Geodesic
On geometric properties of Morrey spaces
Ufa mathematical journal, Tome 13 (2021) no. 1, pp. 131-136 Cet article a éte moissonné depuis la source Math-Net.Ru

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The study of Morrey spaces is motivated by many reasons. Initially, these spaces were introduced in order to understand the regularity of solutions to elliptic partial differential equations [1]. In line with this, many authors study the boundedness of various integral operators on Morrey spaces. In this article, we are interested in their geometric properties, from functional analysis point of view. We show constructively that Morrey spaces are not uniformly non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than earlier results, which showed that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{\mathrm{J}}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{\mathrm{NJ}}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le p$ are equal to $n$.
Keywords: Morrey spaces, uniformly non-$\ell^1_n$-ness, $n$-th James constant
Mots-clés : $n$-th Von Neumann-Jordan constant. 
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H. Gunawan; D. I. Hakim; A. S. Putri. On geometric properties of Morrey spaces. Ufa mathematical journal, Tome 13 (2021) no. 1, pp. 131-136. http://geodesic.mathdoc.fr/item/UFA_2021_13_1_a11/

[1] F. Chiarenza, M. Frasca, “A remark on a paper by C. Fefferman”, Proc. Amer. Math. Soc., 108:2 (1990), 407–409 | MR | Zbl

[2] J. A. Clarkson, “The von Neumann-Jordan constant for the Lebesgue spaces”, Ann. Math., 38:1 (1937), 114–115 | DOI | MR

[3] J. Gao, K.-S. Lau, “On the geometry of spheres in normed linear spaces”, J. Austral. Math. Soc., 48:1 (1990), 101–112 | DOI | MR | Zbl

[4] H. Gunawan, E. Kikianty, Y. Sawano, C. Schwanke, “Three geometric constants for Morrey spaces”, Bull. Korean Math. Soc., 56:6 (2019), 1569–1575 | MR | Zbl

[5] R. C. James, “Uniformly non-square Banach spaces”, Ann. Math., 80:3 (1964), 542–550 | DOI | MR | Zbl

[6] A. Jiménez-Melado, E. Llorens-Fuster, E. Mazcunán-Navarro, “The Dunkl-Williams constant, convexity, smoothness and normal structure”, J. Math. Anal. Appl., 342:1 (2008), 298–310 | DOI | MR | Zbl

[7] P. Jordan, J. Von Neumann, “On inner products in linear, metric spaces”, Ann. Math., 36:3 (1935), 719–723 | DOI | MR

[8] M. Kato, L. Maligranda, Y. Takahashi, “On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces”, Studia Math., 144:3 (2001), 275–295 | DOI | MR | Zbl

[9] M. Kato, Y. Takahashi, K. Hashimoto, “On $n$-th Von Neumann-Jordan constants for Banach spaces”, Bull. Kyushu Inst. Tech., 1998, no. 45, 25–33 | MR

[10] L. Maligandra, L. I. Nikolova, L.-E. Persson, T. Zachariades, “On $n$-th James and Khintchine constants of Banach spaces”, Math. Ineq. Appl., 11:1 (2007), 1–22 | MR

[11] A. Muta'zili, H. Gunawan, On geometric constants for (small) Morrey spaces, 2019, arXiv: 1904.01712 | MR

[12] Y. Sawano, “A thought on generalized Morrey spaces”, J. Indones. Math. Soc., 25:3 (2019), 210–281 | DOI | MR | Zbl