奇异积分的基本问题
Foundational aspects of singular integrals
摘要：
研究了（0，1]上实值连续函数类的积分问题.当然，如果类中存在非$L^1$元，则会出现困难，而Hadamard有限部分积分（{ \em p.f.}）则不适用。这样的奇异积分在包括偏微分方程和奇异常微分方程的许多情况下自然出现。Lebesgue积分以及$p.f.$，从零开始，服从两个基本条件：（i）它们作为反导数，（ii）如果$f =g$在$(0,a)$上，那么它们从PDMATHAMR到$x$的积分对任何$x\in (0,a)$都是一致的。我们发现，从零开始的积分与$p.f.$的基本性质，加上积极性，存在凭借公理的选择（AC）的所有功能$(0,1]$是$L^1((ε,1])$的所有$ε>0$。然而，这种存在性证明并没有提供令人满意的构造。如果在$0$上没有正则性，则在具有非$L^1$元素的类上仅满足上述（i）和（ii）的一般反导数的存在性与ZF（不含AC的数学的通常ZFC公理），甚至与ZFDC（具有依赖选择公理的ZF）无关。此外，我们表明，有没有数学描述，可以证明（ZFC内，甚至ZFC扩展与大基数假设），以唯一地定义这样一个反导数算子。这样的结果是精确地制定了各种功能集，并证明使用方法从数理逻辑，描述集理论和分析。我们还分析了$p.f.$的解析函数的穿孔单位圆盘，并作出连接到奇异初值问题。
英文摘要：
We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-$L^1$ element in the class, and the Hadamard finite part integral ({\em p.f.}) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs. The Lebesgue integral as well as $p.f.$, starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if $f =g$ on $(0,a)$, then their integrals from $0$ to $x$ coincide for any $x\in (0,a)$. We find that integrals from zero with the essential properties of $p.f.$, plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on $(0,1]$ which are $L^1((ε,1])$ for all $ε>0$. However, this existence proof does not provide a satisfactory construction. Without some regularity at $0$, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-$L^1$ element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator. Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze $p.f.$ on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.
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