Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials

We establish the following fractional Trudinger-Moser type inequality with logarithmic convolution potential $$\sup_{u\in W^{\frac{1}{2},2}_0(I),\|u\|_{W_0^{\frac{1}{2},2}}\leq1}\int_{I} \int_{I} \log \frac{1}{|x-y|} G(u(x))G(u(y)) \, dx \, dy<+\infty,$$ where $G(s)\leq C\frac{e^{\pi s^{2}}}{(1 + |s|)^{\gamma}}~ \forall s\in\mathbb{R}$ with some constant $C>0,\gamma\geq1$, the domain $I\subset\mathbb{R}$ is a bounded interval. This type of inequality in the entire space $\mathbb{R}$ is also considered. Moreover, we study the existence of corresponding extremal functions. In addition, by the moving plane method, we obtain the radial symmetry and radial decreasing property of positive solutions to the corresponding Euler-Lagrange equation.