I think that the powerfunctions with exponent sufficiently large (depending on  alpha) are the simplest examples. Aren't they? If yes, then  It is really an interesting question if there are examples outside the (suitable) closure of the  linear span of the power functions.

PS. Obviously, the problem is dependent of the definition of the integral and differential operators. I have assumed they as being  defined by the convolution with power functions  on the space of  functions with positive numbers as a domain.

Dear Professor,

Joachim Domsta,

Kindly, illustrate by example so that D^{alpha}J^{alpha}=J^{alpha}D^{alpha}.

Dear Professor Ganga Ram,

my answer's goal was just a suggestion for you to start the solution by trying power function, with strong emphasize on the definitions you want to apply. In your answer no particular chosen by you  definitionappeared. My quess about your aim is based on the simplest examples known as Liouville left-sided derivative, result of which for power function is a power function. Same for Riemann-Liouville left-sided integrals. Hopefully, the equality of your question can be now verified by simple calculus for power functions of the form  f(x) = x^p,  x > 0.  I just conjecture that this leads to examples you are asking for. Please consult the review at

https://www.hindawi.com/journals/mpe/2014/238459/

Best Regards, Joachim Domsta

Dear Joachim,

Thank you for your comments regarding the domain for the  commutator D^{alpha}J^{alpha}=J^{alpha}D^{alpha} with alpha being fractional.  

Since the domain contains the set of polynomials, it may extend to the set of analytic functions along with appropriate closure, which will depend on how to compute the commutator [D^{alpha}, J^{alpha}] in certain integral form.

The review article "A Review of Definitions for Fractional Derivatives and Integral" also gives detailed exposition about the calculus-wise origin for the definition of the fractional derivatives.

Best Regards, Shijun 

Dear Joachim,

Thank you for your interest in my work on the spectral multipliers for Schroedinger operators. I have uploaded a pdf of the article on RG.

Best wishes. S.Z.

Dear Ganga,

first of all, I am sorry for answering you so late that perhaps you have already found the answer. Generally the answer depends on the definition of the fractional derivative.

If D^\alpha denotes the Riemann-Liouville fractional derivative on a (finite) interval, then D^\alpha J^\alpha f=f for any integrable function f, that is f is in L_1. On the other hand, for J^\alpha D^\alpha f=f it is enough that f is in the image of L_1 under the fractional integration J^\alpha, that is f is in J^\alpha(L_1). This is Theorem 2.4 of the classical book "Fractional Integrals and Derivaticves, Theory and Applications" by Samko, Kilbas and Marichev. Theorem 2.3 of the same book gives the characterization of the set J^\alpha(L_1) in terms of the absolute continuity: f is in J^\alpha (L_1) if and only if J^{n-\alpha} f is n-times absolutely continuous and the derivatives up to order n-1 of J^{n-\alpha} f vanish at a, which is the left-sided end point of the interval, where the fractional operators are defined. Here n is the integer part of \alpha plus 1.

If D^\alpha is the Caputo derivative, then the operators commute, if f is n-times absolutely continuous and that the derivatives of f up to order n-1 vanish at the left end point of the interval, where the fractional operators are defined. Here n is the least integer which is greater or equal to \alpha.

Best regards, Jukka 

Dear Professors

Thank you for answering.