New $2$-microlocal Besov and Triebel--Lizorkin spaces via the Litllewood--Paley decomposition

K. Saka

Eurasian mathematical journal, Tome 14 (2023) no. 3, pp. 75-111.

Résumé

In this paper we introduce and investigate new 2-microlocal Besov and Triebel–Lizorkin spaces via the Littlewood–Paley decomposition. We establish characterizations of these function spaces by the $\varphi$-transform, the atomic and molecular decomposition and the wavelet decomposition. As applications we prove boundedness of the the Calderón–Zygmund operators and the pseudo-differential operators on the function spaces. Moreover, we give characterizations via oscillations and differences.

Bibliographie

[1] A. Almeida, A. Caetano, “Variable exponent Besov-Morrey spaces”, J. Fourier Anal. Appl., 26:1 (2020), 5

[2] Ts. Batbold, Y. Sawano, “Decomposition for local Morrey spaces”, Eurasian Math. J., 5 (2014), 9–45

[3] J. M. Bony, “Second microlocalization and propagation of singularities for semilinear hyperbolic equations”, Hyper bolic equations and related topics, Academic Press, New York, 1986, 11–49

[4] M. Bownik, “Atomic and molecular decompositions of anisotropic Besov spaces”, Math. Z., 250 (2005), 539–571

[5] M. Bownik, K. P. Ho, “Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces”, Trans. Amer. Math. Soc., 358 (2005), 1469–1510

[6] V. I. Burenkov, H. V. Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces”, Studia Math, 163:2 (2004), 157–17645

[7] A. Caetano, H. Kempka, “Variable exponent Triebel-Lizorkin-Morrey spaces”, J. Math. Anal. Appl., 484 (2020), 123712

[8] A. Caetano, H. Kempka, “Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces”, Constr. Approx, 53 (2021), 201–234

[9] M. Frazier, B. Jawerth, “Decomposition of Besov spaces”, Indiana Univ. Math. J., 34 (1985), 777–799

[10] M. Frazier, B. Jawerth, “A discrete transform and decompositions of distribution spaces”, J. Func. Anal., 93 (1990), 34–171

[11] M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conf. Series in Math., 79, Amer. Math. Soc., 1991

[12] M. Frazier, R. Torres, G. Weiss, “The boundedness of Calderón-Zygmund operators on the spaces $\dot{F}_p^{\alpha,q}$”, Revista Mat. Iberoamericana, 4 (1988), 41–70

[13] H. F. Gončalves, “Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces”, Banach J. Math. Anal., 15:3 (2021), 201–234

[14] H. F. Gončalves, H. Kempka, “Non-smooth atomic decomposition of 2-microlocal spaces and application to point wise multipliers”, J. Math. Anal. Appl., 434 (2016), 1875–1890

[15] H. F. Gončalves, H. Kempka, “Intrinsic atomic characterization of 2-microlocal spaces with variable exponents on domain”, Rev. Mat. Complut, 30:3 (2017), 467–486

[16] H. F. Gončalves, H. Kempka, J. Vybiral, “Franke-Jawerth embeddings for Besov and Triebel-Lizorkin spaces with variable exponents”, Ann. Acad. Sci. Fenn. Math., 43 (2018), 187–209

[17] S. Jaffard, “Pointwise smoothness, two-microlocalization and wavelet coefficients”, Pub. Mat., 35 (1991), 155–168

[18] S. Jaffard, “Pointwise regularity criteria”, C.R. Acad. Sci. Paris, 1339 (2004), 757–762

[19] S. Jaffard, “Wavelet techniques for pointwise regularity”, Annal. Fac. Sci. Toulouse, 15 (2006), 3–33

[20] S. Jaffard, “Pointwise regularity associated with function spaces and multifractal analysis”, Approximation and Probability, Banach Center Publications, 72, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 2006, 93–110

[21] S. Jaffard C. Mélot, “Wavelet analysis of fractal boundaries Part 1 local exponents”, Commun. Math. Phys., 258 (2004), 513–539

[22] S. Jaffard, Y. Meyer, Wavelet methods for pointwise regularity and local oscillations of functions, Memoirs of the AMS, 123, 1996

[23] H. Kempka, “2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability”, Rev. Mat. Complut., 22 (2009), 227–251

[24] H. Kempka, “Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces”, J. Funct. spaces Appl., 8:2 (2010), 129–165

[25] H. Kempka, “Intrinsic characterization, the extension operator in variable exponent function spaces on special Lipschitz domains”, Function spaces and inequalities (New Delhi, India, December 11-15, 2015), Proc. in Math. and Statistics, 206, ed. Jain Pankai, Springer, Singapore, 2017, 175–192

[26] H. Kempka, M. Sch?afer, T. Ullrich, “General coorbit space theory for quasi-Banach spaces and inhomogeneous function spaces with variable smoothness and integrability”, J. Fourier. Anal. Appl., 23:6 (2017), 1348–1407

[27] D. G. Kim, “Littlewood-Paley type estimates for Besov spaces on a cube by wavelet coefficients”, J. Korean Math. Soc., 36:6 (1999), 1075–1090

[28] Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, “Applications of Littlewood-Paley theory for $\dot{B}_\sigma$-Morrey spaces to the boundedness of integral operators”, J. Funct. spaces Appli, 2013, 1–21

[29] G. Kyriazis, “Decomposition systems for function spaces”, Studia Math., 157 (2003), 133–169

[30] J. Levy-Vehel, S. Seuret, “2-microlocal formalism”, Fractal Geomery and Applications. A Jubilee of Benoit Man delbrot, Proceedings of Symposia in Pure Mathematics, 72, no. 2, 2004, 153–215

[31] Y. Meyer, “Wavelet analysis, local Fourier analysis and 2-microlocalization”, Contemporary Math., 189, 1995, 393–401

[32] Y. Meyer, “Wavelets, Vibrations and scalings”, AMS, CRM Monograph Series, 9, 393–401

[33] K. Saka, “2-microlocal spaces associated with Besov type and Triebel-Lizorkin type spaces”, Rev. Mat. Compl, 35 (2022), 923–962

[34] Y. Sawano, H. Tanaka, “Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces”, Math. Z., 257 (2007), 871–905

[35] Y. Sawano, D. Yang, W. Yuan, “New applications of Besov-type and Triebel-Lizorkin-type spaces”, J. Math. Anal. Appl., 363 (2010), 73–85

[36] W. Sickel, D. Yang, W. Yuan, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lect. Notes in Math., 2005, Springer, Berlin, 2010

[37] H. Triebel, Theory of function spaces, Birkhäeuser, Basel, 1983

[38] H. Triebel, Theory of function spaces, v. II, Birkhäeuser, Basel, 1992

[39] A. I. Tyulenev, “On various approaches to Besov-type spaces of variable smoothness”, J. Math. Anal. Appli, 451 (2017), 371–392

[40] S. Wu, D. Yang, W. Yuan, C. Zhuo, “Variable 2-microlocal Besov-Triebel-Lizorkin-type spaces”, Acta Math. Sin. (Engl. Ser.), 34 (2018), 699–748

[41] D. Yang, W. Yuan, “New Besov-type spaces and Triebel-Lizorkin-type spaces”, Math. Z., 265 (2010), 451–480

[42] D. Yang, W. Yuan, “A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces”, J. Funct. Anal., 255 (2008), 2760–2809

[43] D. Yang, C. Zhuo, W. Yuan, “Besov-type spaces with variable smoothness and integrability”, J. Funct. Anal., 269 (2015), 1840–1898

[44] D. Yang C. Zhuo, W. Yuan, “Triebel-Lizorkin type spaces with variable exponents”, Banach J. Math. Anal., 9 (2015), 146–202