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This paper presents a highly non-trivial two-fold study of the fractionaldifferential couples - derivatives ($\nabla^{0<s<1}_+=(-\Delta)^\frac{s}{2}$)and gradients ($\nabla^{0<s<1}_-=\nabla (-\Delta)^\frac{s-1}{2}$) of basicimportance in the theory of fractional advection-dispersion equations: one isto 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 tohandle 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).

Fractional Differential Couples by Sharp Inequalities and Duality Equations