Skip to main content
×
×
Home

Variable Hardy Spaces Associated with Operators Satisfying Davies–Gaffney Estimates

  • Dachun Yang (a1), Junqiang Zhang (a1) and Ciqiang Zhuo (a2)
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).

Copyright
Corresponding author
*Corresponding author.
References
Hide All
1.Acerbi, E. and Mingione, G., Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), 213259.
2.Acerbi, E. and Mingione, G., Gradient estimates for the p(x)-Laplacean system, J. Reine Angew. Math. 584 (2005), 117148.
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. 77136, Proceedings of the Centre for Mathematics and its Applications, Volume 34, ANU, Canberra, 1996.
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.
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), 633654.
6.Auscher, P., Duong, X. T. and McIntosh, A., Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished manuscript, 2005.
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), 265316.
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), 703746.
9.Auscher, P., McIntosh, A. and Russ, E., Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008), 192248.
10.Auscher, P. and Tchamitchian, P., Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172pp.
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), 69129.
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), 11271166.
13.Bui, T. A. and Duong, X. T., Weighted Hardy spaces associated to operators and boundedness of singular integrals, arXiv: 1202.2063.
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), 485501.
15.Calderón, A. P., An atomic decomposition of distributions in parabolic H p spaces, Adv. Math. 25 (1977), 216225.
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), 205224.
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), 304335.
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), 5189.
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. 147156 (University of Seville, 2003).
20.Cruz-Uribe, D. V. and Fiorenza, A., Variable Lebesgue spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, Heidelberg, 2013).
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), 239264.
22.Cruz-Uribe, D. and Wang, L.-A. D., Variable Hardy spaces, Indiana Univ. Math. J. 63 (2014), 447493.
23.Davies, E. B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), 141169.
24.Diening, L., Maximal function on generalized Lebesgue spaces L p(·), Math. Inequal. Appl. 7 (2004), 245253.
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).
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), 14091437.
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), 13751420.
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), 943973.
29.Gaffney, M., The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 111.
30.Haase, M., The functional calculus for sectorial operators, Operator Theory: Advances and Applications, Volume 169 (Birkhäuser Verlag, Basel, 2006).
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), 174197.
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.
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), 497515.
34.Hofmann, S. and Mayboroda, S., Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344 (2009), 37116.
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), 723800.
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), 475503.
37.Jiang, R. and Yang, D., New Orlicz-Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), 11671224.
38.Jiang, R. and Yang, D., Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates, Commun. Contemp. Math. 13 (2011), 331373.
39.Kato, T., Perturbation theory for linear operators, Reprint of the 1980 edition, Classics in Mathematics (Springer-Verlag, Berlin, 1995).
40.Kováčik, O. and Rákosník, J., On spaces L p(x) and W k, p(x), Czechoslovak Math. J. 41(116) (1991), 592618.
41.Ky, L. D., New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integr. Equ. Oper. Theory 78 (2014), 115150.
42.McIntosh, A., Operators which have an H functional calculus, in Miniconference on operator theory and partial differential equations (North Ryde, 1986), pp. 210231, Proceedings of the Centre for Mathematics and its Applications, Volume 14, ANU, Canberra, 1986.
43.Nakai, E. and Sawano, Y., Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), 36653748.
44.Nakano, H., Modulared semi-ordered linear spaces (Maruzen Co. Ltd, Tokyo, 1950).
45.Nakano, H., Topology of linear topological spaces (Maruzen Co. Ltd, Tokyo, 1951).
46.Orlicz, W., Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200211.
47.Ouhabaz, E. M., Analysis of heat equations on domains, London Mathematical Society Monographs Series, Volume 31 (Princeton University Press, Princeton, NJ, 2005).
48.Růička, M., Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, Volume 1748 (Springer-Verlag, Berlin, 2000).
49.Sanchón, M. and Urbano, J., Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009), 63876405.
50.Sawano, Y., Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integr. Equ. Oper. Theory 77 (2013), 123148.
51.Song, L. and Yan, L., Riesz transforms associated to Schrödinger operators on weighted Hardy spaces, J. Funct. Anal. 259 (2010), 14661490.
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), 463484.
53.Strömberg, J. O. and Torchinsky, A., Weighted Hardy spaces, Lecture Notes in Mathematics, Volume 1381 (Springer-Verlag, Berlin, 1989).
54.Yang, D. and Yang, S., Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of ℝn, Rev. Mat. Iberoam. 29 (2013), 237292.
55.Yang, D. and Yang, S., Musielak-Orlicz-Hardy spaces associated with operators and their applications, J. Geom. Anal. 24 (2014), 495570.
56.Yang, D., Yuan, W. and Zhuo, C., Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces, Rev. Mat. Complut. 27 (2014), 93157.
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), 357398.
58.Yang, D., Zhuo, C. and Nakai, E., Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), 245270.
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), 11611200.
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), 1642.
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), 15411577.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 26 *
Loading metrics...

Abstract views

Total abstract views: 112 *
Loading metrics...

* Views captured on Cambridge Core between 21st May 2018 - 21st September 2018. This data will be updated every 24 hours.