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A convergent finite difference method for computing minimal Lagrangian graphs
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 |
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.
References:
[1] |
H. Aimar, L. 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. Akcoglu, R. 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. 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. |
[5] |
J. J. Betancor, R. 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. 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. |
[15] |
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. |
[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-Cuerva, G. Mauceri, P. 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. Harboure, R. A. Macías, M. 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. Harboure, J. 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. Hong, W. 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. 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. |
[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. Jones, R. Kaufman, J. 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. 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. |
[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. Ma, J. 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. Menárguez, S. 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. Oberlin, A. Seeger, T. Tao, C. 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. Aimar, L. 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. Akcoglu, R. 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. 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. |
[5] |
J. J. Betancor, R. 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. 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. |
[15] |
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. |
[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-Cuerva, G. Mauceri, P. 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. Harboure, R. A. Macías, M. 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. Harboure, J. 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. Hong, W. 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. 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. |
[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. Jones, R. Kaufman, J. 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. 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. |
[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. Ma, J. 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. Menárguez, S. 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. Oberlin, A. Seeger, T. Tao, C. 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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