## 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
Full Text: