Dual identities in fractional difference calculus within Riemann

We Investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences we have to use both the nabla and delta operators. The solution representation for higher order Riemann fractional difference equation is obtained as well

Non-local Gehring lemmas in spaces of homogeneous type and applications

We prove a self-improving property for reverse H{\"o}lder inequalities with non-local right hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations as well as certain fractional equations. We also consider non-local extensions of A$\infty$ weights. We write our results in spaces of homogeneous type.Comment: Revised version. Changed title. Application to a more relevant fractional elliptic equation given in the final section. 40 page