论文标题
在某些非加倍的歧管上，耐寒类型空间的包含和非包含
Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
论文作者
论文摘要
在本文中，我们在非障碍的riemannian歧管上的各种耐强型空间之间建立了夹杂物和非嵌入，其ricci曲率从下方界定，正面注射率半径和光谱差距。 我们的第一个主要结果指出，如果$ \ mathscr {l} $是$ m $上的正laplace-beltrami操作员，则riesz-hardy space $ h^1_ \ mathscr {r}（m）$是Goldberg Space $ \ Mathfrak $ \ Mathfrak的同构图像$ \ MATHSCR {l}^{1/2}（\ Mathscr {i} + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {l}）具体来说，$ h^1_ \ mathscr {r}（m）$与Hardy Type Space $ \ Mathfrak {x}^{1/2} {m）$ cys;结果，我们证明$ \ mathfrak {h}^1（m）$不接受原子特征。 在特殊情况下，非含义大多是在歧管是damek-ricci space $ s $的特殊情况下证明的。 Our second main result states that $H^1_\mathscr{R}(S)$, the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.
In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds $M$ with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if $\mathscr{L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H^1_\mathscr{R}(M)$ is the isomorphic image of the Goldberg type space $\mathfrak{h}^1(M)$ via the map $\mathscr{L}^{1/2} (\mathscr{I} + \mathscr{L})^{-1/2}$, a fact that is false in $\mathbb{R}^n$. Specifically, $H^1_\mathscr{R}(M)$ agrees with the Hardy type space $\mathfrak{X}^{1/2}(M)$ recently introduced by the the first three authors; as a consequence, we prove that $\mathfrak{h}^1(M)$ does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek-Ricci space $S$. Our second main result states that $H^1_\mathscr{R}(S)$, the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.
