论文标题
Riesz对Riemannian歧管的反向不平等
Reverse inequality for the riesz transforms on Riemannian manifolds
论文作者
论文摘要
让$ m $成为一个完整的Riemannian流形，满足了地球球球的加倍体积条件，以及$ l^Q $缩放的庞加莱不平等现象，在合适的远程球上，以$ q <2 $。 We prove the inequality $\left\Vert Δ^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all $p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when $M$ has a finite number of Euclidean结束。 这项工作的第二部分涉及类似分形的电缆系统中的类似问题。在这个框架中，陈，库洪，费内鲁尔和第二作者已经证明了在维卡克电缆系统中，不等式$ \ weft \ welet \vertΔ^{1/2} f \ right \ right \ vert_p_p \ vert_p \ lyssim \ sillssim \ sillssim \ sillssim \ sillssim \ lyse \ left \ vert \ nabla f \ nabla f \ right per $ $ per $ $ in [1 in]在两位作者和杨的最新联合工作之后，我们研究了形式的不等式的有效性，$ \ welet \ vert \vertΔ^γ^{ - δ} f \ right \ right \ vert_p \ vert_p \ sillssim \ silltsim \ left \ left \ vert \ nabla f \ nabla f \ f \ right \ right \ vert_p $。在Vicsek案例中，我们给出了这种不平等所能达到的最佳$ P $。
Let $M$ be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and $L^q$ scaled Poincaré inequalities on suitable remote balls for some $q<2$. We prove the inequality $\left\Vert Δ^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all $p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when $M$ has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest. The second part of this work deals with analogous questions in fractal-like cable systems. In this framework, it was already proved by Chen, Coulhon, Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert Δ^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ may be false for all $p\in [1,2)$. Following a recent joint work by the two authors and Yang, we examine the validity of inequalities of the form $\left\Vert Δ^γe^{-Δ}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$. In the Vicsek case, we give the optimal range of $p$ for which this inequality holds.
