On bisingular operators in the space \(L_ p(\mathbb{R}^ 2)\) with some invariance conditions (Q1893625)

This paper is concerned with the invertibility criterion of the operators of the form \[ A= aP_{++}+ bP_{+-}+ cP_{-+}+ dP_{--},\quad A= (P_{++}a+P_{+-} b+ P_{-+} c+ P_{--}d)I \] in the space \(L_p(\mathbb{R}^2)\), \(1< p< \infty\), where \(P_{\pm\pm}= P_\pm\otimes P_\pm\) are projectors defined by \(P_\pm= {1\over 2}(I\pm S)\) and \(S\) is the Hilbert transform, and where \(a\), \(b\), \(c\), \(d\) are piecewise- constant functions of the form \[ a(x, y)= a_+ \chi_+(y- x)+ a_- \chi_-(y- x),\;\chi_+(t)= \chi_{[0,\infty]}(t),\;\chi_-(t)= \chi_{[-\infty,0]}(t) \] and \(a_+\), \(a_-\) are complex number. Let \[ f_\pm(x, y)= \begin{cases} c_\pm \chi_{[- \infty, 0]} (y)+ d_\pm \chi_{[0, x]}(y)+ b_\pm \chi_{[x, \infty]}(y),\quad x> 0,\\ c_\pm \chi_{[- \infty, x]}(y)+ a_\pm \chi_{[x, 0]}(y)+ b_\pm \chi_{[0, \infty]}(y),\quad x< 0.\end{cases} \] For a picewise-constant function \(f(t)\) put \[ f_p(t, \xi)= f(t- 0) {1+ S_p(\xi)\over 2}+ f(t+ 0) {1- S_p(\xi)\over 2},\quad t\in\dot\mathbb{R},\quad \xi\in \overline{\mathbb{R}}, \] where \(\dot\mathbb{R}\) and \(\overline{\mathbb{R}}\) are the one- point and two-point compactifications of \(\mathbb{R}\), respectively, and \[ S_p(\xi)= (e^{2\pi(i/p+ \xi)}+ 1)/(e^{2\pi(i/p+ \xi)}- 1). \] The author shows that \(A\) is invertible in the space \(L_p(\mathbb{R}^2)\) if and only if the following conditions are satisfied. (1) \(f_\pm(\pm 1, y)\neq 0\) \(\forall y\in \dot\mathbb{R}\), (2) \(g_p(\pm 1, y, \xi)\neq 0\) \(\forall y\in \dot\mathbb{R}\), \(\forall \xi\in\overline{\mathbb{R}}\), where \(g(\pm 1, y)= f_+(\pm 1, y)/ f_-(\pm 1, y)\), (3) \(\text{ind } g_p(\pm 1, y, \xi)= 0\).