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

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.**Jorge J. Betancor and Krzysztof Stempak,*Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform*, Tohoku Math. J. (2)**53**(2001), no. 1, 109–129. MR**1808644**, 10.2748/tmj/1178207534**2.**John E. Gilbert,*Maximal theorems for some orthogonal series. I*, Trans. Amer. Math. Soc.**145**(1969), 495–515. MR**0252941**, 10.1090/S0002-9947-1969-0252941-3**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), no. 2, 457–464. MR**1096211**, 10.1090/S0002-9939-1992-1096211-0**4.**N. N. Lebedev,*Special functions and their applications*, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR**0350075****5.**Benjamin Muckenhoupt,*Transplantation theorems and multiplier theorems for Jacobi series*, Mem. Amer. Math. Soc.**64**(1986), no. 356, iv+86. MR**858466**, 10.1090/memo/0356**6.**Benjamin Muckenhoupt, Richard L. Wheeden, and Wo-Sang Young,*Sufficiency conditions for 𝐿^{𝑝} multipliers with general weights*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 463–502. MR**876462**, 10.1090/S0002-9947-1987-0876462-0**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**, 10.1090/S0002-9947-1965-0199636-9**8.**Krzysztof Stempak,*On connections between Hankel, Laguerre and Jacobi transplantations*, Tohoku Math. J. (2)**54**(2002), no. 4, 471–493. MR**1936265****9.**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****10.**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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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:
http://dx.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.