Complex interpolation of Besov-type spaces on domains. (English) Zbl 1480.46034
Summary: Let \(\Omega\subset\mathbb{R}^d\) \((d\geq 2)\) be a bounded Lipschitz domain. In this article, the author mainly studies complex interpolation of Besov-type spaces on the domain \(\Omega\), namely, we investigate the interpolation
\[
[B_{p_0,q_0}^{s_0,\tau_0}(\Omega),B_{p_1,q_1}^{s_1,\tau_1}(\Omega)]_\Theta=B_{p,q}^{\diamond s,\tau}(\Omega)
\]
under certain conditions on the parameters, where \(B_{p,q}^{\diamond s,\tau}(\Omega)\) denotes the so-called diamond space associated with the Besov-type space. To this end, we first establish the equivalent characterization of the diamond space \(B_{p,q}^{\diamond s,\tau}(\mathbb{R}^d)\) in terms of Littlewood-Paley decomposition and differences. Via some examples, we also show that this interpolation result does not hold under some other assumptions on the parameters or when \(\Omega=\mathbb{R}^d\).
MSC:
| 46B70 | Interpolation between normed linear spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Besov-type space; diamond space; extension operator; Lipschitz domains; complex interpolationReferences:
| [1] | J. Bergh and J. Löfström,Interpolation Spaces. An Introduction.Grundlehren der Mathematischen Wissenschaften 223. Springer, Berlin-New York, 1976 · Zbl 0344.46071 |
| [2] | H.-Q. Bui, Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures. J. Funct. Anal.55(1984), 39-62 · Zbl 0638.46029 |
| [3] | H.-Q. Bui, M. Paluszy´nski and M. Taibleson, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The caseq < 1.J. Fourier Anal. Appl.3(1997), 837-846 · Zbl 0897.42010 |
| [4] | H.-Q. Bui, M. Paluszy´nski and M. H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces.Studia Math.119(1996), 219-246 · Zbl 0861.42009 |
| [5] | A.-P. Calderón, Intermediate spaces and interpolation, the complex method.Studia Math.24 (1964), 113-190 · Zbl 0204.13703 |
| [6] | D. I. Hakim, Complex interpolation of certain closed subspaces of generalized Morrey spaces. Tokyo J. Math.41(2018), 487-514 · Zbl 1419.46019 |
| [7] | D. I. Hakim, S. Nakamura and Y. Sawano, Complex interpolation of smoothness Morrey subspaces.Constr. Approx.46(2017), 489-563 · Zbl 1385.42023 |
| [8] | D. I. Hakim, T. Nogayama and Y. Sawano, Complex interpolation of smoothness Triebel- Lizorkin-Morrey spaces.Math. J. Okayama Univ.61(2019), 99-128 · Zbl 1414.42029 |
| [9] | D. I. Hakim and Y. Sawano, Interpolation of generalized Morrey spaces.Rev. Mat. Complut. 29(2016), 295-340 · Zbl 1353.46019 |
| [10] | D. I. Hakim and Y. Sawano, Calderón’s first and second complex interpolations of closed subspaces of Morrey spaces.J. Fourier Anal. Appl.23(2017), 1195-1226 · Zbl 1393.46015 |
| [11] | D. I. Hakim and Y. Sawano, Complex interpolation of vanishing Morrey spaces.Ann. Funct. Anal.11(2020), 643-661 · Zbl 1460.46013 |
| [12] | D. I. Hakim and Y. Sawano, Complex interpolation of various subspaces of Morrey spaces. Sci. China Math.63(2020), 937-964 · Zbl 1461.46016 |
| [13] | M. Hovemann and W. Sickel, Besov-type spaces and differences.Eurasian Math. J.11(2020), 25-56 · Zbl 1463.46057 |
| [14] | N. Kalton, S. Mayboroda and M. Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel- Lizorkin spaces and applications to problems in partial differential equations. InInterpolation theory and applications, pp. 121-177, Contemp. Math. 445, American Mathematical Society, Providence, 2007 · Zbl 1158.46013 |
| [15] | S. G. Kre˘ın, Y. I. Petun¯ın and E. M. Semënov,Interpolation of linear operators. Translations of Mathematical Monographs 54, American Mathematical Society, Providence, 1982 · Zbl 0493.46058 |
| [16] | P. G. Lemarié-Rieusset, Multipliers and Morrey spaces.Potential Anal.38(2013), 741-752 · Zbl 1267.42024 |
| [17] | P. G. Lemarié-Rieusset, Erratum to: Multipliers and Morrey spaces [mr3034598].Potential Anal.41(2014), 1359-1362 · Zbl 1302.42031 |
| [18] | Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, New characterizations of Besov- Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets.J. Fourier Anal. Appl.18 (2012), 1067-1111 · Zbl 1260.46026 |
| [19] | 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.489(2013), 114 · Zbl 1283.46027 |
| [20] | Y. Lu, D. Yang and W. Yuan, Interpolation of Morrey spaces on metric measure spaces.Canad. Math. Bull.57(2014), 598-608 · Zbl 1316.46021 |
| [21] | V. S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains.J. London Math. Soc. (2)60(1999), 237-257 · Zbl 0940.46017 |
| [22] | Y. Sawano and H. Tanaka, Decompositions of Besov-Morrey spaces and Triebel-Lizorkin- Morrey spaces.Math. Z.257(2007), 871-905 · Zbl 1133.42041 |
| [23] | V. A. Šestakov, On complex interpolation of Banach spaces of measurable functions.Vestnik Leningrad. Univ.19(1974), 64-68 · Zbl 0297.46028 |
| [24] | V. A. Šestakov, The interpolation of linear operators in spaces of measurable functions. Funkcional. Anal. i Priložen.8(1974), 91-92 · Zbl 0304.47038 |
| [25] | W. Sickel, Smoothness spaces related to Morrey spaces—a survey. I.Eurasian Math. J.3 (2012), 110-149 · Zbl 1274.46074 |
| [26] | W. Sickel, L. Skrzypczak and J. Vybíral, Complex interpolation of weighted Besov and Lizorkin-Triebel spaces.Acta Math. Sin. (Engl. Ser.)30(2014), 1297-1323 · Zbl 1314.46048 |
| [27] | W. Sickel, L. Skrzypczak and J. Vybíral, Complex interpolation of weighted Besov and Lizorkin-Triebel spaces (extended version). arXiv:1212.1614. · Zbl 1314.46048 |
| [28] | L. Tang and J. Xu, Some properties of Morrey type Besov-Triebel spaces.Math. Nachr.278 (2005), 904-917 · Zbl 1074.42011 |
| [29] | H. Triebel,Interpolation theory, function spaces, differential operators. North-Holland Mathematical Library 18, North-Holland Publishing, Amsterdam-New York, 1978 · Zbl 0387.46033 |
| [30] | H. Triebel,Theory of function spaces. Monographs in Mathematics 78, Birkhäuser, Basel, 1983 · Zbl 0546.46028 |
| [31] | H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers.Rev. Mat. Complut.15(2002), 475-524 · Zbl 1034.46033 |
| [32] | H. Triebel,Function spaces and wavelets on domains. EMS Tracts in Mathematics 7, European Mathematical Society (EMS), Zürich, 2008 · Zbl 1158.46002 |
| [33] | S. Wu, D. Yang and W. Yuan, Equivalent quasi-norms of Besov-Triebel-Lizorkin-type spaces via derivatives.Results Math.72(2017), 813-841 · Zbl 1385.46025 |
| [34] | D. Yang, W. Yuan and C. Zhuo, Complex interpolation on Besov-type and Triebel-Lizorkintype spaces.Anal. Appl. (Singap.)11(2013), 1350021, 45 · Zbl 1294.46020 |
| [35] | W. Yuan, D. D. Haroske, L. Skrzypczak and D. Yang, Embedding properties of Besov-type spaces.Appl. Anal.94(2015), 319-341 · Zbl 1320.46036 |
| [36] | W. Yuan, W. Sickel and D. Yang,Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics 2005, Springer, Berlin, 2010 · Zbl 1207.46002 |
| [37] | W. Yuan, W. Sickel and D. Yang, Interpolation of Morrey-Campanato and related smoothness spaces.Sci. China Math.58(2015), 1835-1908 · Zbl 1337.46030 |
| [38] | C. Zhuo, M. Hovemann and W. Sickel, Complex interpolation of Lizorkin-Triebel-Morrey spaces on domains.Anal. Geom. Metr. Spaces8(2020), 268-304 · Zbl 1478.46017 |
| [39] | C. Zhuo, W. Sickel, D. Yang and W. Yuan, Characterizations of Besov-type and Triebel- Lizorkin-type spaces via averages on balls.Canad. Math. Bull.60(2017) · Zbl 1372.42020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
