关于Almeida和Hästö的变量可积与可和空间的注记
A note on the spaces of variable integrability and summability of Almeida and Hästö
摘要：
我们解决一个开放的问题最近提出的阿尔梅达和Hästö在\cite{AlHa10}。他们定义的空间$\ellqp$的可变可积性和可求和性，并表明$\|\cdot|\ellqp\|$是一个规范，如果$q$是常数几乎无处不在或如果$\esssup_{x\in\R^n}1/p(x)+1/q(x)\le 1$。然而，自然的猜想（也用\cite{AlHa10}表示）是，如果$p(x),q(x)\ge 1$几乎处处成立，则该表达式是范数。证明了$\|\cdot|\ellqp\|$是一个范数，如果几乎所有的$x\in\R^n.$都有$1\le q(x)\le p(x)$.此外，我们构造了一个$p(x)$和$q(x)$的例子，其中对每个$x\in\R^n$都有$\min(p(x),q(x))\ge 1$，使得三角不等式对$\|\cdot|\ellqp\|$不成立.
英文摘要：
We address an open problem posed recently by Almeida and Hästö in \cite{AlHa10}. They defined the spaces $\ellqp$ of variable integrability and summability and showed that $\|\cdot|\ellqp\|$ is a norm if $q$ is constant almost everywhere or if $\esssup_{x\in\R^n}1/p(x)+1/q(x)\le 1$. Nevertheless, the natural conjecture (expressed also in \cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show, that $\|\cdot|\ellqp\|$ is a norm, if $1\le q(x)\le p(x)$ for almost every $x\in\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\min(p(x),q(x))\ge 1$ for every $x\in\R^n$ such that the triangle inequality does not hold for $\|\cdot|\ellqp\|$.
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