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Variation operators for semigroups associated with Fourier-Bessel expansions
1. | Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Sánchez, s/n, 38721 La Laguna (Sta. Cruz de Tenerife), Spain |
2. | Department of Mathematics, Nazarbayev University, Kabanbay Batyr Ave. 53, Nur-Sultan 010000 Kazakhstan |
3. | Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway |
In this paper we establish Lp-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.
References:
[1] |
M. A. Akcoglu, R. L. Jones and P. O. Schwartz,
Variation in probability, ergodic theory and analysis, Illinois J. Math., 42 (1998), 154-177.
|
[2] |
V. Almeida, J. J. Betancor, E. Dalmasso and L. Rodríguez-Mesa, Lp-boundedness of Stein's square functions associated with Fourier-Bessel expansions, Mediterr. J. Math., 18 (2021), 40 pp.
doi: 10.1007/s00009-021-01800-x. |
[3] |
J. J. Betancor, A. J. Castro, J. Curbelo, J. C. Fariña and L. Rodríguez-Mesa,
Square functions in the Hermite setting for functions with values in UMD spaces, Ann. Mat. Pura Appl., 193 (2014), 1397-1430.
doi: 10.1007/s10231-013-0335-9. |
[4] |
J. J. Betancor, J. C. Fariña, E. Harboure and L. Rodríguez-Mesa,
Lp-boundedness properties of variation operators in the Schrödinger setting, Rev. Mat. Complut., 26 (2013), 485-534.
doi: 10.1007/s13163-012-0094-y. |
[5] |
J. J. Betancor, E. Harboure, A. Nowak and B. Viviani,
Mapping properties of fundamental operators in harmonic analysis related to Bessel operators, Studia Math., 197 (2010), 101-140.
doi: 10.4064/sm197-2-1. |
[6] |
J. J. Betancor and K. Stempak,
Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform, Tohoku Math. J., 53 (2001), 109-129.
doi: 10.2748/tmj/1178207534. |
[7] |
J. J. Betancor and K. Stempak,
On Hankel conjugate functions, Studia Sci. Math. Hungar., 41 (2004), 59-91.
doi: 10.1556/SScMath.41.2004.1.4. |
[8] |
J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math., 5–45. |
[9] |
T. A. Bui, X. T. Duong and F. K. Ly, Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications, J. Funct. Anal., 278 (2020), 108423, 55 pp.
doi: 10.1016/j.jfa.2019.108423. |
[10] |
J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl,
Oscillation and variation for the Hilbert transform, Duke Math. J., 105 (2000), 59-83.
doi: 10.1215/S0012-7094-00-10513-3. |
[11] |
J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl,
Oscillation and variation for singular integrals in higher dimensions, Trans. Amer. Math. Soc., 355 (2003), 2115-2137.
doi: 10.1090/S0002-9947-02-03189-6. |
[12] |
O. Ciaurri and L. Roncal,
Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions, J. Funct. Anal., 258 (2010), 2173-2204.
doi: 10.1016/j.jfa.2009.12.014. |
[13] |
O. Ciaurri and K. Stempak,
Transplantation and multiplier theorems for Fourier-Bessel expansions, Trans. Amer. Math. Soc., 358 (2006), 4441-4465.
doi: 10.1090/S0002-9947-06-03885-2. |
[14] |
R. Crescimbeni, R. A. Macías, T. Menárguez, J. L. Torrea and B. Viviani,
The ρ-variation as an operator between maximal operators and singular integrals, J. Evol. Equ., 9 (2009), 81-102.
doi: 10.1007/s00028-009-0003-0. |
[15] |
J. Dziubański, M. Preisner, L. Roncal and P. R. Stinga,
Hardy spaces for Fourier-Bessel expansions, J. Anal. Math., 128 (2016), 261-287.
doi: 10.1007/s11854-016-0009-9. |
[16] |
H. Hochstadt,
The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218.
doi: 10.1137/1009034. |
[17] |
T. P. Hytönen, M. T. Lacey and C. Pérez,
Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc., 45 (2013), 529-540.
doi: 10.1112/blms/bds114. |
[18] |
R. L. Jones, R. Kaufman, J. M. Rosenblatt and M. Wierdl,
Oscillation in ergodic theory, Ergod. Theor. Dynam. Syst., 18 (1998), 889-935.
doi: 10.1017/S0143385798108349. |
[19] |
R. L. Jones and K. Reinhold,
Oscillation and variation inequalities for convolution powers, Ergod. Theor. Dynam. Syst., 21 (2001), 1809-1829.
doi: 10.1017/S0143385701001869. |
[20] |
R. L. Jones, A. Seeger and J. Wright,
Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc., 360 (2008), 6711-6742.
doi: 10.1090/S0002-9947-08-04538-8. |
[21] |
R. L. Jones and G. Wang,
Variation inequalities for the Fejér and Poisson kernels, Trans. Amer. Math. Soc., 356 (2004), 4493-4518.
doi: 10.1090/S0002-9947-04-03397-5. |
[22] |
B. Langowski and A. Nowak, Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework, J. Math. Anal. Appl., 499 (2021), 125061, 36 pp.
doi: 10.1016/j.jmaa.2021.125061. |
[23] |
C. Le Merdy and Q. Xu,
Strong q-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble), 62 (2012), 2069-2097.
doi: 10.5802/aif.2743. |
[24] |
N. N. Lebedev, Special Functions and their Applications, Dover Publications, Inc., New York, 1972. |
[25] |
D. Lépingle,
La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316.
doi: 10.1007/BF00532696. |
[26] |
R. Macías, C. Segovia and J. L. Torrea,
Heat-diffusion maximal operators for Laguerre semigroups with negative parameters, J. Funct. Anal., 229 (2005), 300-316.
doi: 10.1016/j.jfa.2005.02.005. |
[27] |
J. Małecki, G. Serafin and T. Zorawik,
Fourier-Bessel heat kernel estimates, J. Math. Anal. Appl., 439 (2016), 91-102.
doi: 10.1016/j.jmaa.2016.02.051. |
[28] |
B. Muckenhoupt and E. M. Stein,
Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17-92.
doi: 10.2307/1993944. |
[29] |
A. Nowak and L. Roncal,
On sharp heat and subordinated kernel estimates in the Fourier-Bessel setting, Rocky Mountain J. Math., 44 (2014), 1321-1342.
doi: 10.1216/RMJ-2014-44-4-1321. |
[30] |
A. Nowak and L. Roncal,
Sharp heat kernel estimates in the Fourier-Bessel setting for a continuous range of the type parameter, Acta Math. Sin. (Engl. Ser.), 30 (2014), 437-444.
doi: 10.1007/s10114-014-2512-1. |
[31] |
A. Nowak and P. Sjögren,
The multi-dimensional pencil phenomenon for {L}aguerre heat-diffusion maximal operators, Math. Ann., 344 (2009), 213-248.
doi: 10.1007/s00208-008-0305-5. |
[32] |
J. Qian,
The p-variation of partial sum processes and the empirical process, Ann. Probab., 26 (1998), 1370-1383.
doi: 10.1214/aop/1022855756. |
[33] |
J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea,
Calderón-Zygmund theory for operator-valued kernels, Adv. Math., 62 (1986), 7-48.
doi: 10.1016/0001-8708(86)90086-1. |
[34] |
F. J. Ruiz and J. L. Torrea,
Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Studia Math., 88 (1988), 221-243.
doi: 10.4064/sm-88-3-221-243. |
[35] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. |
[36] |
K. Stempak, On convergence and divergence of Fourier-Bessel series, 14 (2002), 223–235 |
[37] |
J. L. Torrea and C. Zhang,
Fractional vector-valued Littlewood-Paley-Stein theory for semigroups, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 637-667.
doi: 10.1017/S0308210511001302. |
[38] |
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
![]() ![]() |
[39] |
H. Wu, D. Yang and J. Zhang,
Oscillation and variation for semigroups associated with Bessel operators, J. Math. Anal. Appl., 443 (2016), 848-867.
doi: 10.1016/j.jmaa.2016.05.044. |
[40] |
K. Yosida, Functional Analysis, Springer-Verlag New York Inc., New York, 1968. |
show all references
References:
[1] |
M. A. Akcoglu, R. L. Jones and P. O. Schwartz,
Variation in probability, ergodic theory and analysis, Illinois J. Math., 42 (1998), 154-177.
|
[2] |
V. Almeida, J. J. Betancor, E. Dalmasso and L. Rodríguez-Mesa, Lp-boundedness of Stein's square functions associated with Fourier-Bessel expansions, Mediterr. J. Math., 18 (2021), 40 pp.
doi: 10.1007/s00009-021-01800-x. |
[3] |
J. J. Betancor, A. J. Castro, J. Curbelo, J. C. Fariña and L. Rodríguez-Mesa,
Square functions in the Hermite setting for functions with values in UMD spaces, Ann. Mat. Pura Appl., 193 (2014), 1397-1430.
doi: 10.1007/s10231-013-0335-9. |
[4] |
J. J. Betancor, J. C. Fariña, E. Harboure and L. Rodríguez-Mesa,
Lp-boundedness properties of variation operators in the Schrödinger setting, Rev. Mat. Complut., 26 (2013), 485-534.
doi: 10.1007/s13163-012-0094-y. |
[5] |
J. J. Betancor, E. Harboure, A. Nowak and B. Viviani,
Mapping properties of fundamental operators in harmonic analysis related to Bessel operators, Studia Math., 197 (2010), 101-140.
doi: 10.4064/sm197-2-1. |
[6] |
J. J. Betancor and K. Stempak,
Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform, Tohoku Math. J., 53 (2001), 109-129.
doi: 10.2748/tmj/1178207534. |
[7] |
J. J. Betancor and K. Stempak,
On Hankel conjugate functions, Studia Sci. Math. Hungar., 41 (2004), 59-91.
doi: 10.1556/SScMath.41.2004.1.4. |
[8] |
J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math., 5–45. |
[9] |
T. A. Bui, X. T. Duong and F. K. Ly, Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications, J. Funct. Anal., 278 (2020), 108423, 55 pp.
doi: 10.1016/j.jfa.2019.108423. |
[10] |
J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl,
Oscillation and variation for the Hilbert transform, Duke Math. J., 105 (2000), 59-83.
doi: 10.1215/S0012-7094-00-10513-3. |
[11] |
J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl,
Oscillation and variation for singular integrals in higher dimensions, Trans. Amer. Math. Soc., 355 (2003), 2115-2137.
doi: 10.1090/S0002-9947-02-03189-6. |
[12] |
O. Ciaurri and L. Roncal,
Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions, J. Funct. Anal., 258 (2010), 2173-2204.
doi: 10.1016/j.jfa.2009.12.014. |
[13] |
O. Ciaurri and K. Stempak,
Transplantation and multiplier theorems for Fourier-Bessel expansions, Trans. Amer. Math. Soc., 358 (2006), 4441-4465.
doi: 10.1090/S0002-9947-06-03885-2. |
[14] |
R. Crescimbeni, R. A. Macías, T. Menárguez, J. L. Torrea and B. Viviani,
The ρ-variation as an operator between maximal operators and singular integrals, J. Evol. Equ., 9 (2009), 81-102.
doi: 10.1007/s00028-009-0003-0. |
[15] |
J. Dziubański, M. Preisner, L. Roncal and P. R. Stinga,
Hardy spaces for Fourier-Bessel expansions, J. Anal. Math., 128 (2016), 261-287.
doi: 10.1007/s11854-016-0009-9. |
[16] |
H. Hochstadt,
The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218.
doi: 10.1137/1009034. |
[17] |
T. P. Hytönen, M. T. Lacey and C. Pérez,
Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc., 45 (2013), 529-540.
doi: 10.1112/blms/bds114. |
[18] |
R. L. Jones, R. Kaufman, J. M. Rosenblatt and M. Wierdl,
Oscillation in ergodic theory, Ergod. Theor. Dynam. Syst., 18 (1998), 889-935.
doi: 10.1017/S0143385798108349. |
[19] |
R. L. Jones and K. Reinhold,
Oscillation and variation inequalities for convolution powers, Ergod. Theor. Dynam. Syst., 21 (2001), 1809-1829.
doi: 10.1017/S0143385701001869. |
[20] |
R. L. Jones, A. Seeger and J. Wright,
Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc., 360 (2008), 6711-6742.
doi: 10.1090/S0002-9947-08-04538-8. |
[21] |
R. L. Jones and G. Wang,
Variation inequalities for the Fejér and Poisson kernels, Trans. Amer. Math. Soc., 356 (2004), 4493-4518.
doi: 10.1090/S0002-9947-04-03397-5. |
[22] |
B. Langowski and A. Nowak, Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework, J. Math. Anal. Appl., 499 (2021), 125061, 36 pp.
doi: 10.1016/j.jmaa.2021.125061. |
[23] |
C. Le Merdy and Q. Xu,
Strong q-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble), 62 (2012), 2069-2097.
doi: 10.5802/aif.2743. |
[24] |
N. N. Lebedev, Special Functions and their Applications, Dover Publications, Inc., New York, 1972. |
[25] |
D. Lépingle,
La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316.
doi: 10.1007/BF00532696. |
[26] |
R. Macías, C. Segovia and J. L. Torrea,
Heat-diffusion maximal operators for Laguerre semigroups with negative parameters, J. Funct. Anal., 229 (2005), 300-316.
doi: 10.1016/j.jfa.2005.02.005. |
[27] |
J. Małecki, G. Serafin and T. Zorawik,
Fourier-Bessel heat kernel estimates, J. Math. Anal. Appl., 439 (2016), 91-102.
doi: 10.1016/j.jmaa.2016.02.051. |
[28] |
B. Muckenhoupt and E. M. Stein,
Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17-92.
doi: 10.2307/1993944. |
[29] |
A. Nowak and L. Roncal,
On sharp heat and subordinated kernel estimates in the Fourier-Bessel setting, Rocky Mountain J. Math., 44 (2014), 1321-1342.
doi: 10.1216/RMJ-2014-44-4-1321. |
[30] |
A. Nowak and L. Roncal,
Sharp heat kernel estimates in the Fourier-Bessel setting for a continuous range of the type parameter, Acta Math. Sin. (Engl. Ser.), 30 (2014), 437-444.
doi: 10.1007/s10114-014-2512-1. |
[31] |
A. Nowak and P. Sjögren,
The multi-dimensional pencil phenomenon for {L}aguerre heat-diffusion maximal operators, Math. Ann., 344 (2009), 213-248.
doi: 10.1007/s00208-008-0305-5. |
[32] |
J. Qian,
The p-variation of partial sum processes and the empirical process, Ann. Probab., 26 (1998), 1370-1383.
doi: 10.1214/aop/1022855756. |
[33] |
J. L. Rubio de Francia, F. J. Ruiz and J. L. Torrea,
Calderón-Zygmund theory for operator-valued kernels, Adv. Math., 62 (1986), 7-48.
doi: 10.1016/0001-8708(86)90086-1. |
[34] |
F. J. Ruiz and J. L. Torrea,
Vector-valued Calderón-Zygmund theory and Carleson measures on spaces of homogeneous nature, Studia Math., 88 (1988), 221-243.
doi: 10.4064/sm-88-3-221-243. |
[35] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. |
[36] |
K. Stempak, On convergence and divergence of Fourier-Bessel series, 14 (2002), 223–235 |
[37] |
J. L. Torrea and C. Zhang,
Fractional vector-valued Littlewood-Paley-Stein theory for semigroups, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 637-667.
doi: 10.1017/S0308210511001302. |
[38] |
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
![]() ![]() |
[39] |
H. Wu, D. Yang and J. Zhang,
Oscillation and variation for semigroups associated with Bessel operators, J. Math. Anal. Appl., 443 (2016), 848-867.
doi: 10.1016/j.jmaa.2016.05.044. |
[40] |
K. Yosida, Functional Analysis, Springer-Verlag New York Inc., New York, 1968. |
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