On Riemann boundary value problem in Hardy classes with variable summability exponent (Q2910535)

Let \(p:[-\pi,\pi]\to[1,+\infty]\) be some Lebesgue measurable function, \(L_0\) the class of all functions measurable on \([-\pi,\pi]\), NEWLINE\[NEWLINEI_p(f)\overset\text{def}\equiv\int_{-\pi}^\pi|f(t)|^{p(t)}\,dt, \quad \text{L}\equiv\big\{f\in\text{L}_0:I_p(f)<+\infty\big\}\quad\text{and}\quad p^\pm=\underset{[-\pi,\pi]}{\sup\,\text{vrai}}\,p(t)^{\pm 1}.NEWLINE\]NEWLINENEWLINEWith the condition \(1\leq p^-\leq p^+<+\infty\) and with the norm NEWLINE\[NEWLINE\| f\|_{p(\cdot)}\overset\text{def}\equiv\inf\left\{\lambda>0:I_p\left(\frac{f}{\lambda}\right)\leq1\right\},NEWLINE\]NEWLINE \(L\) is a Banach space denoted by \(L_{p(\cdot)}\).NEWLINENEWLINELet \(\mathbb B=\{z:|z|<1\}\) be the unit disk in the complex plane and \(\partial\mathbb B\) be the unit circle. The Hardy class \(H^+_{p(\cdot)}\) is defined by NEWLINE\[NEWLINEH^+_{p(\cdot)}\equiv\left\{f:f\,\text{analytic in }\mathbb B\,\text{ and }\, \| f\|_{H^+_{p(\cdot)}}<+\infty \right\},NEWLINE\]NEWLINE whereNEWLINE NEWLINE\[NEWLINE\| f\|_{H^+_{p(\cdot)}}\overset\text{def}\equiv\underset{0<r<1} \sup\| f(re^{it})\|_{p(\cdot)}.NEWLINE\]NEWLINE \(H^+_{p(\cdot)}\) is a Banach space if \(1\leq p^-\leq p^+<+\infty\).NEWLINENEWLINELet \(f(z)\) be an analytic function on \(\mathbb C\setminus\overline{\mathbb B}\) which is of finite order \(m_0\leq m\) at infinity, i.e. \(f(z)=f_1(z)+f_2(z)\) where \(f_1(z)\) is a polynomial of degree \(m_0\), \(f_2(z)\) is the right part of the Laurent series expansion of \(f(z)\) in a neighborhood of infinity. If the function \(\varphi(z)=\overline{f_2\left(1/\overline z\right)}\) belongs to the class \(H^+_{p(\cdot)}\), we say that the function \(f(z)\) belongs to the class \(_{m}H^-_{p(\cdot)}\).NEWLINENEWLINEThe author considers the Riemann boundary value problem in the Hardy class \(H^+_{p(\cdot)}\oplus\, _{m}H^-_{p(\cdot)}\) with a variable summability exponent \(p(\cdot)\): NEWLINE\[NEWLINE\begin{cases} F^+(\tau)-G(\tau)\,F^-(\tau)=f(\tau),\quad\tau\in\partial\mathbb B,\\ F^+(\tau)\in H^+_{p(\cdot)},\quad F^-(\tau)\in\, _{m}H^-_{p(\cdot)},\end{cases}NEWLINE\]NEWLINE where \(f(\tau)\in L_{p(\cdot)}\) and a piecewise Hölder function \(\alpha\) on \((-\pi,\pi]\) are given, and \(G(\tau)\equiv e^{2i\alpha(\text{arg}\,\tau)}\). Under certain conditions on the functions \(\alpha\), \(f\) and \(p\), solvability conditions are given and general solutions of the Riemann problem are constructed.