Hadamard-type integrals as \(G\)-transforms. (English) Zbl 1043.26004

Kilbas, Anatoly A.; Trujillo, Juan J.

Integral Transforms Spec. Funct. 14, No. 5, 413-427 (2003).

Summary: This paper is devoted to the study of four integral operators, which are generalizations and modifications of Hadamard integrals, in the space \(X^p_c\) of Lebesgue measurable functions \(f\) on \(\mathbb{R}_+=(0,\infty)\) such that for \(c\in\mathbb{R}=(-\infty,\infty)\) \[ \int^\infty_0\bigl| u^cf(u)\bigr |^p \frac{d_u}{u}< \infty\;(1\leq p<\infty), \quad\text{ess} \sup_{u>0}\bigl[ u^c | f(u) |\bigr]< \infty\;(p=\infty). \] Representations for the operators are given in the form of integral transforms involving the Meijer \(G\)-function in the kernels. The mapping properties such as the boundedness, representation and range are established.

MSC:

26A33	Fractional derivatives and integrals
47B38	Linear operators on function spaces (general)
47G10	Integral operators

Keywords:

Hadamard-type integration; integral transforms with Meijer’s \(G\)-function kernels; spaces of \(p\)-summable functions; Mellin transform; Hadamard-type fractional integral operators