A NOTE ON THE SPACES OF VARIABLE INTEGRABILITY AND SUMMABILITY OF ALMEIDA AND HASTO

作者：Kempka Henning^*; Vybiral Jan

来源：Proceedings of the American Mathematical Society, 2013, 141(9): 3207-3212.

DOI：10.1090/S0002-9939-2013-11605-9

We address an open problem posed recently by Almeida and Hasto. They defined the spaces l(q)(.)(L-p(.)) of variable integrability and summability and showed that broken vertical bar broken vertical bar . |lq(.)(L-p(.)) broken vertical bar broken vertical bar is a norm if q %26gt;= 1 is constant almost every-where or if 1/p(x)+1/q(x) = 1 for almost every x. R-n. Nevertheless, the natural conjecture (expressed also by Almeida and Hasto) is that the expression is a norm if p(x), q(x) = 1 almost everywhere. We show that broken vertical bar broken vertical bar.broken vertical bar| ( q(.)(L-p(.)) broken vertical bar broken vertical bar is a norm if 1 %26lt;= q(x) %26lt;= p(x) for almost every x is an element of R-n. Furthermore, we construct an example of p(x) and q(x) with min(p(x), q(x)) %26gt;= 1 for every x. Rn such that the triangle inequality does not hold for broken vertical bar broken vertical bar. |broken vertical bar l(q)(.)(Lp(.)).

• 出版日期2013-9