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Parabolic Muckenhoupt weights characterized by parabolic fractional maximal and integral operators with time lag
Part of:
General theory of linear operators
Harmonic analysis in several variables
Parabolic equations and systems
Published online by Cambridge University Press: 17 March 2025
Abstract
In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.
Keywords
MSC classification
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900), the National Natural Science Foundation of China (Grant Nos. 12431006 and 12371093), and the Fundamental Research Funds for the Central Universities (Grant No. 2233300008).
References
Adams, D. R. and Hedberg, L. I., Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 314, Springer-Verlag, Berlin, 1996.Google Scholar
Aimar, H.,
Elliptic and parabolic BMO and Harnack’s inequality
. Trans. Amer. Math. Soc. 306(1988), 265–276.Google Scholar
Aimar, H., Forzani, L., and Martín-Reyes, F. J.,
On weighted inequalities for singular integrals
. Proc. Amer. Math. Soc. 125(1997), 2057–2064.Google Scholar
Aliev, I. A. and Rubin, B.,
Parabolic potentials and wavelet transforms with the generalized translation
. Studia Math. 145(2001), 1–16.Google Scholar
Aliev, I. A. and Rubin, B.,
Parabolic wavelet transforms and Lebesgue spaces of parabolic potentials, Conference on Special Functions (Tempe, AZ, 2000)
. Rocky Mountain J. Math. 32(2002), 391–408.Google Scholar
Andersen, K. F. and Sawyer, E. T.,
Weighted norm inequalities for the Riemann–Liouville and Weyl fractional integral operators
. Trans. Amer. Math. Soc. 308(1988), 547–558.Google Scholar
Bergh, J. and Löfström, J., Interpolation spaces, an introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin and New York, 1976.Google Scholar
Berkovits, L.,
Parabolic Muckenhoupt weights in the Euclidean space
. J. Math. Anal. Appl. 379(2011), 524–537.Google Scholar
Berra, F.,
Restricted weak type inequalities for the one-sided Hardy–Littlewood maximal operator in higher dimensions
. Czechoslovak Math. J. 72(2022), no. 147, 1003–1017.Google Scholar
Bhat, M. A. and Kosuru, G. S. R.,
Trace principle for Riesz potentials on Herz-type spaces and applications
. J. Inequal. Appl. 2024(2024), Paper No. 113, 13 pp.Google Scholar
Bögelein, V., Duzaar, F., Kinnunen, J., and Scheven, C.,
Higher integrability for doubly nonlinear parabolic systems
. J. Math. Pures Appl. (9) 143(2020), 31–72.Google Scholar
Bögelein, V., Duzaar, F., and Liao, N.,
On the Hölder regularity of signed solutions to a doubly nonlinear equation
. J. Funct. Anal. 281(2021), Paper No. 109173, 58 pp.Google Scholar
Bögelein, V., Duzaar, F., Liao, N., and Schätzler, L.,
On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II
. Rev. Mat. Iberoam. 39(2023), 1005–1037.Google Scholar
Bögelein, V., Duzaar, F., and Scheven, C.,
Higher integrability for doubly nonlinear parabolic systems
. Partial Differ. Equ. Appl. 3(2022), Paper No. 74, 41 pp.Google Scholar
Bögelein, V., Heran, A., Schätzler, L., and Singer, T.,
Harnack’s inequality for doubly nonlinear equations of slow diffusion type
. Calc. Var. Partial Differential Equations 60(2021), Paper No. 215, 35 pp.Google Scholar
Chen, Y., Jia, H., and Yang, D.,
Boundedness of fractional integrals on Hardy spaces associated with ball quasi-Banach function spaces
. Tokyo J. Math. 47(2024), 19–59.Google Scholar
Chill, R. and Król, S.,
Weighted inequalities for singular integral operators on the half-line
. Studia Math. 243(2018), 171–206.Google Scholar
Cruz-Uribe, D. and Fiorenza, A., Variable Lebesgue spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013.Google Scholar
Cruz-Uribe, D. and MacLellan, J.,
Necessary conditions for two-weight inequalities for singular integral operators
. Studia Math. 261(2021), 187–205.Google Scholar
Cruz-Uribe, D., Martell, J. M., and Pérez, C., Extrapolation from
${A}_{\infty}\!\!$
weights and applications
. J. Funct. Anal. 213(2004), 412–439.Google Scholar
${A}_{\infty}\!\!$
weights and applications
. J. Funct. Anal. 213(2004), 412–439.Google ScholarCruz-Uribe, D., Martell, J. M., and Pérez, C., Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, 215, Birkhäuser/Springer Basel AG, Basel, 2011.Google Scholar
Cruz-Uribe, D., Martell, J. M., and Pérez, C.,
Sharp weighted estimates for classical operators
. Adv. Math. 229(2012), 408–441.Google Scholar
Cruz-Uribe, D. and Moen, K.,
One and two weight norm inequalities for Riesz potentials
. Illinois J. Math. 57(2013), 295–323.Google Scholar
Cruz-Uribe, D. and Myyryläinen, K., Two-weight norm inequalities for parabolic fractional maximal functions. Preprint, arXiv:2410.01012.Google Scholar
Cruz-Uribe, D., Neugebauer, C. J., and Olesen, V.,
The one-sided minimal operator and the one-sided reverse Hölder inequality
. Studia Math. 116(1995), 255–270.Google Scholar
Cruz-Uribe, D. and Suragan, D.,
Hardy–Leray inequalities in variable Lebesgue spaces
. J. Math. Anal. Appl. 530(2024), Paper No. 127747, 14 pp.Google Scholar
Fabes, E. B. and Garofalo, N.,
Parabolic B.M.O. and Harnack’s inequality
. Proc. Amer. Math. Soc. 95(1985), 63–69.Google Scholar
Fabes, E. B., Kenig, C. E., and Serapioni, R. P.,
The local regularity of solutions of degenerate elliptic equations
. Comm. Partial Differential Equations 7(1982), 77–116.Google Scholar
Forzani, L., Martín-Reyes, F. J., and Ombrosi, S.,
Weighted inequalities for the two-dimensional one-sided Hardy–Littlewood maximal function
. Trans. Amer. Math. Soc. 363(2011), 1699–1719.Google Scholar
Ghosh, A. and Mohanty, P.,
Weighted inequalities for higher dimensional one-sided Hardy–Littlewood maximal function in Orlicz spaces
. Expo. Math. 40(2022), 23–44.Google Scholar
Ghosh, A. and Shuin, K.,
Local one-sided maximal function on fractional Sobolev spaces
. Math. Inequal. Appl. 22(2019), 519–530.Google Scholar
Gianazza, U. and Vespri, V.,
A Harnack inequality for solutions of doubly nonlinear parabolic equations
. J. Appl. Funct. Anal. 1(2006), 271–284.Google Scholar
Gopala Rao, V. R.,
A characterization of parabolic function spaces
. Amer. J. Math. 99(1977), 985–993.Google Scholar
Grafakos, L., Classical Fourier analysis. 3rd ed., Graduate Texts in Mathematics 249, Springer, New York, 2014.Google Scholar
Guliyev, V. S.,
Characterizations for the fractional maximal operator and its commutators on total Morrey spaces
. Positivity 28(2024), Paper No. 51, 20 pp.Google Scholar
Hajłasz, P. and Koskela, P.,
Sobolev met Poincaré
. Mem. Amer. Math. Soc. 145(2000), no. 688, x+101 pp.Google Scholar
Hebisch, W. and Saloff-Coste, L.,
On the relation between elliptic and parabolic Harnack inequalities
. Ann. Inst. Fourier (Grenoble) 51(2001), 1437–1481.Google Scholar
Hytönen, T.,
The sharp weighted bound for general Calderón–Zygmund operators
. Ann. of Math. (2) 175(2012), 1473–1506.Google Scholar
Hytönen, T.,
The two-weight inequality for the Hilbert transform with general measures
. Proc. Lond. Math. Soc. (3) 117(2018), 483–526.Google Scholar
Hytönen, T. and Pérez, C.,
Sharp weighted bounds involving
${A}_{\infty }$
. Anal. PDE 6(2013), 777–818.Google Scholar
${A}_{\infty }$
. Anal. PDE 6(2013), 777–818.Google ScholarHytönen, T., Pérez, C., and Rela, E., Sharp reverse Hölder property for
${A}_{\infty }\!\!$
weights on spaces of homogeneous type
. J. Funct. Anal. 263(2012), 3883–3899.Google Scholar
${A}_{\infty }\!\!$
weights on spaces of homogeneous type
. J. Funct. Anal. 263(2012), 3883–3899.Google ScholarJones, B. F.,
Lipschitz spaces and the heat equation
. J. Math. Mech. 18(1968/69), 379–409.Google Scholar
Kairema, A.,
Sharp weighted bounds for fractional integral operators in a space of homogeneous type
. Math. Scand. 114(2014), 226–253.Google Scholar
Kinnunen, J. and Kuusi, T.,
Local behaviour of solutions to doubly nonlinear parabolic equations
. Math. Ann. 337(2007), 705–728.Google Scholar
Kinnunen, J. and Myyryläinen, K.,
Characterizations of parabolic Muckenhoupt classes
. Adv. Math. 444(2024), Paper No. 109612, 57 pp.Google Scholar
Kinnunen, J. and Myyryläinen, K., Characterizations of parabolic reverse Hölder classes. Preprint, arXiv:2310.00370.Google Scholar
Kinnunen, J., Myyryläinen, K., and Yang, D.,
John–Nirenberg inequalities for parabolic BMO
. Math. Ann. 387(2023), 1125–1162.Google Scholar
Kinnunen, J., Myyryläinen, K., Yang, D., and Zhu, C.,
Parabolic Muckenhoupt weights with time lag on spaces of homogeneous type with monotone geodesic property
. Potential Anal. 60(2024), 1513–1569.Google Scholar
Kinnunen, J. and Saari, O.,
On weights satisfying parabolic Muckenhoupt conditions
. Nonlinear Anal. 131(2016), 289–299.Google Scholar
Kinnunen, J. and Saari, O.,
Parabolic weighted norm inequalities and partial differential equations
. Anal. PDE 9(2016), 1711–1736.Google Scholar
Kokilashvili, V., Meskhi, A., and Zaighum, M.,
Sharp weighted bounds for fractional integrals via the two-weight theory
. Banach J. Math. Anal. 12(2018), 673–692.Google Scholar
Kong, W., Yang, D., Yuan, W., and Zhu, C.,
Real-variable characterizations of optimal parabolic Campanato spaces with time lag related to almost increasing functions
, Submitted.Google Scholar
Kuusi, T., Siljander, J., and Urbano, J. M.,
Local Hölder continuity for doubly nonlinear parabolic equations
. Indiana Univ. Math. J. 61(2012), 399–430.Google Scholar
Lacey, M. and Li, J.,
Compactness of commutator of Riesz transforms in the two weight setting
. J. Math. Anal. Appl. 508(2022), Paper No. 125869, 11 pp.Google Scholar
Lacey, M., Moen, K., Pérez, C., and Torres, R.,
Sharp weighted bounds for fractional integral operators
. J. Funct. Anal. 259(2010), 1073–1097.Google Scholar
Lerner, A. and Ombrosi, S.,
A boundedness criterion for general maximal operators
. Publ. Mat. 54(2010), 53–71.Google Scholar
Li, J. and Wick, B. D.,
The two-weight inequality for the Poisson operator in the Bessel setting
. J. Math. Anal. Appl. 489(2020), 124178, 15 pp.Google Scholar
Lorente, M.,
A characterization of two weight norm inequalities for one-sided operators of fractional type
. Canad. J. Math. 49(1997), 1010–1033.Google Scholar
Lorente, M. and Martín-Reyes, F. J.,
A note on weighted inequalities for a one-sided maximal operator in
${\mathbb{R}}^n$
. Rev. Un. Mat. Argentina 58(2017), 253–258.Google Scholar
${\mathbb{R}}^n$
. Rev. Un. Mat. Argentina 58(2017), 253–258.Google ScholarLu, S., Ding, Y., and Yan, D., Singular integrals and related topics, World Scientific Publishing Co Pte. Ltd., Hackensack, NJ, 2007.Google Scholar
Ma, J., He, Q., and Yan, D.,
Weighted characterization of parabolic fractional maximal operator
. Front. Math. 18(2023), 185–196.Google Scholar
Macías, R. A. and Riveros, M. S.,
One-sided extrapolation at infinity and singular integrals
. Proc. Roy. Soc. Edinburgh Sect. A 130(2000), 1081–1102.Google Scholar
Martín-Reyes, F. J.,
New proofs of weighted inequalities for the one-sided Hardy–Littlewood maximal functions
. Proc. Amer. Math. Soc. 117(1993), 691–698.Google Scholar
Martín-Reyes, F. J., Ortega Salvador, P., and de la Torre, A.,
Weighted inequalities for one-sided maximal functions
. Trans. Amer. Math. Soc. 319(1990), 517–534.Google Scholar
Martín-Reyes, F. J., Pick, L., and de la Torre, A.,
${A}_{\infty}^{+}$
condition
. Canad. J. Math. 45(1993), 1231–1244.Google Scholar
${A}_{\infty}^{+}$
condition
. Canad. J. Math. 45(1993), 1231–1244.Google ScholarMartín-Reyes, F. J. and Rivera-Ríos, I. P., A note on one sided extrapolation of compactness and applications. Preprint, arXiv: 2406.11663.Google Scholar
Martín-Reyes, F. J. and de la Torre, A.,
Two weight norm inequalities for fractional one-sided maximal operators
. Proc. Amer. Math. Soc. 117(1993), 483–489.Google Scholar
Martín-Reyes, F. J. and de la Torre, A.,
One-sided BMO spaces
. J. London Math. Soc. (2) 49(1994), 529–542.Google Scholar
Martín-Reyes, F. J. and de la Torre, A.,
Sharp weighted bounds for one-sided maximal operators
. Collect. Math. 66(2015), 161–174.Google Scholar
Melchiori, L. and Pradolini, G.,
Potential operators and their commutators acting between variable Lebesgue spaces with different weights
. Integral Transforms Spec. Funct. 29(2018), 909–926.Google Scholar
Moser, J.,
On Harnack’s theorem for elliptic differential equations
. Comm. Pure Appl. Math. 14(1961), 577–591.Google Scholar
Moser, J.,
A Harnack inequality for parabolic differential equations
. Comm. Pure Appl. Math. 17(1964), 101–134.Google Scholar
Moser, J.,
Correction to: “A Harnack inequality for parabolic differential equations”
. Comm. Pure Appl. Math. 20(1967), 231–236.Google Scholar
Moser, J.,
On a pointwise estimate for parabolic differential equations
. Comm. Pure Appl. Math. 24(1971), 727–740.Google Scholar
Muckenhoupt, B.,
Weighted norm inequalities for the Hardy maximal function
. Trans. Amer. Math. Soc. 165(1972), 207–226.Google Scholar
Muckenhoupt, B. and Wheeden, R. L.,
Weighted norm inequalities for fractional integrals
. Trans. Amer. Math. Soc. 192(1974), 261–274.Google Scholar
Myyryläinen, K. and Yang, D.,
Parabolic John–Nirenberg spaces with time lag
. Math. Z. 306(2024), Paper No. 46, 20 pp.Google Scholar
Ombrosi, S.,
Weak weighted inequalities for a dyadic one-sided maximal function in
${\mathbb{R}}^n$
. Proc. Amer. Math. Soc. 133(2005), 1769–1775.Google Scholar
${\mathbb{R}}^n$
. Proc. Amer. Math. Soc. 133(2005), 1769–1775.Google ScholarOmbrosi, S. and de Rosa, L.,
Boundedness of the Weyl fractional integral on one-sided weighted Lebesgue and Lipschitz spaces
. Publ. Mat. 47(2003), 71–102.Google Scholar
Pérez, C.,
Two weighted inequalities for potential and fractional type maximal operators
. Indiana Univ. Math. J. 43(1994), 663–683.Google Scholar
Riveros, M. S. and de la Torre, A., On the best ranges for
${A}_p^{+}$
and
$R{H}_r^{+}$
. Czechoslovak Math. J. 51 (2001), no. 126, 285–301.Google Scholar
${A}_p^{+}$
and
$R{H}_r^{+}$
. Czechoslovak Math. J. 51 (2001), no. 126, 285–301.Google Scholarde Rosa, L. and de la Torre, A., On conditions for the boundedness of the Weyl fractional integral on weighted
${L}^p\!\!$
spaces
. Comment. Math. Univ. Carolin. 45(2004), 17–36.Google Scholar
${L}^p\!\!$
spaces
. Comment. Math. Univ. Carolin. 45(2004), 17–36.Google ScholarSaari, O.,
Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations
. Rev. Mat. Iberoam. 32(2016), 1001–1018.Google Scholar
Saari, O.,
Parabolic BMO and the forward-in-time maximal operator
. Ann. Mat. Pura Appl. (4) 197(2018), 1477–1497.Google Scholar
Sawyer, E.,
Weighted inequalities for the one-sided Hardy–Littlewood maximal functions
. Trans. Amer. Math. Soc. 297(1986), 53–61.Google Scholar
Sawyer, E.,
A characterization of two weight norm inequalities for fractional and Poisson integrals
. Trans. Amer. Math. Soc. 308(1988), 533–545.Google Scholar
Sawyer, E. and Wheeden, R. L.,
Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces
. Amer. J. Math. 114(1992), 813–874.Google Scholar
Trudinger, N. S.,
Pointwise estimates and quasilinear parabolic equations
. Comm. Pure Appl. Math. 21(1968), 205–226.Google Scholar
Turesson, B. O., Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, 1736, Springer-Verlag, Berlin, 2000.Google Scholar
Vespri, V.,
On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations
. Manuscripta Math. 75(1992), 65–80.Google Scholar
Welland, G. V.,
Weighted norm inequalities for fractional integrals
. Proc. Amer. Math. Soc. 51(1975), 143–148.Google Scholar