On Fractional Orlicz-Hardy Inequalities

Abstract

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 $$\Phi (a+b)\le \lambda \Phi (a)+\frac{C(\Phi ,\Lambda )}{(\lambda -1{)}^{p_{\Phi }^{+}-1}}\Phi (b),\forall a,b\in [0,\infty ),\forall \lambda \in (1,\Lambda ],$$ where $p_{\Phi }^{+}:=\sup \{t\phi (t)/\Phi (t):t>0\},$ $\phi$ is the right derivatives of $\Phi$ and $C(\Phi ,\Lambda )$ is a positive constant that depends only on $\Phi$ and $\Lambda .$

24 pages, 3 figures