We want to know if there exists a fundamental theorem of fractional calculus for the Riesz Derivative (a type of fractional Laplacian), e.g. there exists an operator $L$ such that $-L_a^b((-\Delta)^\beta f)=f(b)-f(a)$ or something similar, where the fractional laplacian is defined via a Fourier space:

\begin{eqnarray} -(-\Delta)^\beta f = -{\mathcal{F}}^{-1}((|q|^{2\beta}\mathcal{F}(f)), \end{eqnarray} where $q$ is the variable in the Fourier domain.

Secondly, although the Riesz derivative is a left inverse for the Riesz potential, is there a way to get the right inverse? We need to solve some simple fractional differential equations, namely:

\begin{eqnarray} -(-\Delta)^\beta H&=&H,\;\lim_{|x|\to\infty}H=0, \end{eqnarray} and another equation that we try to solve is \begin{eqnarray} -(-\Delta)^\beta H&=&c,\; c\in\mathbb{R}. \end{eqnarray} Do we need to use other definitions of the fractional derivative for this to work nicely? One criterion that we need is that the fractional derivative should approach the classical derivative when the fractional exponent approaches an integer.