Kyoto Journal of Mathematics

Characterization of variable Besov-type spaces by ball means of differences

Douadi Drihem

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

With the help of the maximal function characterizations of Besov-type spaces with variable smoothness and integrability we prove the characterization by ball means of differences for these function spaces.

Article information

Source
Kyoto J. Math., Volume 56, Number 3 (2016), 655-680.

Dates
Received: 27 March 2015
Revised: 13 April 2015
Accepted: 16 April 2015
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1471872286

Digital Object Identifier
doi:10.1215/21562261-3600220

Mathematical Reviews number (MathSciNet)
MR3542780

Zentralblatt MATH identifier
1355.46039

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis

Keywords
Besov-type space maximal function ball means of differences Peetre maximal function variable exponent

Citation

Drihem, Douadi. Characterization of variable Besov-type spaces by ball means of differences. Kyoto J. Math. 56 (2016), no. 3, 655--680. doi:10.1215/21562261-3600220. https://projecteuclid.org/euclid.kjm/1471872286


Export citation

References

  • [1] A. Almeida and P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258 (2010), 1628–1655.
  • [2] Y. Chen, S. Levine, and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383–1406.
  • [3] D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, The boundedness of classical operators on variable $L^{p}$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), 239–264.
  • [4] L. Diening, Maximal function on generalized Lebesque spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), 245–253.
  • [5] L. Diening, P. Hästö, and S. Roudenko, Function spaces of variable smoothness and integrability, J. Funct. Anal. 256 (2009), 1731–1768.
  • [6] L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura, Maximal functions in variable exponent spaces: limiting cases of the exponent, Ann. Acad. Sci. Fenn. Math. 34 (2009), 503–522.
  • [7] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
  • [8] D. Drihem, Some embeddings and equivalent norms of the $\mathcal{L}_{p,q}^{\lambda,s}$ spaces, Funct. Approx. Comment. Math. 41 (2009), 15–40.
  • [9] D. Drihem, Atomic decomposition of Besov spaces with variable smoothness and integrability, J. Math. Anal. Appl. 389 (2012), 15–31.
  • [10] D. Drihem, Characterizations of Besov-type and Triebel–Lizorkin-type spaces by differences, J. Funct. Spaces Appl. 2012, no. 328908.
  • [11] D. Drihem, Atomic decomposition of Besov-type and Triebel–Lizorkin-type spaces, Sci. China Math. 56 (2013), 1073–1086.
  • [12] D. Drihem, Some characterizations of variable Besov-type spaces, Ann. Funct. Anal. 6 (2015), 255–288.
  • [13] D. Drihem, Some properties of variable Besov-type spaces, Funct. Approx. Comment. Math. 52 (2015), 193–221.
  • [14] J. Fu and J. Xu, Characterizations of Morrey type Besov and Triebel–Lizorkin spaces with variable exponents, J. Math. Anal. Appl. 381 (2011), 280–298.
  • [15] P. Harjulehto, P. Hästö, Ú. V. Lê, and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), 4551–4574.
  • [16] H. Kempka and J. Vybíral, Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences, J. Fourier Anal. Appl. 18 (2012), 852–891.
  • [17] H. Kempka and J. Vybíral, A note on the spaces of variable integrability and summability of Almeida and Hästö, Proc. Amer. Math. Soc. 141 (2013), 3207–3212.
  • [18] O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), 592–618.
  • [19] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier–Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations 19 (1994), 959–1014.
  • [20] Y. Liang, D. Yang, W. Yuan, Y. Sawano, and T. Ullrich, A new framework for generalized Besov-type and Triebel–Lizorkin-type spaces, Dissertationes Math. (Rozprawy Mat.) 489 (2013), 1–114.
  • [21] A. L. Mazzucato, Besov–Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1297–1364.
  • [22] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166.
  • [23] Y. V. Netrusov, Some imbedding theorems for spaces of Besov–Morrey type (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 139 (1984), 139–147.
  • [24] S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Grundlehren Math. Wiss. 205, Springer, New York, 1975.
  • [25] J. Peetre, On the theory of $\mathcal{L}_{p,\lambda}$ spaces, J. Funct. Anal. 4 (1969), 71–87.
  • [26] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000.
  • [27] I. I. Šarapudinov, The topology of the space $L^{p(t)}([0,1])$ (in Russian), Mat. Zametki 26 (1979), no. 4, 613–632, 655.
  • [28] Y. Sawano, Wavelet characterizations of Besov–Morrey and Triebel–Lizorkin–Morrey spaces, Funct. Approx. Comment. Math. 38 (2008), 93–107.
  • [29] Y. Sawano, A note on Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1223–1242.
  • [30] Y. Sawano, Brezis–Gallouët–Wainger type inequality for Besov–Morrey spaces, Studia Math. 196 (2010), 91–101.
  • [31] Y. Sawano, D. Yang, and W. Yuan, New applications of Besov-type and Triebel–Lizorkin-type spaces, J. Math. Anal. Appl. 363 (2010), 73–85.
  • [32] W. Sickel, On pointwise multipliers for $F_{p,q}^{s}(\mathbb{R}^{n})$ in case $\sigma_{p,q}<s<n/p$, Ann. Mat. Pura Appl. (4) 176 (1999), 209–250.
  • [33] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989.
  • [34] H. Triebel, Theory of Function Spaces, Monogr. Math. 78, Birkhäuser, Basel, 1983.
  • [35] H. Triebel, Theory of Function Spaces, II, Monogr. Math. 84, Birkhäuser, Basel, 1992.
  • [36] H. Triebel, Local Function Spaces, Heat and Navier–Stokes Equations, EMS Tracts in Math. 20, Eur. Math. Soc. (EMS), Zürich, 2013.
  • [37] J. Xu, Variable Besov and Triebel–Lizorkin spaces, Ann. Acad. Sci. Fenn. Math. 33 (2008), 511–522.
  • [38] J. Xu, An atomic decomposition of variable Besov and Triebel–Lizorkin spaces, Armen. J. Math. 2 (2009), 1–12.
  • [39] D. Yang and W. Yuan, A new class of function spaces connecting Triebel–Lizorkin spaces and $Q$ spaces, J. Funct. Anal. 255 (2008), 2760–2809.
  • [40] D. Yang and W. Yuan, New Besov-type spaces and Triebel–Lizorkin-type spaces including $Q$ spaces, Math. Z. 265 (2010), 451–480.
  • [41] D. Yang and W. Yuan, Relations among Besov-type spaces, Triebel–Lizorkin-type spaces and generalized Carleson measure spaces, Appl. Anal. 92 (2013), 549–561.
  • [42] D. Yang, C. Zhuo, and W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal. 269 (2015), 1840–1898.
  • [43] W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer, Berlin, 2010.
  • [44] W. Yuan, W. Sickel, and D. Yang, On the coincidence of certain approaches to smoothness spaces related to Morrey spaces, Math. Nachr. 286 (2013), 1571–1584.
  • [45] W. Yuan, D. D. Haroske, L. Skrzypczak, and D. Yang, Embedding properties of Besov-type spaces, Appl. Anal. 94 (2015), 319–341.