Integral transforms with h-function kernels on L _v,r -Spaces

Abstract

Integral transforms (Hf )involving Fox's H-functions as kernels are studied in the spaces L ν,r of functions f such thatMapping properties such as the boundedness, the representation and the range of the transforms H are given.

MODIFIED H-TRANSFORMS IN £v,r-SPACES

“…Similarly to the previous section the boundedness, the representation and the range of the // 2 -transform (1.2) can be established by virtue of the relation (2.2), Lemma 2.1(a) and the corresponding results for the Htransform (1.7) given in [12]- [13] and [6]- [7].…”

“…It is known ( [12]- [13], [6]- [7]) that from the existence of the if-transform (1.7) on the space £^2 the transform can be extended to £ Utr for 1 < r < 00 such that H G [£i/, r , £1-1/,s] for a certain range of the value 5. Moreover, the range of H on £" )r is characterized in terms of the Erdelyi-Kober type fractional integral operators ^o+;<t,?7 (2-15) and in ( Such results can also be proved for the H 1 -and ii 2 -transforms on the basis of their existence on the space £",2 guaranteed in Theorems 3.1 and 3.2.…”

“…and elementary operators M£ and W& defined for £ e C and S > 0 by (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) …”

“…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].…”

“…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.…”