论文标题
$ p $变化及其应用的模量
The modulus of $p$-variation and its applications
论文作者
论文摘要
在本说明中，我们介绍了$ p $变量的模量的概念，以实现真实变量的函数，并表明它至少有两个重要问题，即，傅立叶系列的统一收敛和某些$ k $ unctionals的计算。要更具体地说，让$ν$是一个非核心的正实数凹面顺序和$ 1 \ leq p <\ infty $。使用我们的新工具，我们首先定义一个Banach空间，表示$ v_p [ν] $，该空间介于Wiener类$ bv_p $和$ l^\ infty $之间，并证明它满足了Helly-type选择原则。我们还证明，可以根据$ p $变量的模量表示，可以表达夫妇$（l^\ infty，bv_p）$的Peetre $ k $功能。接下来，我们获得了同等的尖锐条件，以在每个类别中的所有功能的傅立叶系列均匀收敛$ v_p [ν] $和$ h^ω\ cap v_p [ν] $中，其中$ω$是连续性的模量，$ h^ω$表示其相关的lipschitz类。最后，我们将最佳的嵌入到$ v_p [ν] $中的$ v_p [ν] $的各个函数函数的函数。作为后者结果的副产品，我们推断出某些对称序列空间的嵌入结果。
In this note, we introduce the notion of modulus of $p$-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain $K$-functionals. To be more specific, let $ν$ be a nondecreasing concave sequence of positive real numbers and $1\leq p<\infty$. Using our new tool, we first define a Banach space, denoted $V_p[ν]$, that is intermediate between the Wiener class $BV_p$ and $L^\infty$, and prove that it satisfies a Helly-type selection principle. We also prove that the Peetre $K$-functional for the couple $(L^\infty,BV_p)$ can be expressed in terms of the modulus of $p$-variation. Next, we obtain equivalent sharp conditions for the uniform convergence of the Fourier series of all functions in each of the classes $V_p[ν]$ and $H^ω\cap V_p[ν]$, where $ω$ is a modulus of continuity and $H^ω$ denotes its associated Lipschitz class. Finally, we establish optimal embeddings into $V_p[ν]$ of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.
