Journal of the
Korean Mathematical Society JKMS

The paper is devoted to the study of fractional integration and differentiation on a finite interval $[a,b]$ of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration $\int^{x}_{a}(t/x)^{\mu}f(t)dt/t$ and the modified differentiation $\delta+\mu$ $(\delta =xD,\ D=d/dx)$ with real $\mu$, being taken $n$ times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space $X_{c}^{p}(a,b)$ of Lebesgue measurable functions $f$ on ${\bf R}_{+}=(0,\infty)$ such that
$$
\int^{b}_{a}\vert t^{c}f(t) \vert ^{p}{\frac{dt}{t}}<\infty \
(1\leq p <\infty),\ \ {\rm ess\ sup}_{a\leq t\leq b}[u^{c}\vert f(t)\vert]
< \infty\ (p=\infty),
$$ for $c \in {\bf R}=(-\infty,\infty)$, in particular in the space $L^{p}(0,\infty)$ $(1\leq p\leq \infty)$. The existence almost everywhere is established for the corresponding Hadamard-type fractional derivative for a function $g(x)$ such that $x^{\mu}g(x)$
have $\delta$ derivatives up to order $n-1$ on $[a,b]$ and $\delta^{n-1}[x^{\mu}g(x)]$ is absolutely continuous on $[a,b]$. Semigroup and reciprocal properties for the above operators are proved.

Keywords: Hadamard-type fractional integration and di?erentiation, weighted spaces of summable and absolutely continuous functions