On pseudoanalyticity of continuous functions with constant \(\sigma\)- extension (Q1332083)

Defnitions and notations. \(D\) is a domain in the complex plane, \(\sigma= p- iq\in C(D)\), \(p(z)> 0\), \(\forall z\in D\), a function \(f=u+ iv\) of class \(C^ 1(D)\) is called \(\sigma\)-analytic in \(D\) if it satisfies one of the three equivalent conditions: (1) \({{\partial f} \over {\partial \overline {z}}} + {{\sigma-1} \over {\sigma+1}} {{\partial \overline {f}} \over {\partial \overline {z}}} =0\); (2) \(\sigma {{\partial u} \over {\partial \overline {z}}} +i {{\partial v} \over {\partial \overline {z}}} =0\), (3) \(p {{\partial u} \over {\partial y}} +q {{\partial u} \over {\partial y}} - {{\partial v} \over {\partial x}} =0,\;-q {{\partial u} \over {\partial x}} +p {{\partial u} \over {\partial y}} + {{\partial v} \over {\partial x}} =0\). \(f= u+iv\) satisfies the condition \(K_ \sigma^{\prime\prime}\) at \(a\in D\) if there exists three rays \(t_ 1(a)\), \(t_ 2(a)\), and \(t_ 3(a)\), originating from \(a\) and lying on different straight lines, such that the limit \[ R(a)= \lim _{\substack{ z\to a\\ z\in t_ \nu(a) }} \Biggl| {{\sigma(z) [u(z)- u(a)]+ i[v(z)- v(a)]} \over {z-a}} \Biggr| \] exists along all these lines. Furthermore \(f= u+iv\) is said to be straight at the point \(a\in D\) if there exist two sequences \(\{ z'_ n\}^ \infty_{n=1}\), \(\{z_ n^{\prime\prime} \}^ \infty_{n=1}\) in \(D\) which converge to the point \(a\) and have semitangent lines \(\ell'\), \(\ell''\) at the point \(a\) (lying on different straight lines) such that all points \(w'_ n= f(z'_ n)\), \(w_ n^{\prime\prime}= f(z_ n^{\prime\prime})\) differ from \(b= f(a)\) and the sequences \(\{w'_ n\}^ \infty_{n=1}\), \(\{w_ n^{\prime\prime}\}^ \infty _{n=1}\) have semitangent lines \(L'\), \(L''\) at the point \(b\) with the following property: If \(0< {\mathcal L}(\ell', \ell'')<\pi\) and this angle is counted from \(\ell'\) in the positive direction, then \(0\leq {\mathcal L} (L', L'')<\pi\) if this angle is counted from \(L'\). If \(f\) is straight at every point \(a\) of a certain subset \(A\subset D\), then we say that this function is straight on \(A\). Finally \(C(D)\) denotes the class of continuous functions and \(C^ k_ \alpha (D)\) is the class of functions which have \(k\) partial derivatives Hölder continuous with exponent \(\alpha\), \(0<\alpha \leq 1\). The principal result of the paper is: Theorem. Let \(D\) be a domain in the complex plane, \(\sigma= p- iq\in C^ 1_ \alpha (D)\), and let \(f\) be a continuous function possessing the \(K_ \sigma^{\prime\prime}\) property at every point \(a\in D\) except possibly at a countable set. If the function \(f\) is straight at almost every point \(a\in D\), then it is \(\sigma\)-analytic on \(D\) and has second partials in the class \(C_ \gamma (D)\) with index \(\gamma\) arbitrarily close to \(\alpha\). Furthermore, (a) if \(\sigma\in C^ k_ \alpha (D)\), \(k\geq 1\), then \(f\in C_ \gamma^{k+1} (D)\), where \(\gamma\) is arbitrarily close to \(\alpha\); (b) if \(C^ \infty (D)\), then \(f\in C^ \infty (D)\); (c) if \(\sigma\) is an analytic function of \(x\) and \(y\), then \(f\) is an analytic function of \(x\) and \(y\).