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Abstract

Let $X$ be a quasi-Banach space. We prove that there exists $K>0$ such that for every function $w:\mathbb R \to X$ satisfying $$ \|w(s+t)-w(s)-w(t) \| \leq \varepsilon (|s|+|t|) \quad\ \hbox{for } s,t \in \mathbb R, $$ there exists a unique additive function $a:\mathbb R \to X$ such that $a(1)=0$ and $$ \|w(s)-a(s)-s \theta(\log_2|s|)\| \leq K\varepsilon|s| \quad\ \hbox{for } s \in \mathbb R, $$ where $\theta :\mathbb R \to X$ is defined by $\theta(k):=w(2^k)/2^k$ for $k \in \mathbb Z$ and extended in a piecewise linear way over the rest of $\mathbb R$.