Variation operators for semigroups associated with Fourier-Bessel expansions

Jorge J. Betancor; Alejandro J. Castro; Marta De León-Contreras

doi:10.3934/cpaa.2021176

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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 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 L^p-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.

Keywords: Variation operators, Fourier-Bessel expansions, Poisson semigroups, Hardy spaces, L^p-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 & Applied Analysis, 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. Google Scholar
[2]	V. Almeida, J. J. Betancor, E. Dalmasso and L. Rodríguez-Mesa, L_p-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. Google Scholar
[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. Google Scholar
[4]	J. J. Betancor, J. C. Fariña, E. Harboure and L. Rodríguez-Mesa, L_p-boundedness properties of variation operators in the Schrödinger setting, Rev. Mat. Complut., 26 (2013), 485-534. doi: 10.1007/s13163-012-0094-y. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math., 5–45. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	O. Ciaurri and L. Roncal, Littlewood-Paley-Stein g_k-functions for Fourier-Bessel expansions, J. Funct. Anal., 258 (2010), 2173-2204. doi: 10.1016/j.jfa.2009.12.014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218. doi: 10.1137/1009034. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	N. N. Lebedev, Special Functions and their Applications, Dover Publications, Inc., New York, 1972. Google Scholar
[25]	D. Lépingle, La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316. doi: 10.1007/BF00532696. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[36]	K. Stempak, On convergence and divergence of Fourier-Bessel series, 14 (2002), 223–235 Google Scholar
[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. Google Scholar
[38]	G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Google Scholar
[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. Google Scholar
[40]	K. Yosida, Functional Analysis, Springer-Verlag New York Inc., New York, 1968. Google Scholar

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. Google Scholar
[2]	V. Almeida, J. J. Betancor, E. Dalmasso and L. Rodríguez-Mesa, L_p-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. Google Scholar
[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. Google Scholar
[4]	J. J. Betancor, J. C. Fariña, E. Harboure and L. Rodríguez-Mesa, L_p-boundedness properties of variation operators in the Schrödinger setting, Rev. Mat. Complut., 26 (2013), 485-534. doi: 10.1007/s13163-012-0094-y. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math., 5–45. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	O. Ciaurri and L. Roncal, Littlewood-Paley-Stein g_k-functions for Fourier-Bessel expansions, J. Funct. Anal., 258 (2010), 2173-2204. doi: 10.1016/j.jfa.2009.12.014. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	H. Hochstadt, The mean convergence of Fourier-Bessel series, SIAM Rev., 9 (1967), 211-218. doi: 10.1137/1009034. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	N. N. Lebedev, Special Functions and their Applications, Dover Publications, Inc., New York, 1972. Google Scholar
[25]	D. Lépingle, La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316. doi: 10.1007/BF00532696. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[36]	K. Stempak, On convergence and divergence of Fourier-Bessel series, 14 (2002), 223–235 Google Scholar
[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. Google Scholar
[38]	G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Google Scholar
[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. Google Scholar
[40]	K. Yosida, Functional Analysis, Springer-Verlag New York Inc., New York, 1968. Google Scholar

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