The spherical maximal operator of order $\mu$ was introduced by El Kohen in the setting of real hyperbolic spaces, where its $L^p$-boundedness properties were studied. In this paper, we extend this notion to rank-one Riemannian symmetric spaces of noncompact type. We establish sufficient conditions on the parameter $\mu$ for the corresponding spherical maximal operator to be bounded on $L^p$ for $1<p\leq\infty$. We also obtain a necessary condition for $L^p$-boundedness when $\mu$ is real and $1<p<\infty$. In particular, for $1<p\leq2$, the necessary condition agrees with the sufficient condition, yielding the sharp range of real $\mu$ in this regime.
We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of $\mathbb{R}^n$ with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of $\ell^2$ endowed with a nondegenerate Gaussian measure. To the best of our knowledge, this is the first such eq...
Let $M$ be a complete, non-compact Riemannian manifold. We prove that its Riesz transform is of weak type $(1,1)$, with constant $2$ for real-valued functions. Consequently, it is bounded on $L^p(M)$ for $1<p\leq2$, with constants depending only on $p$, which proves the Coulhon--Duong conjecture. The proof uses an obst...
We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometricall...
Let $(X,d,p)$ be a noncollapsed pointed Gromov--Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge4$. We assume that, on each bounded ball, the integral of the $3/2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume...
We establish the uniform doubling property for $G$-invariant metrics on a wide class of compact Riemannian homogeneous spaces $(G/K,g)$, describing explicitly the volume growth of the metric balls $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$...
We establish a noncommutative version of a familiar Johnson--Maurey--Schechtman--Tzafriri Theorem, by showing that for any $0<p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1) $ containing the function $t\mapsto t^{...
Jing-Hao Huang, M. Junge, F. Sukochev et al.· 1 citation
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