Transference Results for Multipliers, Maximal Multipliers and Transplantation Operators Associated with Fourier-bessel Expansions and Hankel Transform

Our objective in this survey is to present some results concerning to transference of multipliers, maximal multipliers and transplantation operators between Fourier-Bessel series and Hankel integrals. Also we list some related problems that can be interesting and that have not been studied yet. From August 31st to September 3rd, 2004, was held in Merlo (San Luis, Argentine) the congress ”VII Encuentro de Analistas Alberto Calderón y I Encuentro Conjunto Hispano-Argentino de Análisis”. This meeting was dedicated to Professor Roberto Maćıas in his 60th birthday. In these notes we include the main results that were commented in the talk presented by the author there. That talk was devoted to Professor Roberto Maćıas who has been an example for many Spanish and Argentine mathematicians and so is this note. Our purpose is to present some results about transference of boundedness of multipliers, maximal operators associated to multipliers and transplantation operators relating mainly Hankel transforms and Fourier-Bessel expansions settings. The author knew these topics when he wrote the papers [10] and [11] jointly with Professor Krzysztof Stempak who introduced him in the questions of transference. The author would like to thank to Professor Stempak all the very fruitfull and nice discussions about these and other mathematical topics. 1. Transference of boundedness for multipliers. Although all the following definitions and results related to Fourier integrals and series can be given in dimension n ≥ 1, to simplify we will write them for n = 1. As it is wellknown the Fourier transform on R is defined by F(f)(y) = 1 √ 2π ∫ ∞ −∞ f(x)e−iyxdx, y ∈ R, provided that f ∈ L(R). Moreover Hausdorff-Young Theorem says that the Fourier transform can be extended to L(R) as a bounded operator from L(R) into L ′ (R), for every 1 ≤ p ≤ 2, where p′ denotes the exponent conjugated to p, that is, p′ = p p−1 . Partially supported by Grant PI2003/068 and MTM2004/05878.