论文标题
关键Sobolev空间功能的John-Nirenberg不平等的改善
An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
论文作者
论文摘要
众所周知，在具有关键指数的Sobolev空间中的功能嵌入了有限平均振荡的功能空间中，因此满足了John-Nirenberg的不平等和相应的指数可集成性估计。尽管这些不平等对于有界平均振荡的一般函数是最佳的，但本文的主要结果是改善了一系列关键Sobolev空间的功能。确切地说，我们证明了不等式\ [\ mathcal {h}^β_ {\ infty}（\ {x \ inω：|i_αf（x）|> t \}）\ leq ce^{ - 美元$ l^{n/α，q}（ω）$是lorentz空间，$ q \ in（1，\ infty] $，$ q'= q/（q-1）$是hölderconjugate to $ q $，$i_αf$ ex $ f $ f $ f $ f $ f $ f $ f $ f $ n in（0，n）$。
It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality \[\mathcal{H}^β_{\infty}(\{x\in Ω:|I_αf(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/α,q}(Ω)}\leq 1$ and any $β\in (0,N]$, where $Ω\subset \mathbb{R}^N$, $\mathcal{H}^β_{\infty}$ is the Hausdorff content, $L^{N/α,q}(Ω)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the Hölder conjugate to $q$, and $I_αf$ denotes the Riesz potential of $f$ of order $α\in (0,N)$.
