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January  2022, 21(1): 239-273. doi: 10.3934/cpaa.2021176

Variation operators for semigroups associated with Fourier-Bessel expansions

Jorge J. Betancor 1, , Alejandro J. Castro 2, and  Marta De León-Contreras 3,,

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

* Corresponding author

Received  April 2021 Revised  September 2021 Published  January 2022 Early access  November 2021

Fund Project: J. J. B. was partially supported by PID2019-106093GB-I00, A. J. C. by the Nazarbayev University FDCRGP 110119FD4544 and M. D L-C by EPSRC Research Grant EP/S029486/1 and the ERCIM 'Alain Bensoussan' Fellowship Programme

In this paper we establish Lp-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.

Keywords: Variation operators, Fourier-Bessel expansions, Poisson semigroups, Hardy spaces, Lp-boundedness properties.
Mathematics Subject Classification: Primary: 42B20, 42B25; Secondary: 42B30.
Citation: Jorge J. Betancor, Alejandro J. Castro, Marta De León-Contreras. Variation operators for semigroups associated with Fourier-Bessel expansions. Communications on Pure and Applied Analysis, 2022, 21 (1) : 239-273. doi: 10.3934/cpaa.2021176
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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