Article
Ufa Mathematical Journal
Volume 13, Number 1, pp. 131-136
On geometric properties of Morrey spaces
Gunawan H., Hakim D.I., Putri A.S.
DOI:10.13108/2021-13-1-131
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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 \cite{CF}. 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
$C_{{\rm J}}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{{\rm NJ}}^{(n)}(X)$
for a Banach space $X$, and obtain that both constants for any Morrey space
$\mathcal{M}^p_q(\mathds{R}^d)$ with $1\le p