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

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
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