Fractional Differential Couples by Sharp Inequalities and Duality Equations

This paper presents a highly non-trivial two-fold study of the fractional differential couples - derivatives ($ abla^{0<s<1}_+=(-Delta)^frac{s}{2}$) and gradients ($ abla^{0<s<1}_-= abla (-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 $ abla^{0<s<1}_pm$; the other is to handle the distributional solutions $u$ of the duality equations $[ abla^{0<s<1}_pm]^ast u=mu$ (a nonnegative Radon measure) and $[ abla^{0<s<1}_pm]^ast u=f$ (a Morrey function).