doi: 10.3934/cpaa.2021176
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

* 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 Lp-boundedness properties for variation operators defined by semigroups associated with Fourier-Bessel expansions.

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. AkcogluR. 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, 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.  Google Scholar

[3]

J. J. BetancorA. J. CastroJ. CurbeloJ. 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. BetancorJ. C. FariñaE. 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.  Google Scholar

[5]

J. J. BetancorE. HarboureA. 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. CampbellR. L. JonesK. 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. CampbellR. L. JonesK. 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 gk-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. CrescimbeniR. A. MacíasT. MenárguezJ. 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. DziubańskiM. PreisnerL. 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. HytönenM. 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. JonesR. KaufmanJ. 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. JonesA. 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. MacíasC. 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. MałeckiG. 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 FranciaF. 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. WuD. 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. AkcogluR. 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, 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.  Google Scholar

[3]

J. J. BetancorA. J. CastroJ. CurbeloJ. 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. BetancorJ. C. FariñaE. 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.  Google Scholar

[5]

J. J. BetancorE. HarboureA. 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. CampbellR. L. JonesK. 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. CampbellR. L. JonesK. 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 gk-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. CrescimbeniR. A. MacíasT. MenárguezJ. 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. DziubańskiM. PreisnerL. 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. HytönenM. 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. JonesR. KaufmanJ. 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. JonesA. 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. MacíasC. 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. MałeckiG. 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 FranciaF. 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. WuD. 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

[1]

Carlo Bardaro, Ilaria Mantellini. Boundedness properties of semi-discrete sampling operators in Mellin–Lebesgue spaces. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021031

[2]

Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435

[3]

Markus Kunze, Abdallah Maichine, Abdelaziz Rhandi. Vector-valued Schrödinger operators in Lp-spaces. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1529-1541. doi: 10.3934/dcdss.2020086

[4]

Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469

[5]

Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87

[6]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389

[7]

Alexandre Thorel. A biharmonic transmission problem in Lp-spaces. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3193-3213. doi: 10.3934/cpaa.2021102

[8]

Jesse Goodman, Daniel Spector. Some remarks on boundary operators of Bessel extensions. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 493-509. doi: 10.3934/dcdss.2018027

[9]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203

[10]

Dachun Yang, Sibei Yang. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2135-2160. doi: 10.3934/cpaa.2016031

[11]

Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure & Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893

[12]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069

[13]

V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481

[14]

Danilo Costarelli, Gianluca Vinti. Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators. Mathematical Foundations of Computing, 2020, 3 (1) : 41-50. doi: 10.3934/mfc.2020004

[15]

Li Zhang, Xiaofeng Zhou, Min Chen. The research on the properties of Fourier matrix and bent function. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 571-578. doi: 10.3934/naco.2020052

[16]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[17]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[18]

Víctor Almeida, Jorge J. Betancor. Variation and oscillation for harmonic operators in the inverse Gaussian setting. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021183

[19]

Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577

[20]

Jiecheng Chen, Dashan Fan, Lijing Sun. Asymptotic estimates for unimodular Fourier multipliers on modulation spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 467-485. doi: 10.3934/dcds.2012.32.467

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (30)
  • HTML views (29)
  • Cited by (0)

[Back to Top]