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Complex interpolation of predual spaces of general local Morrey-type spaces. (English) Zbl 1412.46040

The author studies the interpolation properties of the predual spaces of generalized local Morrey spaces. The local or central Morrey space \(LM^\lambda_p({\mathbb R}^n)\) is the set of all functions \(f\in L_p^{loc}({\mathbb R}^n)\) such that \(\|f|LM^\lambda_p\| := \sup_{r>0} \|f\chi_{B(r)}\|_p < \infty\), where \(B(r)\) denotes the open ball centred at the origin of radius \(r > 0\). The condition that defines its generalized version \(LM_{p, q,w}({\mathbb R}^n)\) is given by \(\|f|LM_{p, q,w}\| := \|w(r) \|f\chi_{B(r)}\|\, \|_q < \infty\), where \(w\) is a non-negative measurable function on \(]0,1[\) and \(1< p, q\leq\infty\). The author describes the predual spaces to the generalized Morrey-type spaces \(LM_{p, q,w}({\mathbb R}^n)\). The spaces are the generalized version of so-called local block spaces. It is proved that the local block spaces behave well with respect to the complex Calderón interpolation method. Two interpolation formulas are proved with some technical assumption concerning the weight function \(w\) and the coefficients \(p\) and \(q\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B70 Interpolation between normed linear spaces
42B35 Function spaces arising in harmonic analysis
46M35 Abstract interpolation of topological vector spaces
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References:

[1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. · Zbl 0647.46057
[2] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, New York, 1976.
[3] O. Blasco, A. Ruiz, and L. Vega, Non-interpolation in Morrey-Campanato and block spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 (1999), no. 1, 31–40. · Zbl 0955.46013
[4] V. I. Burenkov, D. I. Hakim, E. Nakai, Y. Sawano, T. Sobukawa, and T. V. Tararykova, Complex interpolation of the predual of Morrey spaces over measure spaces, in preparation.
[5] V. I. Burenkov, P. Jain, and T. V. Tararykova, On boundedness of the Hardy operator in Morrey-type spaces, Eurasian Math. J. 2 (2011), no. 1, 52–80. · Zbl 1321.47080
[6] V. I. Burenkov and E. D. Nursultanov, Description of interpolation spaces for local Morrey-type spaces (in Russian), Tr. Mat. Inst. Steklova 269 (2010), 52–62; English translation in Proc. Steklov Inst. Math. 269 (2010), no. 1, 46–56. · Zbl 1216.46019
[7] V. I. Burenkov, E. D. Nursultanov, and D. K. Chigambayeva, Description of the interpolation spaces for a pair of local Morrey-type spaces and their generalizations (in Russian), Tr. Mat. Inst. Steklova 284 (2014), 105–137; English translation in Proc. Steklov Inst. Math. 284 (2014), no. 1, 97–128. · Zbl 1304.42055
[8] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. · Zbl 0204.13703
[9] F. Cobos, J. Peetre, and L. E. Persson, On the connection between real and complex interpolation of quasi-Banach spaces, Bull. Sci. Math. 122 (1998), no. 1, 17–37. · Zbl 0953.46037
[10] A. Gogatishvili and R. Mustafayev, Dual spaces of local Morrey-type spaces, Czechoslovak Math. J. 61(136) (2011), no. 3, 609–622. · Zbl 1249.46020
[11] V. S. Guliyev, S. G. Hasanov, and Y. Sawano, Decompositions of local Morrey-type spaces, Positivity 21 (2017), no. 3, 1223–1252. · Zbl 1386.42007
[12] D. I. Hakim, S. Nakamura, Y. Sawano, and T. Sobukawa, Complex interpolation of \(B^{u}_{w}\)-spaces, Complex Var. Elliptic Equ. (2017). · Zbl 1388.42069
[13] D. I. Hakim and Y. Sawano, Interpolation of generalized Morrey spaces, Rev. Mat. Complut. 29 (2016), no. 2, 295–340. · Zbl 1353.46019
[14] 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), no. 5, 1195–1226. · Zbl 1393.46015
[15] P. G. Lemarié-Rieusset, Multipliers and Morrey spaces, Potential Anal. 38 (2013), no. 3, 741–752. Erratum, Potential Anal. 41 (2014), no. 4, 1359–1362. · Zbl 1267.42024
[16] Y. Lu, D. Yang, and W. Yuan, Interpolation of Morrey spaces on metric measure spaces, Canad. Math. Bull. 57 (2014), no. 3, 598–608. · Zbl 1316.46021
[17] C. B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, no. 1 (1938), 126–166. · Zbl 0018.40501
[18] E. Nakai and T. Sobukawa, \(B_{w}^{u}\)-function spaces and their interpolation, Tokyo J. Math. 39 (2016), no. 2, 483–517. · Zbl 1365.42016
[19] S. Polidoro and M. A. Ragusa, Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal. 14 (2001), no. 4, 341–350. · Zbl 0980.35081
[20] A. Ruiz and L. Vega, Unique continuation for Schrödinger operators with potential in Morrey spaces, Publ. Mat. 35 (1991), no. 1, 291–298. Corrigendum, Publ. Mat. 39 (1995), no. 2, 405–411. · Zbl 0809.47046
[21] Y. Sawano and H. Tanaka, The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo 22 (2015), no. 3, 663–683. · Zbl 1334.42051
[22] G. Stampacchia, \({\mathcal{L}}^{(p,λ)}\)-spaces and interpolation, Comm. Pure Appl. Math. 17 (1964), 293–306. · Zbl 0149.09201
[23] W. Yuan, Complex interpolation for predual spaces of Morrey-type spaces, Taiwanese J. Math. 18 (2014), no. 5, 1527–1548. · Zbl 1357.46018
[24] W. Yuan, W. Sickel, and D. Yang, Interpolation of Morrey-Campanato and related smoothness spaces, Sci. China Math. 58 (2015), no. 9, 1835–1908. · Zbl 1337.46030
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