February  2022, 21(2): 419-470. doi: 10.3934/cpaa.2021183

Variation and oscillation for harmonic operators in the inverse Gaussian setting

Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Sánchez, s/n, 38721 La Laguna (Sta. Cruz de Tenerife), Spain

* Corresponding Author

Received  March 2021 Revised  September 2021 Published  February 2022 Early access  November 2021

Fund Project: This paper is partially supported by grant PID2019-106093GB-I00 from the Spanish Government

We prove variation and oscillation $ L^p $-inequalities associated with fractional derivatives of certain semigroups of operators and with the family of truncations of Riesz transforms in the inverse Gaussian setting. We also study these variational $ L^p $-inequalities in a Banach-valued context by considering Banach spaces with the UMD-property and whose martingale cotype is fewer than the variational exponent. We establish $ L^p $-boundedness properties for weighted difference involving the semigroups under consideration.

Citation: Víctor Almeida, Jorge J. Betancor. Variation and oscillation for harmonic operators in the inverse Gaussian setting. Communications on Pure and Applied Analysis, 2022, 21 (2) : 419-470. doi: 10.3934/cpaa.2021183
References:
[1]

H. AimarL. Forzani and R. Scotto, On Riesz transforms and maximal functions in the context of Gaussian harmonic analysis, Trans. Amer. Math. Soc., 359 (2007), 2137-2154.  doi: 10.1090/S0002-9947-06-04100-6.

[2]

M. A. AkcogluR. L. Jones and P. O. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math., 42 (1998), 154-177. 

[3]

J. J. Betancor, A. Castro and M. de León Contreras, The hardy-littlewood property and maximal operators associated with the inverse gauss measure, arXiv: 2010.01341.

[4]

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.

[5]

J. J. BetancorR. Crescimbeni and J. L. Torrea, The $\rho$-variation of the heat semigroup in the Hermitian setting: behaviour in $L^\infty$, Proc. Edinb. Math. Soc., 54 (2011), 569-585.  doi: 10.1017/S0013091510000556.

[6]

J. J. Betancor and L. Rodríguez, Higher order riesz transforms in the inverse gauss setting and UMD banach spaces, arXiv: 2011.11285.

[7]

O. Blasco and P. Villarroya, Transference of vector-valued multipliers on weighted $L^p$-spaces, Canad. J. Math., 65 (2013), 510-543.  doi: 10.4153/CJM-2012-041-0.

[8]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.  doi: 10.1007/BF02384306.

[9]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes \'Etudes Sci. Publ. Math., 69 (1989), 5–45.

[10]

T. Bruno, Endpoint results for the Riesz transform of the Ornstein-Uhlenbeck operator, J. Fourier Anal. Appl., 25 (2019), 1609-1631.  doi: 10.1007/s00041-018-09648-8.

[11]

T. Bruno, Singular integrals and Hardy type spaces for the inverse Gauss measure, J. Geom. Anal., 31 (2021), 6481-6528.  doi: 10.1007/s12220-020-00541-9.

[12]

T. Bruno and P. Sjögren, On the Riesz transforms for the inverse Gauss measure, Ann. Fenn. Math., 46 (2021), 433-448.  doi: 10.5186/aasfm.2021.4609.

[13]

D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser. Wadsworth, Belmont, CA, 1983, pp. 270–286.

[14]

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.

[15]

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.

[16]

Z. Chao and J. Torrea, Boundedness of differential transforms for heat semigroups generated by schrödinger operators, Cand. J. Math. (2020), 1–34. doi: 10.4153/S0008414X20000097.

[17]

L. Deleaval and C. Kriegler, Maximal and q-variational hörmander functional calculus., Preprint, 2019.

[18]

Y. Do and M. Lacey, Weighted bounds for variational Fourier series, Studia Math., 211 (2012), 153-190.  doi: 10.4064/sm211-2-4.

[19]

J. García-CuervaG. MauceriP. Sjögren and J. L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math., 78 (1999), 281-305.  doi: 10.1007/BF02791138.

[20]

E. HarboureR. A. MacíasM. T. Menárguez and J. L. Torrea, Oscillation and variation for the Gaussian Riesz transforms and Poisson integral, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 85-104.  doi: 10.1017/S0308210500003772.

[21]

E. HarboureJ. L. Torrea and B. Viviani, Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup, J. Anal. Math., 91 (2003), 1-29.  doi: 10.1007/BF02788780.

[22]

G. HongW. Liu and T. Ma, Vector-valued $q$-variational inequalities for averaging operators and the Hilbert transform, Arch. Math. (Basel), 115 (2020), 423-433.  doi: 10.1007/s00013-020-01472-1.

[23]

G. Hong and T. Ma, Vector valued $q$-variation for differential operators and semigroups I, Math. Z., 286 (2017), 89-120.  doi: 10.1007/s00209-016-1756-0.

[24]

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.

[25]

Jr. Jodeit and M., Restrictions and extensions of Fourier multipliers, Studia Math., 34 (1970), 215-226.  doi: 10.4064/sm-34-2-215-226.

[26]

R. L. JonesR. KaufmanJ. M. Rosenblatt and M. Wierdl, Oscillation in ergodic theory, Ergodic Theory Dynam. Syst., 18 (1998), 889-935.  doi: 10.1017/S0143385798108349.

[27]

R. L. Jones and K. Reinhold, Oscillation and variation inequalities for convolution powers, Ergodic Theory Dynam. Syst., 21 (2001), 1809-1829.  doi: 10.1017/S0143385701001869.

[28]

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.

[29]

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.

[30]

C. Le Merdy and Q. Xu, Strong $q$-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble), 62 (2012), 2069–2097 (2013). doi: 10.5802/aif.2743.

[31]

D. Lépingle, La variation d'ordre $p$ des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316.  doi: 10.1007/BF00532696.

[32]

T. MaJ. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals in higher dimensions, Sci. China Math., 60 (2017), 1419-1442.  doi: 10.1007/s11425-016-9012-7.

[33]

A. Mas and X. Tolsa, Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs, Proc. Lond. Math. Soc., 105 (2012), 49-86.  doi: 10.1112/plms/pdr061.

[34]

T. MenárguezS. Pérez and F. Soria, The Mehler maximal function: a geometric proof of the weak type 1, J. London Math. Soc., 61 (2000), 846-856.  doi: 10.1112/S0024610700008723.

[35]

B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc., 139 (1969), 243-260.  doi: 10.2307/1995317.

[36]

R. OberlinA. SeegerT. TaoC. Thiele and J. Wright, A variation norm Carleson theorem, J. Eur. Math. Soc., 2 (2012), 421-464.  doi: 10.4171/JEMS/307.

[37]

S. Pérez and F. Soria, Operators associated with the Ornstein-Uhlenbeck semigroup, J. London Math. Soc., 3 (2000), 857-871.  doi: 10.1112/S0024610700008917.

[38]

G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math., 20 (1975), 326-350.  doi: 10.1007/BF02760337.

[39]

G. Pisier and Q. H. Xu, The strong $p$-variation of martingales and orthogonal series, Probab. Theory Related Fields, 4 (1988), 497-514.  doi: 10.1007/BF00959613.

[40]

J. Qian, The $p$-variation of partial sum processes and the empirical process, Ann. Probab., 3 (1998), 1370-1383.  doi: 10.1214/aop/1022855756.

[41]

F. Salogni, Harmonic Bergman Spaces, Hardy-Type Spaces and Harmonic Analysis of a Symetric Diffusion Semigroup on $\mathbb R^n$, PhD thesis, Università degli Studi di Milano-Bicocca, 2013.

[42]

J. Teuwen, On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroupì, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 19 (2016), 1650030, 13 pp. doi: 10.1142/S0219025716500302.

show all references

References:
[1]

H. AimarL. Forzani and R. Scotto, On Riesz transforms and maximal functions in the context of Gaussian harmonic analysis, Trans. Amer. Math. Soc., 359 (2007), 2137-2154.  doi: 10.1090/S0002-9947-06-04100-6.

[2]

M. A. AkcogluR. L. Jones and P. O. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math., 42 (1998), 154-177. 

[3]

J. J. Betancor, A. Castro and M. de León Contreras, The hardy-littlewood property and maximal operators associated with the inverse gauss measure, arXiv: 2010.01341.

[4]

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.

[5]

J. J. BetancorR. Crescimbeni and J. L. Torrea, The $\rho$-variation of the heat semigroup in the Hermitian setting: behaviour in $L^\infty$, Proc. Edinb. Math. Soc., 54 (2011), 569-585.  doi: 10.1017/S0013091510000556.

[6]

J. J. Betancor and L. Rodríguez, Higher order riesz transforms in the inverse gauss setting and UMD banach spaces, arXiv: 2011.11285.

[7]

O. Blasco and P. Villarroya, Transference of vector-valued multipliers on weighted $L^p$-spaces, Canad. J. Math., 65 (2013), 510-543.  doi: 10.4153/CJM-2012-041-0.

[8]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat., 21 (1983), 163-168.  doi: 10.1007/BF02384306.

[9]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes \'Etudes Sci. Publ. Math., 69 (1989), 5–45.

[10]

T. Bruno, Endpoint results for the Riesz transform of the Ornstein-Uhlenbeck operator, J. Fourier Anal. Appl., 25 (2019), 1609-1631.  doi: 10.1007/s00041-018-09648-8.

[11]

T. Bruno, Singular integrals and Hardy type spaces for the inverse Gauss measure, J. Geom. Anal., 31 (2021), 6481-6528.  doi: 10.1007/s12220-020-00541-9.

[12]

T. Bruno and P. Sjögren, On the Riesz transforms for the inverse Gauss measure, Ann. Fenn. Math., 46 (2021), 433-448.  doi: 10.5186/aasfm.2021.4609.

[13]

D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser. Wadsworth, Belmont, CA, 1983, pp. 270–286.

[14]

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.

[15]

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.

[16]

Z. Chao and J. Torrea, Boundedness of differential transforms for heat semigroups generated by schrödinger operators, Cand. J. Math. (2020), 1–34. doi: 10.4153/S0008414X20000097.

[17]

L. Deleaval and C. Kriegler, Maximal and q-variational hörmander functional calculus., Preprint, 2019.

[18]

Y. Do and M. Lacey, Weighted bounds for variational Fourier series, Studia Math., 211 (2012), 153-190.  doi: 10.4064/sm211-2-4.

[19]

J. García-CuervaG. MauceriP. Sjögren and J. L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math., 78 (1999), 281-305.  doi: 10.1007/BF02791138.

[20]

E. HarboureR. A. MacíasM. T. Menárguez and J. L. Torrea, Oscillation and variation for the Gaussian Riesz transforms and Poisson integral, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 85-104.  doi: 10.1017/S0308210500003772.

[21]

E. HarboureJ. L. Torrea and B. Viviani, Vector-valued extensions of operators related to the Ornstein-Uhlenbeck semigroup, J. Anal. Math., 91 (2003), 1-29.  doi: 10.1007/BF02788780.

[22]

G. HongW. Liu and T. Ma, Vector-valued $q$-variational inequalities for averaging operators and the Hilbert transform, Arch. Math. (Basel), 115 (2020), 423-433.  doi: 10.1007/s00013-020-01472-1.

[23]

G. Hong and T. Ma, Vector valued $q$-variation for differential operators and semigroups I, Math. Z., 286 (2017), 89-120.  doi: 10.1007/s00209-016-1756-0.

[24]

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.

[25]

Jr. Jodeit and M., Restrictions and extensions of Fourier multipliers, Studia Math., 34 (1970), 215-226.  doi: 10.4064/sm-34-2-215-226.

[26]

R. L. JonesR. KaufmanJ. M. Rosenblatt and M. Wierdl, Oscillation in ergodic theory, Ergodic Theory Dynam. Syst., 18 (1998), 889-935.  doi: 10.1017/S0143385798108349.

[27]

R. L. Jones and K. Reinhold, Oscillation and variation inequalities for convolution powers, Ergodic Theory Dynam. Syst., 21 (2001), 1809-1829.  doi: 10.1017/S0143385701001869.

[28]

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.

[29]

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.

[30]

C. Le Merdy and Q. Xu, Strong $q$-variation inequalities for analytic semigroups, Ann. Inst. Fourier (Grenoble), 62 (2012), 2069–2097 (2013). doi: 10.5802/aif.2743.

[31]

D. Lépingle, La variation d'ordre $p$ des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295-316.  doi: 10.1007/BF00532696.

[32]

T. MaJ. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals in higher dimensions, Sci. China Math., 60 (2017), 1419-1442.  doi: 10.1007/s11425-016-9012-7.

[33]

A. Mas and X. Tolsa, Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs, Proc. Lond. Math. Soc., 105 (2012), 49-86.  doi: 10.1112/plms/pdr061.

[34]

T. MenárguezS. Pérez and F. Soria, The Mehler maximal function: a geometric proof of the weak type 1, J. London Math. Soc., 61 (2000), 846-856.  doi: 10.1112/S0024610700008723.

[35]

B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc., 139 (1969), 243-260.  doi: 10.2307/1995317.

[36]

R. OberlinA. SeegerT. TaoC. Thiele and J. Wright, A variation norm Carleson theorem, J. Eur. Math. Soc., 2 (2012), 421-464.  doi: 10.4171/JEMS/307.

[37]

S. Pérez and F. Soria, Operators associated with the Ornstein-Uhlenbeck semigroup, J. London Math. Soc., 3 (2000), 857-871.  doi: 10.1112/S0024610700008917.

[38]

G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math., 20 (1975), 326-350.  doi: 10.1007/BF02760337.

[39]

G. Pisier and Q. H. Xu, The strong $p$-variation of martingales and orthogonal series, Probab. Theory Related Fields, 4 (1988), 497-514.  doi: 10.1007/BF00959613.

[40]

J. Qian, The $p$-variation of partial sum processes and the empirical process, Ann. Probab., 3 (1998), 1370-1383.  doi: 10.1214/aop/1022855756.

[41]

F. Salogni, Harmonic Bergman Spaces, Hardy-Type Spaces and Harmonic Analysis of a Symetric Diffusion Semigroup on $\mathbb R^n$, PhD thesis, Università degli Studi di Milano-Bicocca, 2013.

[42]

J. Teuwen, On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroupì, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 19 (2016), 1650030, 13 pp. doi: 10.1142/S0219025716500302.

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