## Homogeneous Besov spaces.(English)Zbl 1437.42034

Summary: This note is based on a series of lectures delivered at Kyoto University in 2015. This note surveys the homogeneous Besov space $$\dot{B}^s_{pq}$$ on $$\mathbb{R}^n$$ with $$1\le p$$, $$q\le \infty$$ and $$s\in\mathbb{R}$$ in a rather self-contained manner. Among other results, we show that $$\mathcal{S}'_{\infty}$$ and $$\mathcal{S}^\prime/\mathcal{P}$$ are isomorphic, and we also discuss the realizations in $$\dot{B}^s_{pq}$$. The fact that $$\mathcal{S}^\prime_{\infty}$$ and $$\mathcal{S}^\prime/\mathcal{P}$$ are isomorphic can be found in textbooks. The realization of $$\dot{B}^s_{pq}$$ can be found in works by H. Bahouri et al. [Fourier analysis and nonlinear partial differential equations. Berlin: Heidelberg (2011; Zbl 1227.35004)] and by G. Bourdaud [Ark. Mat. 26, No. 1, 41–54 (1988; Zbl 0661.46026)] for example. Here, we prove these facts using fundamental results in functional analysis such as the Hahn-Banach extension theorem.

### MSC:

 42B35 Function spaces arising in harmonic analysis 46A04 Locally convex Fréchet spaces and (DF)-spaces 30H05 Spaces of bounded analytic functions of one complex variable 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

### Citations:

Zbl 1227.35004; Zbl 0661.46026
Full Text:

### References:

 [1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure. Appl. Math. XVII (1959), no. 4, 623-727. · Zbl 0093.10401 [2] N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Tech. Rep. 14, Univ. Kansas, Lawrence, 1954, 77-94. [3] N. Aronszajn and K. T. Smith, Theory of Bessel potentials, I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385-475. · Zbl 0102.32401 [4] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss. 343, Springer, Heidelberg, 2011. · Zbl 1227.35004 [5] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. · Zbl 0344.46071 [6] O. V. Besov, On some families of functional spaces: Imbedding and extension theorems, Dokl. Akad. Nauk SSSR 126 (1959), 1163-1165. · Zbl 0097.09701 [7] O. V. Besov, Investigation of a class of function spaces in connection with imbedding and extension theorems, Tr. Mat. Inst. Steklova 60 (1961), 42-81. [8] O. V. Besov, V. P. Il’in, and S. M. Nikol’skiĭ, Integral Representations of Functions and Imbedding Theorems, I, Wiley, New York, 1978. II, 1979. [9] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Éc. Norm. Supér. (4) 14 (1981), no. 2, 209-246. · Zbl 0495.35024 [10] G. Bourdaud, Réalisations des espaces de Besov homogènes, Ark. Mat. 26 (1988), no. 1, 41-54. · Zbl 0661.46026 [11] G. Bourdaud, Realizations of homogeneous Sobolev spaces, Complex Var. Elliptic Equ. 56 (2011), no. 10-11, 857-874. · Zbl 1232.46030 [12] V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte Math. 137, Teubner, Stuttgart, 1998. · Zbl 0893.46024 [13] A. P. Calderón, “Lebesgue spaces of differentiable functions and distributions” in Partial Differential Equations, Proc. Sympos. Pure Math. 4, Amer. Math. Soc., Providence, 1961, 33-49. [14] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math. 29, Amer. Math. Soc., Providence, 2001. · Zbl 0969.42001 [15] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Math. 120, Cambridge Univ. Press, Cambridge, 1996. · Zbl 0865.46020 [16] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Ergeb. Math. Grenzgeb. (3) 90, Springer, Berlin, 1977. [17] M. Frazier, Y.-S. Han, B. Jawerth, and G. Weiss, “The $$T1$$ theorem for Triebel-Lizorkin spaces” in Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Math. 1384, Springer, Berlin, 1989, 168-181. · Zbl 0679.46026 [18] O. D. Gabisonija, On the absolute convergence of double Fourier series and Fourier integrals, Soobšč. Akad. Nauk. Gruzin. SSR 42 (1966), 3-9. · Zbl 0144.06401 [19] E. Gagliardo, Proprietà di alcune classi di funzioni in più variabli, Ric. Mat. 7 (1958), 102-137. · Zbl 0089.09401 [20] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985. · Zbl 0578.46046 [21] A. G. Georgiadis, J. Johnsen, and M. Nielsen, Wavelet transforms for homogeneous mixed-norm Triebel-Lizorkin spaces, Monatsh. Math. 183 (2017), no. 4, 587-624. · Zbl 1380.46030 [22] K. K. Golovkin and V. A. Solonnikov, Estimates of convolution operators (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 6-86. · Zbl 0205.14602 [23] L. Grafakos, Modern Fourier Analysis, 2nd ed., Grad. Texts in Math. 250, Springer, New York, 2009. · Zbl 1158.42001 [24] B. Grevholm, On the structure of the spaces $${\mathcal{L}}^{p,\lambda }_k$$, Math. Scand. 26 (1970), 241-254. · Zbl 0212.46002 [25] M. Gubinelli, P. Imkeller, and N. Perkowski, A Fourier approach to pathwise stochastic integration, Electron. J. Probab. 21 (2016), no. 2. · Zbl 1338.60139 [26] D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbk. Math., Eur. Math. Soc., Zürich, 2008. · Zbl 1133.46001 [27] M. Holschneider, Wavelets: An Analysis Tool, Oxford Math. Monogr., Oxford Univ. Press, New York, 1995. · Zbl 0874.42020 [28] B. Jawerth, Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand. 40 (1977), no. 1, 94-104. · Zbl 0358.46023 [29] G. Kyriazis, Decomposition systems for function spaces, Studia Math. 157 (2003), no. 2, 133-169. · Zbl 1050.42027 [30] W. R. Madych, Absolute summability of Fourier transforms on $${\mathbb{R}}^n$$, Indiana Univ. Math. J. 25 (1976), no. 5, 467-479. · Zbl 0327.42008 [31] R. E. Megginson, An Introduction to Banach Space Theory, Grad. Texts in Math. 183, Springer, New York, 1998. · Zbl 0910.46008 [32] C. Miranda, Partial Differential Equations of Elliptic Type, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, New York, 1970. · Zbl 0198.14101 [33] K. Morii, T. Sato, Y. Sawano, and H. Wadade, Sharp constants of Brézis-Gallouët-Wainger type inequalities with a double logarithmic term on bounded domains in Besov and Triebel-Lizorkin spaces, Bound. Value Probl. 2010, art. ID 584521. · Zbl 1214.46021 [34] S. Nakamura, T. Noi, and Y. Sawano, Generalized Morrey spaces and trace operator, Sci. China Math. 59 (2016), no. 2, 281-336. · Zbl 1344.42020 [35] S. M. Nikol’skiĭ, Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables (in Russian), Tr. Mat. Inst. Steklova 38 (1951), 244-278. [36] J. Peetre, Remarques sur les espaces de Besov; Le cas \(0
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