Transplantation and multiplier theorems for Fourier-Bessel expansions

Authors:
Óscar Ciaurri and Krzysztof Stempak

Journal:
Trans. Amer. Math. Soc. **358** (2006), 4441-4465

MSC (2000):
Primary 42C10; Secondary 44A20

DOI:
https://doi.org/10.1090/S0002-9947-06-03885-2

Published electronically:
February 20, 2006

MathSciNet review:
2231384

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Abstract | References | Similar Articles | Additional Information

Abstract: Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.

**1.**J. J. Betancor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform,*Tohoku Math. J.*,**53**(2001), 109-129. MR**1808644 (2002b:42035)****2.**J. E. Gilbert, Maximal theorems for some orthogonal series I,*Trans. Amer. Math. Soc.*,**145**(1969), 495-515. MR**0252941 (40:6156)****3.**J. J. Guadalupe, M. Pérez, F. J. Ruiz and J. L. Varona, Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems,*Proc. Amer. Math. Soc.*,**116**(1992), 457-464. MR**1096211 (92m:42032)****4.**N. N. Lebedev, Special functions and its applications, Dover, New York, 1972. MR**0350075 (50:2568)****5.**B. Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series,*Mem. Amer. Math. Soc.*,**356**(1986). MR**0858466 (88c:42035)****6.**B. Muckenhoupt, R. L. Wheeden and W.-S. Young, Sufficiency conditions for multipliers with general weights,*Trans. Amer. Math. Soc.*,**300**(1987), 463-502. MR**0876462 (88e:42016b)****7.**B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions,*Trans. Amer. Math. Soc.*,**118**(1965), 17-92. MR**0199636 (33:7779)****8.**K. Stempak, On connections between Hankel, Laguerre and Jacobi transplantation,*Tohoku Math. J.*,**54**(2002), 471-493. MR**1936265 (2003h:42045)****9.**G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1958. MR**1349110 (96i:33010)**(review of the second (1944) edition)**10.**A. Zygmund, Trigonometric series, Vols. I and II, Cambridge University Press, Cambridge, 1959. MR**0107776 (21:6498)**

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Additional Information

**Óscar Ciaurri**

Affiliation:
Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J. L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain

Email:
oscar.ciaurri@dmc.unirioja.es

**Krzysztof Stempak**

Affiliation:
Instytut Matematyki i Informatyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

Email:
stempak@pwr.wroc.pl

DOI:
https://doi.org/10.1090/S0002-9947-06-03885-2

Keywords:
Fourier-Bessel expansions,
transplantation,
multipliers,
weighted norm inequalities,
fractional integrals

Received by editor(s):
February 16, 2004

Received by editor(s) in revised form:
August 24, 2004

Published electronically:
February 20, 2006

Additional Notes:
The research of the first author was supported by grant BFM2003-06335-603-03 of the DGI

The research of the second author was supported by KBN grant #2 P03A 028 25

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.