论文标题
关于达到其加权规范的全体形态功能
On holomorphic functions attaining their weighted norms
论文作者
论文摘要
我们研究了达到加权规范的全体形态功能及其与实现霍明型功能的经典理论的联系。我们证明，$ \ ell_p $上有多项式，但他们的加权却没有，但没有达到他们的超级规范和Viceversa。然而，我们还证明，在固定程度的多项式中，两种规范实际上都是等效的。这使我们遇到了本文的主要问题，即，达到其加权规范的全体形态功能是否密集。尽管我们展示了一个不存在的示例，因为作为本文的主要定理，我们证明了稠度，只要域空间均匀地凸出。实际上，我们在这种情况下提供了Bollobás类型定理。为了证明这种结果，我们开发了一种新的几何技术。
We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their supremum norm and viceversa. Nevertheless, we also prove that in the context of polynomials of fixed degree both norms are in fact equivalent. This leads us to the main problem of the paper, namely, whether the holomorphic functions attaining their weighted norm are dense. Although we exhibit an example where this does not hold, as the main theorem of our paper, we prove the denseness provided the domain space is uniformly convex. In fact, we provide a Bollobás type theorem in this setting. For the proof of such a result we develop a new geometric technique.
