On the range and invertibility of a class of Mellin multiplier transforms. I. (English) Zbl 0736.44004
This paper is the first in an announced series of papers by the authors in which they study operators whose (Mellin) multipliers involve products and quotients of gamma functions and which include as special cases operators with \(G\)-function kernels (studied by P. G. Rooney in a series of papers during the seventies and the eighties).
The present paper concentrates on the range and invertibility of operators with multipliers of type \(\Gamma(\eta+s/m)\) and \(\Gamma(\xi- s/\ell)\) on weighted \(L^ p\)-spaces. The analysis is long, because of the various cases to be considered; however, a simplification is obtained by considering these operators on certain spaces of generalised functions (in which the weighted \(L^ p\)-spaces under consideration are embedded).
The present paper concentrates on the range and invertibility of operators with multipliers of type \(\Gamma(\eta+s/m)\) and \(\Gamma(\xi- s/\ell)\) on weighted \(L^ p\)-spaces. The analysis is long, because of the various cases to be considered; however, a simplification is obtained by considering these operators on certain spaces of generalised functions (in which the weighted \(L^ p\)-spaces under consideration are embedded).
Reviewer: B.P.Duggal (Lesotho)
MSC:
| 44A15 | Special integral transforms (Legendre, Hilbert, etc.) |
| 46F12 | Integral transforms in distribution spaces |
| 42A45 | Multipliers in one variable harmonic analysis |
Keywords:
Mellin multiplier transform; gamma functions; \(G\)-function kernels; range; invertibility; weighted \(L^ p\)-spaces; spaces of generalised functionsReferences:
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