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I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...

There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....

I have asked the same question on MathSE. I was thinking about the following problem. Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...

Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$. How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$? Recall that, $$|u|^p_{W^{s,p}(\Omega)}= ...

I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. Consider the following problem: $$ \Delta_xu+\frac{a}{y}u_y+u_{yy}=0, $...