## Real Analysis Exchange

- Real Anal. Exchange
- Volume 22, Number 1 (1996), 122-141.

### 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.

#### Article information

**Source**

Real Anal. Exchange, Volume 22, Number 1 (1996), 122-141.

**Dates**

First available in Project Euclid: 1 June 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1338515208

**Mathematical Reviews number (MathSciNet)**

MR1433601

**Zentralblatt MATH identifier**

0879.26024

**Subjects**

Primary: 26A33: Fractional derivatives and integrals 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]

Secondary: 26A39: Denjoy and Perron integrals, other special integrals

**Keywords**

Fractional integral Fractional derivative Hausdorff integral Hausdorff derivative Lipschitz number singular function Cantor type set

#### Citation

Fu, Shusheng. On some theorems in fractional calculus for singular functions. Real Anal. Exchange 22 (1996), no. 1, 122--141. https://projecteuclid.org/euclid.rae/1338515208