On the boundedness of periodic Fourier integral operators in Lebesgue spaces with variable exponent

The aim of this paper is to investigate the boundedness of periodic Fourier integral operators in Lebesgue spaces with variable exponent $L^{p(\cdot )}$ on the $n$-dimensional torus. We deal with operators of type $(\rho ,\delta )$ which symbols belong to the Hörmander class $S_{\rho ,\delta }^m({\mathbb{T}}^n\times {\mathbb{Z}}^n)$ for $0\le \delta <\rho \le 1.$