On Fractional Orlicz-Hardy Inequalities

We establish the weighted fractional Orlicz-Hardy inequalities for various Orlicz functions. Further, we identify the critical cases for each Orlicz function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Orlicz function $\Phi$ and for any $\Lambda>1$, the following inequality is established $$\Phi(a+b)\leq \lambda\Phi(a)+\frac{C( \Phi, \Lambda )}{(\lambda-1)^{p_\Phi^+-1}}\Phi(b),\;\;\;\forall\,a,b\in [0,\infty),\,\forall\,\lambda\in (1,\Lambda],$$ where $p_\Phi^+:=\sup\big\{t\varphi(t)/\Phi(t):t>0\big\},$ $\varphi$ is the right derivatives of $\Phi$ and $C( \Phi, \Lambda )$ is a positive constant that depends only on $\Phi$ and $\Lambda.$