Bessel-type integral transforms on \({\mathcal L}_{\nu,r}\)-spaces
The authors investigate the two Bessel-type integral transforms \(K^\rho_r\) and \(L_r^{(n)}\) in Banach spaces \({\mathfrak L}_{\nu,r}\) with \(1\leq r< \infty\) and \(\nu\in \mathbb{R}=(-\infty, \infty)\) of Lebesgue measurable complex-valued functions on \(\mathbb{R}_+\) with \[ \int^\infty_0 \bigl|t^\nu f(t)\bigr |^r t^{-1} dt<\infty \] The two Bessel-type integral transforms were introduced by \textit{E. Krätzel} [see Proc. Conf., Varna 1975, 148-155 (1979; Zbl 0403.44003) and Wiss. Z. Univ. Jena, Math.-Naturw. Reihe 14, 369-381 (1965; Zbl 0168.10204)]. It was shown that these transforms reduce to well-known transforms. An operational calculus for these transforms was given by \textit{J. Rodrigues}, \textit{J. J. Trujillo} and \textit{M. Rivero} [Lect. Notes Pure Appl. Math. 118, 613-620 (1989; Zbl 0681.34007)]. The composition of the operator \(K^\rho_r\) with the operators \(R-L\) of fractional calculus and applications to the solutions of second-order ordinary differential equations were given by \textit{A. A. Kilbas} and \textit{S. A. Shlapakov} [Dokl. Akad. Nauk Belarusi 37, No. 4, 10-14 (1993; Zbl 0791.44005); Differ. Uravn. 30, No. 2, 256-268 (1994; Zbl 0824.47025)]. The above results were proved for the function \(f\in {\mathfrak L}_1^{\text{loc}}(\mathbb{R}_+)\) with special behaviour at \(0+\) and \(+\infty\). It is proved that the Bessel-type integral transforms \(K_r^\rho\) and \(L^{(n)}_r\) are special cases of the \(H\)-transform [see \textit{A. A. Prodnikov}, \textit{Y. A. Brychkov} and \textit{O. I. Marichev}, Integrals and series, Vol. 3: More special functions, Gordon and Breach Sci. Publ. NY (1990; Zbl 0606.33001)]. The authors in their series of papers investigated mapping properties such as the boundedness, the representation and the range of \(H\)-transforms on \({\mathfrak L}_{\nu,r}\)-spaces. In this paper they apply these results to investigate such properties of Bessel-type integral transforms on \({\mathfrak L}_{\nu,r}\)-spaces.
scientific article; zbMATH DE number 1267813