr)^s F^\sim (\lambda), \end{matrix}\quad \lambda\in R. \] One defines a subspace \(H^1_s(G//K)\) of \(L^1(G//K)\) (which is analogous to the Hardy space \(H^1(R))\) as \[ H^1_s(G//K)= \bigl\{f\in L^1_{\text{loc}}(G//K); \quad {\mathcal Q}_s {\mathcal C}^1_+F^1_f \in H^1(R) \bigr\}. \] By a detailed analysis of the Harish-Chandra expansion and the Gangolli expansion of the zonal spherical functions of \(G\), the author obtains some basic properties of \(f\) satisfying \({\mathcal C}^1_+ F^1_f \in H^1(R)\) and some criteria by which one can judge whether a radial maximal operator from \(H^1_s (G//K)\) to \(L^1(G//K)\) is bounded. Then, he applies these criteria to radial maximal operators for suitable dilations, to the heat and Poisson maximal operators. He also studies the Riesz transform. Finally he shows that these operators are in \((L^p, L^p)\) for \(p>1\) and that they are of weak type (1,1) and from \(H^1_s (G//K)\) to \(L^1 (G//K)\) for suitable values of \(s\geq 0\).
scientific article; zbMATH DE number 1211636
scientific article