Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access November 22, 2020

Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains

Ciqiang Zhuo EMAIL logo , Marc Hovemann and Winfried Sickel

Abstract

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

MSC 2010: 46B70; 46E35

References

[1] D. R. Adams, Morrey spaces, Birkhäuser, Cham, 2015.10.1007/978-3-319-26681-7Search in Google Scholar

[2] D. R. Adams, J. Xiao, Morrey spaces in harmonic analysis, Ark. Mat., 2012, 50, 201-230.10.1007/s11512-010-0134-0Search in Google Scholar

[3] J. Bergh, Relation between the two complex methods of interpolation, Indiana Univ. Math. J., 1979, 28(5), 775-778.10.1512/iumj.1979.28.28054Search in Google Scholar

[4] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction. Springer, New York, 1976.10.1007/978-3-642-66451-9Search in Google Scholar

[5] O. Blasco, A. Ruiz, L. Vega, Non-interpolation in Morrey-Campanato and block spaces, Ann. Scuola Norm Sup. Pisa Cl. Sci., 1999, 28(4), 31-40.Search in Google Scholar

[6] B. Bojarski, T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in 𝕉n, Ann. Acad. Sci. Fenn. Ser. AI Math., 1983, 8, 257-324.10.5186/aasfm.1983.0806Search in Google Scholar

[7] H.-Q. Bui, M. Paluszy«ski, M.H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math., 1996, 119(3), 219-246.Search in Google Scholar

[8] H.-Q. Bui, M. Paluszy«ski, M.H. Taibleson, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1, J. Fourier Anal. Appl., 1997, 3, 837-846.10.1007/BF02656489Search in Google Scholar

[9] V.I. Burenkov, D.K. Darbayeva, E.D. Nursultanov, Description of interpolation spaces for general local Morrey-type spaces, Eurasian Math. J., 2013, 4(1), 46-53.10.1134/S0081543810020045Search in Google Scholar

[10] V.I. Burenkov, E.D. Nursultanov, D.K. Chigambayeva, Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations, Trudy Mat. Inst. Steklova, 2014, 284, 105-137.10.1134/S0081543814010064Search in Google Scholar

[11] V.I. Burenkov, A. Ghorbanalizadeh, Y. Sawano, On the equivalence of the K-functional and the modulus of continuity on the Morrey spaces, J. Approx. Theory, 2019, 248, 19 pp.10.1016/j.jat.2019.105295Search in Google Scholar

[12] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math., 1964, 24, 113-190.10.4064/sm-24-2-113-190Search in Google Scholar

[13] G.T. Dchumakeva, A criterion for the imbedding of the Sobolev-Morrey class Wp,Φl in the space C, Mat. Zametki, 1985, 37, 399-406.10.1007/BF01158745Search in Google Scholar

[14] M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal., 1990, 93, 34-170.10.1016/0022-1236(90)90137-ASearch in Google Scholar

[15] J. Gustavsson, On interpolation of weighted Lp-spaces and Ovchinnikov’s theorem, Studia Math., 1982, 72, 237-251.10.4064/sm-72-3-237-251Search in Google Scholar

[16] J. Gustavsson, J. Peetre, Interpolation of Orlicz spaces, Studia Math., 1977, 60, 33-59.10.4064/sm-60-1-33-59Search in Google Scholar

[17] P.M. Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math., 1993, 64, 93-100.10.4064/cm-64-1-93-101Search in Google Scholar

[18] D.I. Hakim, Complex interpolation of certain closed subspaces of generalized Morrey spaces, Tokyo J. Math., 2018, 41(2), 487-514.10.3836/tjm/1502179272Search in Google Scholar

[19] D.I. Hakim, Y. Sawano, Interpolation of generalized Morrey spaces, Rev. Mat. Complut., 2016, 29(2), 295-340.10.1007/s13163-016-0192-3Search in Google Scholar

[20] D.I. Hakim, Y. Sawano, Calderón first and second complex interpolations of closed subspaces of Morrey spaces, J. Fourier Anal. Appl., 2017, 23, 1195-1226.10.1007/s00041-016-9503-9Search in Google Scholar

[21] D.I. Hakim, Y. Sawano, Complex interpolation of vanishing Morrey spaces, Ann. Funct. Anal., 2020, 11, 643-661.10.1007/s43034-019-00045-wSearch in Google Scholar

[22] D.I. Hakim, Y. Sawano, Complex interpolation of various subspaces of Morrey spaces, Sci. China Math., 2020, 63, 937-964.10.1007/s11425-017-9318-0Search in Google Scholar

[23] D.I. Hakim, S. Nakamura, Y. Sawano, Complex interpolation of smoothness Morrey subspaces, Constr. Approx., 2017, 46, 489-563.10.1007/s00365-017-9392-4Search in Google Scholar

[24] D.I. Hakim, T. Nogayama, Y. Sawano, Complex interpolation of smoothness Triebel-Lizorkin-Morrey spaces, Math. J. Okayama Univ., 2019, 61, 99-128.Search in Google Scholar

[25] D.D. Haroske, L. Skrzypczak, On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces, Rev. Mat. Complut., 2014, 27(2), 541-573.10.1007/s13163-013-0143-1Search in Google Scholar

[26] M. Hovemann, Triebel-Lizorkin-Morrey spaces and differences, Math. Nachr. In print.Search in Google Scholar

[27] N. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math., 1986, 84, 297-324.10.4064/sm-84-3-297-324Search in Google Scholar

[28] N. Kalton, S. Mayboroda, M. Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations. Interpolation Theory and Applications, Contemp. Math., 2007, 445, 121-177.10.1090/conm/445/08598Search in Google Scholar

[29] N. Kalton, M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc., 1998, 350, 3903-3922.10.1090/S0002-9947-98-02008-XSearch in Google Scholar

[30] H. Kozono, M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 1994, 19, 959-1014.10.1080/03605309408821042Search in Google Scholar

[31] S.G. Kreĭn, Y.I. Petunin, E.M. Semenov, Interpolation of linear operators. Moscow: Nauka, 1978, engl. translation AMS, Providence, R.I., 1982.Search in Google Scholar

[32] P.G. Lemarié-Rieusset, Multipliers and Morrey spaces, Potential Anal., 2013, 38, 741-752.10.1007/s11118-012-9295-8Search in Google Scholar

[33] P.G. Lemarié-Rieusset, Erratum to “Multipliers and Morrey spaces”, Potential Anal., 2014, 41, 1359-1362.10.1007/s11118-014-9407-8Search in Google Scholar

[34] Y. Liang, D. Yang, W. Yuan, Y. Sawano, T. Ullrich, A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces, Dissertationes Math., 2013, 489, 1-114.10.4064/dm489-0-1Search in Google Scholar

[35] Y. Lu, D. Yang, W. Yuan, Interpolation of Morrey spaces on metric measure spaces, Canad. Math. Bull., 2014, 57, 598-608.10.4153/CMB-2013-009-4Search in Google Scholar

[36] A. Lunardi, Interpolation Theory. Lect. Notes. Pisa: Scuola Normale Superiore Pisa, 2009.Search in Google Scholar

[37] A. Mazzucato, Decomposition of Besov-Morrey spaces. In: Harmonic Analysis at Mount Holyoke 2001. Contemp. Math., 2003, 320, 279-294.10.1090/conm/320/05613Search in Google Scholar

[38] A. Mazzucato, Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 2003, 355, 1297-1369.10.1090/S0002-9947-02-03214-2Search in Google Scholar

[39] S.D. Moura, J.S. Neves, C. Schneider, Spaces of generalized smoothness in the critical case: optimal embeddings, continuity envelopes and approximation numbers, J. Approx. Theory, 2014, 187, 82-117.10.1016/j.jat.2014.07.010Search in Google Scholar

[40] M. Rosenthal, Local means, wavelet bases and wavelet isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces, Math. Nachr., 2013, 286, 59-87.10.1002/mana.201200020Search in Google Scholar

[41] A. Ruiz, L. Vega, Corrigenda to “Unique continuation for Schrödinger operators with potential in Morrey spaces” and a remark on interpolation of Morrey spaces, Publ. Mat., 1995, 3, 405-411.10.5565/PUBLMAT_39295_15Search in Google Scholar

[42] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications 3. Walter de Gruyter & Co., Berlin, 1996.10.1515/9783110812411Search in Google Scholar

[43] V.S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. London Math. Soc., 1999, 60(2), 237-257.10.1112/S0024610799007723Search in Google Scholar

[44] Y. Sawano, Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morrey spaces, Funct. Approx. Comment Math., 2008, 38, 93-107.10.7169/facm/1229624654Search in Google Scholar

[45] Y. Sawano, H. Tanaka, Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Math. Z., 2007, 257, 871-905.10.1007/s00209-007-0150-3Search in Google Scholar

[46] Y. Sawano, H. Tanaka, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures, Math. Nachr., 2009, 282, 1788-1810.10.1002/mana.200610818Search in Google Scholar

[47] V.A. Shestakov, Interpolation of linear operators in spaces of measurable functions, Funktsional Anal. i Prilozhen, 1974, 8, 91-92.10.1007/BF01075709Search in Google Scholar

[48] V.A. Shestakov, On complex interpolation of Banach spaces of measurable functions, Vestnik Leningrad Univ., 1974, 19, 64-68.Search in Google Scholar

[49] W. Sickel, Smoothness spaces related to Morrey spaces - a survey. I, Eurasian Math. J., 2012, 3, 110-149.Search in Google Scholar

[50] W. Sickel, Smoothness spaces related to Morrey spaces-a survey. II, Eurasian Math. J., 2013, 4, 82-124.Search in Google Scholar

[51] W. Sickel, L. Skrzypczak, J. Vybíral, Complex interpolation of weighted Besov- and Lizorkin-Triebel spaces, Acta Math. Sinica, 2014, 30, 1297-1323.10.1007/s10114-014-2762-ySearch in Google Scholar

[52] W. Sickel, L. Skrzypczak, J. Vybíral, Complex interpolation of weighted Besov- and Lizorkin-Triebel spaces (extended version). arxiv: 1212.1614.Search in Google Scholar

[53] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970.10.1515/9781400883882Search in Google Scholar

[54] L. Tang, J. Xu, Some properties of Morrey type Besov-Triebel spaces, Math. Nachr., 2005, 278, 904-914.10.1002/mana.200310281Search in Google Scholar

[55] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Co., Amsterdam, 1978.Search in Google Scholar

[56] H. Triebel, Theory of Function Spaces. Birkhäuser, Basel, 1983.10.1007/978-3-0346-0416-1Search in Google Scholar

[57] H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Revista Mat. Complutense, 2002, 15(2), 475-524.10.5209/rev_REMA.2002.v15.n2.16910Search in Google Scholar

[58] H. Triebel, Function Spaces and Wavelets on Domains. EMS Publishing House, Zürich, 2008.10.4171/019Search in Google Scholar

[59] H. Triebel, Hybrid Function Spaces, Heat and Navier-Stokes Equations. EMS Tracts in Mathematics 24. European Mathematical Society (EMS), Zürich, 2014.10.4171/150Search in Google Scholar

[60] D. Yang, W. Yuan, C. Zhuo, Complex interpolation on Besov-type and Triebel-Lizorkin-type spaces, Anal. Appl., 2013, 11, 45 pp.10.1142/S0219530513500218Search in Google Scholar

[61] W. Yuan, A note on complex interpolation and Calderón product of quasi-Banach spaces, Taiwanese J. Math., 2014, 18(5), 1527-1548.10.11650/tjm.18.2014.4373Search in Google Scholar

[62] W. Yuan, W. Sickel, D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics 2005. Springer, Berlin, 2010.10.1007/978-3-642-14606-0Search in Google Scholar

[63] W. Yuan, W. Sickel, D. Yang, Interpolation of Morrey-Campanato and Related Smoothness Spaces, Sci. China Math., 2015, 58, 1835-1908.10.1007/s11425-015-5047-8Search in Google Scholar

Received: 2020-08-04
Accepted: 2020-10-18
Published Online: 2020-11-22

© 2020 Ciqiang Zhuo et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 4.12.2022 from frontend.live.degruyter.dgbricks.com/document/doi/10.1515/agms-2020-0114/html
Scroll Up Arrow