Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

中文翻译：

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带端的非倍增抛物线流形的最大函数的有界

让$M$是一个有端点的非倍增抛物线流形。首先，本文研究了与热半群相关的极大函数的有界性${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f( x)|$在哪里$\unicode[STIX]{x1D6E5}$是拉普拉斯-贝尔特拉米算子$M$. 然后，结合从属公式和前面的结果，我们得到弱类型$(1,1)$和$L^{p}$最大函数的有界${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{ -t\sqrt{L}}f(x)|$在$L^{p}(M)$为了$1<p\leq\infty$在哪里$k$是一个非负整数并且$L$是满足合适热核上界的非负自伴随算子。关于结果的一个有趣的事情是缺乏两个加倍条件$M$以及算子核的平滑度。