论文标题
Lorentz Space的内核中$ L_P（r^n）$中的卷积最大化器的存在
Existence of convolution maximizers in $L_p(R^n)$ for kernels from Lorentz spaces
论文作者
论文摘要
本文扩展了G.V.〜Kalachev和作者（Sb。Math。2019或Arxiv：1712.08836）的早期结果，该结果是关于存在$ r^n $在$ r^n $之间在$ l_q $，$ l_q $，$ 1 <q <q <q <Q <f inbes的两个lebesgue Space之间的最大卷积操作员的最大化器。鉴于利布（Lieb）1983年的结果是关于强硬的木材 - 贝伯夫（Hardy-Littlewood-Sobolev）不平等的极端变量，因此自然要问是否存在来自弱$ l_q $的任何内核的卷积最大化器。 Lieb在上述引用中给出了否定的答案。在本文中，我们证明了与弱$ l_q $更狭窄的粒子的最大化器，其中包含所有lorentz spaces $ l_ {q，s} $和$ q \ leq s <\ s <\ s <\ infty $。
The paper extends an earlier result of G.V.~Kalachev and the author (Sb. Math. 2019 or arXiv:1712.08836) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on $R^n$ with kernel from some $L_q$, $1<q<\infty$. In view of Lieb's result of 1983 about the existence of an extremizer for the Hardy-Littlewood-Sobolev inequality it is natural to ask whether a convolution maximizer exists for any kernel from weak $L_q$. The answer in the negative was given by Lieb in the above citation. In this paper we prove the existence of maximizers for kernels from a slightly more narrow class than weak $L_q$, which contains all Lorentz spaces $L_{q,s}$ with $q\leq s<\infty$.
