Transference of L p bounds between symmetrized Jacobi expansions and Dunkl transform
- PDF / 1,226,902 Bytes
- 33 Pages / 476.22 x 680.315 pts Page_size
- 72 Downloads / 158 Views
DOI: 0
TRANSFERENCE OF Lp BOUNDS BETWEEN SYMMETRIZED JACOBI EXPANSIONS AND DUNKL TRANSFORM W. S�LOMIAN Faculty of Pure and Applied Mathematics, Wroc�law University of Science and Technology, Wyb. Wyspia´ nskiego 27, 50-370 Wroc�law, Poland e-mail: [email protected] (Received September 30, 2019; revised February 4, 2020; accepted February 5, 2020)
Abstract. We prove two theorems showing connections between multipliers for the symmetrized Jacobi expansions and multipliers for the Dunkl transform on R. These results can be seen as generalizations of the classical Igari’s result relating Jacobi and Hankel multipliers. Moreover, we show a transference theorem relating Jacobi and Dunkl transplantations.
1. Introduction For a bounded sequence (mn )n∈Z we define a “discrete” Lp -Fourier multiplier by Um (f ) = mn �f, ein·�e−inx , f ∈ L2 ∩ Lp (−π, π). n∈Z
If Um is bounded on Lp (−π, π), then its operator norm is denoted by �mn �p,F . The symbol F stands for Fourier since it is the “Fourier” setting. Similarly, for a bounded measurable function we may define a “continuous” Lp –multiplier as follows Um (f ) = F −1 (mF (f )),
f ∈ L2 ∩ Lp (R),
where F and F −1 denote the Fourier–Plancherel transform and its inverse, respectively. If the operator Um is bounded on Lp (R), then its operator norm is denoted by |m|p,F . Igari [3] proved the following theorem relating discrete and continuous multipliers. The paper was authors master thesis written under the supervision of Professor Krzysztof Stempak. Key words and phrases: Dunkl transform, Hankel transform, Jacobi polynomial, Jacobi expansion, symmetrization, multiplier, transplantation. Mathematics Subject Classification: primary 42B10, secondary 42C10, 42A45. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
22
W. W. SL S�LOMIAN OMIAN
Theorem 1.1 [3, Theorem 2]. Let 1 < p < ∞ and let m be a bounded function on R, continuous except on a set of Lebesgue measure zero and such that limε→0+ �m(εn)�p,F is finite. Then Um is bounded on Lp (R) and |m|p,F ≤ lim �m(εn)�p,F . ε→0+
Igari [4] proved a similar result concerning multipliers for the Jacobi polynomial expansions and multipliers for the modified Hankel transform. Using the method introduced by Igari, Stempak [8] proved an analogous relation between multipliers for the Laguerre expansions and multipliers for the Hankel transform. A few years later Stempak and Betancor [1] proved, among others, that a version of the above theorem holds in the case of the Fourier–Bessel expansions and the non-modified Hankel transform. The main goal of this paper is to prove that a relation of the same type holds between multipliers for symmetrized Jacobi expansions and multipliers for the Dunkl transform. Further, we deal with the case p = 1 in which the weak type boundedness of multipliers is considered. Finally, we also investigate a relation between transplantations for the symmetrized Jacobi expansions and that for the modified Dunkl transform (to find some
Data Loading...
