Fractional Differential Couples by Sharp Inequalities and Duality Equations

Abstract

This paper presents a highly non-trivial two-fold study of the fractional differential couples - derivatives ($\nabla^{0 n$) inequalities for $\nabla^{0<s<1}_\pm$; the other is to handle the distributional solutions $u$ of the duality equations $[\nabla^{0<s<1}_\pm]^\ast u=\mu$ (a nonnegative Radon measure) and $[\nabla^{0<s<1}_\pm]^\ast u=f$ (a Morrey function).