Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T07:14:19.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

Part of: Special classes of linear operators Harmonic analysis in several variables Elliptic equations and systems

Published online by Cambridge University Press:  21 May 2018

Dachun Yang ,
Junqiang Zhang  and
Ciqiang Zhuo
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn; zhangjunqiang@mail.bnu.edu.cn)
Junqiang Zhang*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (dcyang@bnu.edu.cn; zhangjunqiang@mail.bnu.edu.cn)
Ciqiang Zhuo
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, People's Republic of China (cqzhuo@mail.bnu.edu.cn)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

Keywords

second-order divergence form elliptic operatorDavies–Gaffney estimatevariable Hardy spacesquare functionmaximal functionmoleculeRiesz transform

MSC classification

Primary: 42B30: $H^p$-spaces
Secondary: 42B35: Function spaces arising in harmonic analysis 42B25: Maximal functions, Littlewood-Paley theory 35J15: Second-order elliptic equations 47B06: Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 759 - 810
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Acerbi, E. and Mingione, G., Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213–259.Google Scholar
2.Acerbi, E. and Mingione, G., Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117–148.Google Scholar
3.Albrecht, D., Duong, X. T. and McIntosh, A., Operator theory and harmonic analysis, in Instructional workshop on analysis and geometry, Part III (Canberra, 1995), pp. 77–136, Proceedings of the Centre for Mathematics and its Applications, Volume 34, ANU, Canberra, 1996.Google Scholar
4.Auscher, P., On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on ℝn and related estimates, Mem. Amer. Math. Soc. 186(871) (2007), xviii+75pp.Google Scholar
5.Auscher, P., Hofmann, S., Lacey, M., McIntosh, A. and Tchamitchian, Ph., The solution of the Kato square root problem for second order elliptic operators on ℝn, Ann. of Math. (2) 156 (2002), 633–654.Google Scholar
6.Auscher, P., Duong, X. T. and McIntosh, A., Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished manuscript, 2005.Google Scholar
7.Auscher, P. and Martell, J. M., Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), 265–316.Google Scholar
8.Auscher, P. and Martell, J. M., Weighted norm inequalities, off-diagonal estimates and elliptic operators. III. Harmonic analysis of elliptic operators, J. Funct. Anal. 241 (2006), 703–746.Google Scholar
9.Auscher, P., McIntosh, A. and Russ, E., Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192–248.Google Scholar
10.Auscher, P. and Tchamitchian, P., Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172pp.Google Scholar
11.Bui, T. A., Cao, J., Ky, L. D., Yang, D. and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Anal. Geom. Metr. Spaces 1 (2013), 69–129.Google Scholar
12.Bui, T. A., Cao, J., Ky, L. D., Yang, D. and Yang, S., Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates, Taiwanese J. Math. 17 (2013), 1127–1166.Google Scholar
13.Bui, T. A. and Duong, X. T., Weighted Hardy spaces associated to operators and boundedness of singular integrals, arXiv: 1202.2063.Google Scholar
14.Bui, T. A. and Li, J., Orlicz-Hardy spaces associated to operators satisfying bounded H ∞ functional calculus and Davies-Gaffney estimates, J. Math. Anal. Appl. 373 (2011), 485–501.Google Scholar
15.Calderón, A. P., An atomic decomposition of distributions in parabolic H p spaces, Adv. Math. 25 (1977), 216–225.Google Scholar
16.Chen, Y., Guo, W., Zeng, Q. and Liu, Y., A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images, Inverse Probl. Imaging 2 (2008), 205–224.Google Scholar
17.Coifman, R. R., Meyer, Y. and Stein, E. M., Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.Google Scholar
18.Cowling, M., Doust, I., McIntosh, A. and Yagi, A., Banach space operators with a bounded H ∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.Google Scholar
19.Cruz-Uribe, D., The Hardy–Littlewood maximal operator on variable-L p spaces, in Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Volume 64, pp. 147–156 (University of Seville, 2003).Google Scholar
20.Cruz-Uribe, D. V. and Fiorenza, A., Variable Lebesgue spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, Heidelberg, 2013).Google Scholar
21.Cruz-Uribe, D., Fiorenza, A., Martell, J. M. and Pérez, C., The boundedness of classical operators on variable L p spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 239–264.Google Scholar
22.Cruz-Uribe, D. and Wang, L.-A. D., Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), 447–493.Google Scholar
23.Davies, E. B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141–169.Google Scholar
24.Diening, L., Maximal function on generalized Lebesgue spaces L p(·), Math. Inequal. Appl. 7 (2004), 245–253.Google Scholar
25.Diening, L., Harjulehto, P., Hästö, P. and Růžička, M., Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Volume 2017 (Springer, Heidelberg, 2011).Google Scholar
26.Duong, X. T. and Li, J., Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal. 264 (2013), 1409–1437.Google Scholar
27.Duong, X. T. and Yan, L., New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), 1375–1420.Google Scholar
28.Duong, X. T. and Yan, L., Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), 943–973.Google Scholar
29.Gaffney, M., The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 1–11.Google Scholar
30.Haase, M., The functional calculus for sectorial operators, Operator Theory: Advances and Applications, Volume 169 (Birkhäuser Verlag, Basel, 2006).Google Scholar
31.Harjulehto, P., Hästö, P. and Latvala, V., Minimizers of the variable exponent, non-uniformly convex Dirichlet energy, J. Math. Pures Appl. (9) 89 (2008), 174–197.Google Scholar
32.Hofmann, S., Lu, G., Mitrea, D., Mitrea, M. and Yan, L., Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214(1007) (2011), vi+78pp.Google Scholar
33.Hofmann, S. and Martell, J. M., L p bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), 497–515.CrossRefGoogle Scholar
34.Hofmann, S. and Mayboroda, S., Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37–116.Google Scholar
35.Hofmann, S., Mayboroda, S. and McIntosh, A., Second order elliptic operators with complex bounded measurable coefficients in L p, Sobolev and Hardy spaces, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 723–800.CrossRefGoogle Scholar
36.Izuki, M., Vector-valued inequalities on Herz spaces and characterizations of Herz-Sobolev spaces with variable exponent, Glas. Mat. Ser. III 45(65) (2010), 475–503.Google Scholar
37.Jiang, R. and Yang, D., New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 1167–1224.CrossRefGoogle Scholar
38.Jiang, R. and Yang, D., Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331–373.Google Scholar
39.Kato, T., Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics (Springer-Verlag, Berlin, 1995).Google Scholar
40.Kováčik, O. and Rákosník, J., On spaces L p(x) and W k, p(x), Czechoslovak Math. J. 41(116) (1991), 592–618.Google Scholar
41.Ky, L. D., New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integr. Equ. Oper. Theory 78 (2014), 115–150.Google Scholar
42.McIntosh, A., Operators which have an H ∞ functional calculus, in Miniconference on operator theory and partial differential equations (North Ryde, 1986), pp. 210–231, Proceedings of the Centre for Mathematics and its Applications, Volume 14, ANU, Canberra, 1986.Google Scholar
43.Nakai, E. and Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), 3665–3748.Google Scholar
44.Nakano, H., Modulared semi-ordered linear spaces (Maruzen Co. Ltd, Tokyo, 1950).Google Scholar
45.Nakano, H., Topology of linear topological spaces (Maruzen Co. Ltd, Tokyo, 1951).Google Scholar
46.Orlicz, W., Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211.Google Scholar
47.Ouhabaz, E. M., Analysis of heat equations on domains, London Mathematical Society Monographs Series, Volume 31 (Princeton University Press, Princeton, NJ, 2005).Google Scholar
48.Růička, M., Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, Volume 1748 (Springer-Verlag, Berlin, 2000).Google Scholar
49.Sanchón, M. and Urbano, J., Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009), 6387–6405.CrossRefGoogle Scholar
50.Sawano, Y., Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integr. Equ. Oper. Theory 77 (2013), 123–148.Google Scholar
51.Song, L. and Yan, L., Riesz transforms associated to Schrödinger operators on weighted Hardy spaces, J. Funct. Anal. 259 (2010), 1466–1490.Google Scholar
52.Song, L. and Yan, L., A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates, Adv. Math. 287 (2016), 463–484.Google Scholar
53.Strömberg, J. O. and Torchinsky, A., Weighted Hardy spaces, Lecture Notes in Mathematics, Volume 1381 (Springer-Verlag, Berlin, 1989).Google Scholar
54.Yang, D. and Yang, S., Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of ℝn, Rev. Mat. Iberoam. 29 (2013), 237–292.Google Scholar
55.Yang, D. and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators and their applications, J. Geom. Anal. 24 (2014), 495–570.Google Scholar
56.Yang, D., Yuan, W. and Zhuo, C., Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces, Rev. Mat. Complut. 27 (2014), 93–157.Google Scholar
57.Yang, D. and Zhuo, C., Molecular characterizations and dualities of variable exponent Hardy spaces associated with operators, Ann. Acad. Sci. Fenn. Math. 41 (2016), 357–398.CrossRefGoogle Scholar
58.Yang, D., Zhuo, C. and Nakai, E., Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), 245–270.CrossRefGoogle Scholar
59.Zhang, J., Cao, J., Jiang, R. and Yang, D., Non-tangential maximal function characterizations of Hardy spaces associated to degenerate elliptic operators, Canad. J. Math. 67 (2015), 1161–1200.Google Scholar
60.Zhuo, C. and Yang, D., Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates, Nonlinear Anal. 141 (2016), 16–42.Google Scholar
61.Zhuo, C., Yang, D. and Liang, Y., Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), 1541–1577.Google Scholar