Rational approximation and weak analyticity. I (Q1057453)

Let R(X) denote the uniform closure on the compact \(X\subset {\mathbb{C}}\) of the set of rational functions having poles off X. Let \(W^{1,p}_{loc}\) denote the Sobolev space of functions \(f: {\mathbb{C}}\to {\mathbb{C}}\) such that f and the first distributional derivatives of f are locally p-th power integrable. Theorem 1: Let \(f\in W^{1,p}_{loc}\) for some p with \(2<p\leq \infty\). Then \(f\in R(X)\) if and only if \(\frac{\partial f}{\partial \bar z}(a)=0\) for almost all nonpeak points \(a\in X.\) Given a measurable set \(E\subset {\mathbb{C}}\), let \(D^{1,p}(E)\) denote the space of \(f\in L_ p(E)\) for which there exist \(f_ 1,f_ 2\in L_ p(E)\) such that, for almost all \(b\in E\) (with respect to area measure \({\mathcal L}^ 2)\), \[ \frac{1}{r^ 3}\int_{| b-z| \leq r}| f(z)-f(b)-(z-b)f_ 1(b)-(z-b)f_ 2(b)| d{\mathcal L}^ 2(z)\to 0\quad as\quad r\downarrow 0. \] For such f, denote \(f_ 2\) by \(\frac{\partial f}{\partial \bar z}.\) Theorem 2: Let \(f\in R(X)\). Let a be a nonpeak point, and \(1\leq p<2\). Then there exists a set E of nonpeak points, having full area density at a, such that \(f\in D^{1,p}(E)\) and \(\frac{\partial f}{\partial \bar z}=0\quad a.e.\) on E.