On fractional Orlicz-Hardy inequalities

Abstract

We establish the weighted fractional Orlicz-Hardy inequalities for various Young functions satisfying the ${△}_2$-condition. Further, we identify the critical cases for such Young 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 Young function Φ satisfying the ${△}_2$-condition and for any $\Lambda >1$, the following inequality is established$\Phi (a+b)\le \lambda \Phi \left(a\right)+\frac{C(\Phi ,\Lambda )}{{(\lambda -1)}^{p_{\Phi }^{+}-1}}\Phi \left(b\right),\forall a,b\in [0,\infty ),\forall \lambda \in (1,\Lambda ],$ where $p_{\Phi }^{+}:=\sup \left\{t\phi \right(t)/\Phi (t):t>0\}$, φ is the right derivatives of Φ and $C(\Phi ,\Lambda )$ is a positive constant that depends only on Φ and Λ.