$p$ -变差的模及其应用
The modulus of $p$-variation and its applications
摘要：
本文引入了实变函数的$p$ -变差模的概念，并证明了它至少在两个重要问题中起作用，即Fourier级数的一致收敛和某些$K$ -泛函的计算.更具体地说，设$ν$是一个非减的正实数凹序列，$1\leq p<\infty$。使用我们的新工具，我们首先定义了一个Banach空间，表示$V_p[ν]$，这是中间的维纳类$BV_p$和$L^\infty$，并证明它满足Helly型选择原则。我们还证明了Peetre $K$ -功能的夫妇$(L^\infty,BV_p)$可以表示的$p$ -变化的模。接下来，我们得到等价的尖锐条件的傅立叶级数的一致收敛的所有功能在每个类$V_p[ν]$和$H^ω\cap V_p[ν]$，其中$ω$是一个模的连续性和$H^ω$表示其相关的Lipschitz类。最后，我们建立最佳嵌入到$V_p[ν]$的各种空间的广义有界变差函数。作为这些后者的结果的一个副产品，我们推断嵌入结果的某些对称序列空间。
英文摘要：
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.
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