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 ${\nabla }_{\pm }^{0<s<1}$; the other is to handle the distributional solutions $u$ of the duality equations $[{\nabla }_{\pm }^{0<s<1}{]}^{\ast }u=\mu$ (a nonnegative Radon measure) and $[{\nabla }_{\pm }^{0<s<1}{]}^{\ast }u=f$ (a Morrey function).

29 pages

References in corpus (1)

• Riesz potential and maximal function for Dunkl transform