May  2021, 20(5): 1851-1866. doi: 10.3934/cpaa.2021045

Jump and variational inequalities for averaging operators with variable kernels

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

* Corresponding author

Received  September 2020 Revised  January 2021 Published  March 2021

Fund Project: The second author is supported by NSF-China grant-11871096 and grant-11471033

In this paper, we prove that the jump function and variation of averaging operators with rough variable kernels are bounded on $ L^{2}(\mathbb{R}^{n}) $ if $ \Omega\in L^{\infty}(\mathbb{R}^{n})\times L^{q}(\mathbb{S}^{n-1}) $ for $ q>2(n-1)/n $ and $ n\geq2 $. Moreover, we obtain the boundedness on weighted $ L^{p}(\mathbb{R}^{n}) $ spaces of the jump function and $ \rho $-variations for averaging operators with smooth variable kernels. Finally, we extend the result to the Morrey spaces.

Citation: Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1851-1866. doi: 10.3934/cpaa.2021045
References:
[1]

K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1981), 19-31.  doi: 10.4064/sm-69-1-19-31.  Google Scholar

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études SCI. Publ. Math., 69 (1989), 5–41. doi: 10.1007/BF02698838.  Google Scholar

[3]

A. P. Calderón and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc., 78 (1955), 209-224.  doi: 10.2307/1992955.  Google Scholar

[4]

A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.  doi: 10.2307/2372441.  Google Scholar

[5]

A. P. Calderón and A. Zygmund, On singular integrals with variable kernels, Applicable Anal., 7 (1978), 221-238.  doi: 10.1080/00036817808839193.  Google Scholar

[6]

Y. Chen, Boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, Acta Math. Sin. (Engl. Ser.), 25 (2009), 983-1000.  doi: 10.1007/s10114-009-7681-y.  Google Scholar

[7]

Y. ChenY. Ding and R. Li, $L^{2}$-boundedness for maximal commutators with rough variable kernels, Rev. Mat. Iberoamericana, 27 (2011), 361-391.  doi: 10.4171/RMI/640.  Google Scholar

[8]

Y. ChenY. DingG. Hong and H. Liu, Weighted jump and variational inequalities for rough operators, J. Funct. Anal., 274 (2018), 2446-2475.  doi: 10.1016/j.jfa.2018.01.009.  Google Scholar

[9]

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

[10]

Y. DingG. Hong and H. Liu, Jump and variational inequalities for rough operators, J. Fourier Anal. Appl., 23 (2017), 679-711.  doi: 10.1007/s00041-016-9484-8.  Google Scholar

[11]

G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241-256.  doi: 10.1006/jfan.1993.1032.  Google Scholar

[12]

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.  Google Scholar

[13]

R. L. JonesJ. M. Rosenblatt and M. Wierdl, Oscillation inequalities for rectangles, Proc. Am. Math. Soc., 129 (2000), 1349-1358.  doi: 10.1090/S0002-9939-00-06032-9.  Google Scholar

[14]

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

[15]

D. Kurtz, Littlewood-Paley and multipliers theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.  doi: 10.2307/1998156.  Google Scholar

[16]

B. Krause and P. Zorin-Kranich, Weighted and vector-valued variational estimates for ergodic averages, Ergodic Theory Dynam. Syst., 38 (2018), 244-256.   Google Scholar

[17]

H. Liu, Jump estimates for operators associated with polynomials, J. Math. Anal. Appl., 467 (2018), 785-806.  doi: 10.1016/j.jmaa.2018.07.012.  Google Scholar

[18]

T. MaJ. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals, J. Funct. Anal., 268 (2015), 376-416.  doi: 10.1016/j.jfa.2014.10.008.  Google Scholar

[19]

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.  Google Scholar

[20]

M. MirekE. M. Stein and B. Trojan, $l^{p}(\mathbb{Z}^{d})$-estimates for discrete operator of Radon type: variational estimates, Invent. Math., 209 (2017), 665-748.  doi: 10.1007/s00222-017-0718-4.  Google Scholar

[21]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87 (1958), 159-172.  doi: 10.1090/S0002-9947-1958-0092943-6.  Google Scholar

show all references

References:
[1]

K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math., 69 (1981), 19-31.  doi: 10.4064/sm-69-1-19-31.  Google Scholar

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études SCI. Publ. Math., 69 (1989), 5–41. doi: 10.1007/BF02698838.  Google Scholar

[3]

A. P. Calderón and A. Zygmund, On a problem of Mihlin, Trans. Amer. Math. Soc., 78 (1955), 209-224.  doi: 10.2307/1992955.  Google Scholar

[4]

A. P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921.  doi: 10.2307/2372441.  Google Scholar

[5]

A. P. Calderón and A. Zygmund, On singular integrals with variable kernels, Applicable Anal., 7 (1978), 221-238.  doi: 10.1080/00036817808839193.  Google Scholar

[6]

Y. Chen, Boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, Acta Math. Sin. (Engl. Ser.), 25 (2009), 983-1000.  doi: 10.1007/s10114-009-7681-y.  Google Scholar

[7]

Y. ChenY. Ding and R. Li, $L^{2}$-boundedness for maximal commutators with rough variable kernels, Rev. Mat. Iberoamericana, 27 (2011), 361-391.  doi: 10.4171/RMI/640.  Google Scholar

[8]

Y. ChenY. DingG. Hong and H. Liu, Weighted jump and variational inequalities for rough operators, J. Funct. Anal., 274 (2018), 2446-2475.  doi: 10.1016/j.jfa.2018.01.009.  Google Scholar

[9]

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

[10]

Y. DingG. Hong and H. Liu, Jump and variational inequalities for rough operators, J. Fourier Anal. Appl., 23 (2017), 679-711.  doi: 10.1007/s00041-016-9484-8.  Google Scholar

[11]

G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 112 (1993), 241-256.  doi: 10.1006/jfan.1993.1032.  Google Scholar

[12]

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.  Google Scholar

[13]

R. L. JonesJ. M. Rosenblatt and M. Wierdl, Oscillation inequalities for rectangles, Proc. Am. Math. Soc., 129 (2000), 1349-1358.  doi: 10.1090/S0002-9939-00-06032-9.  Google Scholar

[14]

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

[15]

D. Kurtz, Littlewood-Paley and multipliers theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc., 259 (1980), 235-254.  doi: 10.2307/1998156.  Google Scholar

[16]

B. Krause and P. Zorin-Kranich, Weighted and vector-valued variational estimates for ergodic averages, Ergodic Theory Dynam. Syst., 38 (2018), 244-256.   Google Scholar

[17]

H. Liu, Jump estimates for operators associated with polynomials, J. Math. Anal. Appl., 467 (2018), 785-806.  doi: 10.1016/j.jmaa.2018.07.012.  Google Scholar

[18]

T. MaJ. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals, J. Funct. Anal., 268 (2015), 376-416.  doi: 10.1016/j.jfa.2014.10.008.  Google Scholar

[19]

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.  Google Scholar

[20]

M. MirekE. M. Stein and B. Trojan, $l^{p}(\mathbb{Z}^{d})$-estimates for discrete operator of Radon type: variational estimates, Invent. Math., 209 (2017), 665-748.  doi: 10.1007/s00222-017-0718-4.  Google Scholar

[21]

E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc., 87 (1958), 159-172.  doi: 10.1090/S0002-9947-1958-0092943-6.  Google Scholar

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