论文标题
重新排列的某些规模的基本功能特性
Basic functional properties of certain scale of rearrangement-invariant spaces
论文作者
论文摘要
令$ x $为非原子$σ$ -finite量$（\ Mathscr {r}，μ）$的重新安排不变空间，让$α\ in（0，\ infty）$。我们定义功能\ begin {equation*} \ | f \ | _ {X^{\langleα\ rangle}} = \ |（（（| F |^α） $ f $是$（\ Mathscr {r}，μ）$和$ \ overline {x}（0，μ（\ Mathscr {r}））$定义的$μ$ - 可容纳标量函数。我们用$ x^{\langleα\ rangle} $ collection $ f $的收集$ f $，使得$ \ | f \ | _ {x^{\langleα\ rangle}} $是有限的。这些空间最近浮出水面，以涉及上ahlfors常规措施的总体Sobolev嵌入目标函数空间的最优性。 我们在这些空间上介绍了各种结果，包括它们的基本功能属性，它们与习惯功能空间的关系和相互嵌入，以及在特定情况下的相关结构的表征。我们发现了从Lebesgue空间到Zygmund类的一个新的单参数路径，我们将其与经典的类别进行了比较。
Let $X$ be a rearrangement-invariant space over a non-atomic $σ$-finite measure space $(\mathscr{R},μ)$ and let $α\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle α\rangle}} = \|((|f|^α)^{**})^{\frac{1}α}\|_{\overline{X}(0,μ(\mathscr{R}))}, \end{equation*} in which $f$ is a $μ$-measurable scalar function defined on $(\mathscr{R},μ)$ and $\overline{X}(0,μ(\mathscr{R}))$ is the representation space of $X$. We denote by $X^{\langle α\rangle}$ the collection of all almost everywhere finite functions $f$ such that $\|f\|_{X^{\langle α\rangle}}$ is finite. These spaces recently surfaced in connection of optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings and, in a particular situation, a characterization of their associate structures. We discover a new one-parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.
