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 ($\nabla^{0<s<1}_+=(-\Delta)^\frac{s}{2}$) and gradients ($\nabla^{0<s<1}_-=\nabla (-\Delta)^\frac{s-1}{2}$) of basic importance in the theory of fractional advection-dispersion equations: one is to discover the sharp Hardy-Rellich ($sp<p<n$) $|$ Adams-Moser ($sp=n$) $|$ Morrey-Sobolev ($sp>n$) 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).