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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.

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The fractional Riez derivative can be written as a sum of Caputo or Reimann-Liouville fractional derivatives, so maybe this could help you?

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The fractional Laplacean $(-\Delta)^{\alpha/2}$ in $\mathbb R^n$ is, up to some constant the Fourier multiplier $\vert\xi\vert^\alpha$. So its inverse is, at least formally, the Fourier multiplier $\vert\xi\vert^{-\alpha}$.

Let us assume that $0<\alpha < n$. Then $\vert\xi\vert^{-\alpha}$ is $L^1_{loc}$ and a tempered distribution homogeneous with degree $-\alpha$. Its Fourier transform is a tempered distribution homogeneous with degree $\alpha-n$ and in fact is also radial so is, up to a constant $\vert x\vert^{\alpha-n}$: a parametrix for $(-\Delta)^{\alpha/2}$ is the convolution by const. $\vert x\vert^{\alpha-n}$. Note that this is the case for $\alpha =2$, when $n\ge 3$. The constants are not uninteresting to compute.

When $\alpha =n$, $\vert \xi\vert^{-n}$ is not $L^1_{loc}$ and is not a tempered Fourier multiplier. The homogeneity is partly lost and you have to perform a direct calculation: for instance for $n=2$ you find $$ \frac{-1}{2\pi}\ln\vert x\vert $$ which is not homogeneous of degree 0 but whose derivatives are homogeneous of degree -1.

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some time ago I defined derivatives that are similar to the Grunwald-Letnikov derivative, but that are two sided. One particular case has Fourier transform equal to |w|^beta.F(w). I showed that it is equivalent to the Riesz and has inverse if |beta| < 1. If you are interested, I can send you my papers:

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