The two-dimensional Cauchy principal value integral \(\Phi\) (f;s,t) of a function f is defined by \[ \Phi (f;s,t)=\lim_{\epsilon_ 1,\epsilon_ 2\to 0\quad +}\iint_{D_{\epsilon_ 1,\epsilon_ 2}}\frac{f(x,y)}\quad {(x-s)(y-s)}w_ 1(x)w_ 2(y)dxdy, \] (s,t)\(\in D\subseteq (-1,1)\) 2, where \[ D_{\epsilon_ 1,\epsilon_ 2}=\{x,y\in I/| x-s| \geq \epsilon_ 1,\quad (y-t)\geq \epsilon_ 2,\quad (\epsilon_ 1>0,\epsilon_ 2>0)\}, \] \(w_ 1(x)=\psi_ 1(x)\prod^{p+1}_{i=0}| x-s_ i|^{\alpha_ i},\) \(p\in N\), \(w_ 2(y)=\psi_ 2(y)\prod^{q+1}_{j=0}| y-t_ j|^{\beta_ j},\) \(q\in N\), \(\alpha_ i>-1\), \(\beta_ j>-1\), \(- 1=s_ 0<s_ 1<...<s_{p+1}=1\), \(-1=t_ 0<t_ 1<...<t_{q+1}=1\), \(\psi_ k\geq 0\), \(\psi_ k^{-1}\) is integrable on \([-I,+I]\) and the moduli of continuity w of \(\psi_ k\) satisfy the condition \(\int^{1}_{0}w(\psi_ k;u)u^{-1}du<\infty\) \((k=1,2)\). Let \(\Delta\) be a closed set such that \(\Delta\subset (-1,1)\) 2 and no points of the straight lines \(x=s_ i\), \(i=1,...,p\) and \(y=t_ j\), \(j=1,...,q\) belongs to \(\Delta\). We denote by C k(I), \(k\geq 0\), the space of all continuous functions on I with continuous partial derivatives of order i, where \(i=0,1,...,k.\)
Let \(\| f\|_ J=\max_{J}| f(x,y)|\) and \(\| f\| =\| f\|_ I\), \(J\subseteq I\). The modulus of continuity of the function \(f\in C(J)\) is defined by \[ w_ J(f;\delta)=\max \{| \Delta_{h_ 1,h_ 2}f(x,y)|:(x,y)\in J,\quad (n+h_ 1,y+h_ 2)\in J,\quad h_ 1+h_ 2<\delta \}, \] \(w(f;\delta)=w_ 1(f;\delta),\) where \(h_ 1,h_ 2\), \(\delta\geq 0\), \(\Delta_{h_ 1,h_ 2}=f(x+h_ 1,y+h_ 2)-f(x,y).\) We write \[ \overline{TD}=\{f\in C(I):\int^{1}_{0}\log u^{-1}w(f;u)u^{-1}du<\infty \}. \] In the main result of the paper the author proves that for any function \(f\in \overline{TD}\) there exists a subsequence of the sequence \(\{\Phi_{m,n}f\}_{(m,n)\in N\times N}\) which converges uniformly to \(\Phi\) f in \(\Delta\).