On some theorems in fractional calculus for singular functions

Abstract

In this paper we show that there are close relationships amongst Cantor bar totality, non-integer integral and Hausdorff integral. If a singular function \(f\) has zero Lipschitz (\(1-\nu\))-numbers on a Cantor set \(C\) with \(H\)-dim~\(C=1-\nu\), \(0\lt \nu\lt 1\), then the \(1-\nu\) order fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Moreover, under strong assumptions on the function \(f\) the \(1-\nu\) order derivative of \(f\) exists everywhere. Consequently, the \(\nu\) order fractional integral of \(f\) equals \(f(0)t^{\nu}\). Using the Concept of th Hausdorff derivative we prove that if a singular function has a Hausdorff derivative on \(C\), then the fractional derivative of \(f\) exists almost everywhere on \([0,T]\). Finally, under some assumptions on \(f\) and \(C\), we establish \begin{align*} (D^{-\nu}f)(t)=&\int_0^t\int_0^uv^{\nu-1}f'_H(u-v)\,dv^{1-\nu}\,du\\ =&\int_0^tu^{1-\nu}J_{\nu}(v^{\nu-1}f'_H(u-v))(u)\,du \end{align*} This identity is similar to the identity \[(D^{-\nu}f)(t)=\frac{1}{B_{\nu}}t^{\nu}(J_{\nu}f)(t)\] which professor R. R. Nigmatullin claimed.