Variable integral and smooth exponent Besov spaces associated to non-negative self-adjoint operators

Abstract

We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators. Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.

Key words： Besov space; variable exponent; maximal function; non-negative self-adjoint operators; atomic decomposition

References

1 Adamowicz T, Harjulehto P, Hästö P. Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math Scand, 2015, 116: 5–22

2 Almeida A, Hästö P. Besov spaces with variable smoothness and integrability. J Funct Anal, 2010, 258: 1628–1655

3 Bui A, Cao J, Ky L D, Yang D, Yang S. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal Geom Metr Spaces, 2014, 1: 69–129

4 Bui A, Duong X T. Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces. Trans Amer Math Soc, 2017, 369: 2109–2150

5 Cao J, Chang D C, Wu H, Yang D. Weak Hardy spaces WHLp (ℝn) associated to operators satisfying k-Davies-Gaffney estimates. J Nonlinear Convex Anal, 2015, 16: 1205–1255

6 Cao J, Chang D C, Yang D, Yang S. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Commun Pure Appl Anal, 2014, 13: 1435–1463

7 Cao J, Yang D, Yang S. Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators. Rev Mat Complut, 2013, 26: 99–114

8 Chen Y, Levine S, Rao R. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66: 1383–1406

9 Christ M. A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq Math, 1990, 60-61: 601–628

10 Cruz-Uribe D, Fiorenza A. Variable Lebesgue Spaces. Heidelberg: Springer, 2013

11 Cruz-Uribe D, Fiorenza A, Martell J, Pérez C. The boundedness of classical operators on variable Lp paces. Ann Acad Sci Fenn Math, 2006, 31: 239–264

12 Cruz-Uribe D, Fiorenza A, Ruzhansky M, Wirth J. Variable Lebesgue Spaces and Hyperbolic Systems. Basel: Birkhäuser, 2014

13 Cruz-Uribe D, Wang L A. Variable Hardy spaces. Indiana Univ Math J, 2014, 63: 447–493

14 Dekel S, Kerkyacharian G, Kyriazis G, Petrushev P. Hardy spaces associated with non-negative self-adjoint operators. Studia Math, 2017, 239: 17–54

15 Diening L, Harjulehto P, Hästö P, Mizuta Y, Shimomura T. Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann Acad Sci Fenn Math, 2009, 34: 503–522

16 Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents. Berlin: Springer, 2011

17 Diening L, Hästö P, Roudenko S. Function spaces of variable smoothness and integrability. J Funct Anal, 2009, 256: 1731–1768

18 Georgiadis A G, Kerkyacharian G, Kyriazis G, Petrushev P. Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators. J Math Anal Appl, 2017, 449: 1382–1412

19 Hu G. Besov and Triebel-Lizorkin spaces associated with non-negative self-adjoint operators. J Math Anal Appl, 2014, 411: 753–772

20 Jiang R, Yang D. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224

21 Jiang R, Yang D. Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun Contemp Math, 2011, 13: 331–373

22 Jiang R, Yang D, Zhou Y. Orlicz-Hardy spaces associated with operators. Sci China Ser A, 2009, 52: 1042–1080

23 Kempka H. 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev Mat Complut, 2009, 22: 227–251

24 Kempka H. Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces. J Funct Spaces Appl, 2010, 8: 129–165

25 Kempka H, Vybíral J. Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences. J Fourier Anal Appl, 2012, 18: 852–891

26 Kempka H, Vybíral J. A note on the spaces of variable integrability and summability of Almeida and Hästö. Proc Amer Math Soc, 2013, 141: 3207–3212

27 Kerkyacharian G, Petrushev P. Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Trans Amer Math Soc, 2015, 367: 121–189

28 Kovacik O, Rakosnik J. On spaces Lp(x) and Wk,p(x).Czechoslovak Math J, 1991, 41: 592–618

29 Krantz S G, Parks H G. A Primer of Real Analytic Functions. Basel: Birkhäuser, 1992

30 Liang Y, Yang D, Yang S. Applications of Orlicz-Hardy spaces associated with operators satisfying Poisson estimates. Sci China Math, 2011, 54: 2395–2426

31 Liu L, Yang D, Yuan W. Besov-type and Triebel-Lizorkin-type spaces associated with heat kernels. Collect Math, 2016, 67: 247–310

32 Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal 2012, 262: 3665–3748

33 Růžička M. Electrorheological Fluids: Modeling and Mathematical Theory. Berlin: Springer, 2000

34 Rychkov V. S. On a theorem of Bui, Paluszyński, and Taibleson. Proc Steklov Inst Math, 1999, 227: 280–292

35 Triebel H. Theory of Function Spaces. Monogr Math, Vol 78. Basel: Birkhäuser, 1983

36 Triebel H. Theory of Function Spaces II. Monogr Math, Vol 84. Basel: Birkhäuser, 1992

37 Triebel H. The Structure of Functions. Monogr Math, Vol 97. Basel: Birkhäuser, 2001

38 Triebel H. Theory of Function Spaces III. Monogr Math, Vol 100. Basel: Birkhäuser, 2006

39 Ullrich T. Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits. J Funct Spaces Appl, 2012, 2012: Article ID 163213 (47 pp)

40 Xu J. Variable Besov spaces and Triebel-Lizorkin spaces. Ann Acad Sci Fenn Math, 2008, 33: 511–522

41 Yang D C, Yang D Y. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators. Front Math China, 2015, 10: 1203–1232

42 Yang D C, Yang S. Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of ℝn.Indiana Univ Math J, 2012, 61: 81–129

43 Yang D C, Yang S. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55: 1677–1720

44 Yang D C, Yang S. Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of ℝn.Rev Mat Iberoam, 2013, 29: 237–292

45 Yang D C, Yang S. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014, 24: 495–570

46 Yang D C, Yang S. Second-order Riesz transforms and maximal inequalities associated to magnetic Schrödinger operators. Canad Math Bull, 2015, 58: 432–448

47 Zhang J, Cao J, Jiang R, Yang D. Non-tangential maximal function characterizations of Hardy spaces associated to degenerate elliptic operators. Canad J Math, 2015, 67: 1161–1200