Integral transforms with fox'sH-function in spaces of summable functions

Weighted norm inequalities for the ℋ‐transformation

“…It is sufficient to take into account that ].9(x)[ < Cox -, x > 0, with c < a </3 and for certain Co > 0 By using this inequality instead of (2 5) It is clear that f ,.v. and one has dz < w"zO"-ldx (15) llfli,.. …”

“…A. Kilbas, M. Saigo and S. A. Shlapakov 15], [16] and 17]), investigated the behavior of transformation (1) in certain weighted Lp spaces introduced by P. G. Rooney [21 ].…”

MODIFIED H-TRANSFORMS IN £v,r-SPACES

“…In this paper we apply the results of [11]- [13] and [6]- [7] to investigate such properties of the H 1 -and ii" 2 -transforms (1.1) and (1.2). Section 2 contains some auxiliary results, definitions and notations.…”

“…The integral transform (1.7) was first considered by Fox [4] while investigating G-and //-functions as symmetrical Fourier kernels. Many authors investigated the properties of the ii-transforms in Z>i(0, oo), ¿2(0, 00) and some special function spaces (see a short survey and bibliography in [11]- [13]). Mapping properties such as the boundedness, the representation and the range of the if-transform (1.7) were proved independently by the authors together with Shlapakov in [11]- [13] and with Shlapakov and Glaeske in [6]- [7] and by Betancor and Jerez Diaz [1], The invertibility of (1.7) in £ Uir was given by the authors together with Shlapakov in [24].…”

“…Many authors investigated the properties of the ii-transforms in Z>i(0, oo), ¿2(0, 00) and some special function spaces (see a short survey and bibliography in [11]- [13]). Mapping properties such as the boundedness, the representation and the range of the if-transform (1.7) were proved independently by the authors together with Shlapakov in [11]- [13] and with Shlapakov and Glaeske in [6]- [7] and by Betancor and Jerez Diaz [1], The invertibility of (1.7) in £ Uir was given by the authors together with Shlapakov in [24]. We note that the investigation of the boundedness, representation and range of the if-transform (1.7) is based on the technique of Mellin transform developed by Rooney [20] for the G-transform, while the invertibility of such a transform is due to the asymptotic behavior of the function H™ q n (z) in (1.4) considered by the authors in [9].…”