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Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent functionsatisfying the globally log-H\"older continuous condition. In this article, theauthors first obtain a decomposition for any distribution of the variable weakHardy space into "good" and "bad" parts and then prove the following realinterpolation theorem between the variable Hardy space $H^{p(\cdot)}(\mathbbR^n)$ and the space $L^{\infty}(\mathbb R^n)$: \begin{equation*}(H^{p(\cdot)}(\mathbb R^n),L^{\infty}(\mathbb R^n))_{\theta,\infty}=W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n),\quad \theta\in(0,1), \end{equation*}where $W\!H^{p(\cdot)/(1-\theta)}(\mathbb R^n)$ denotes the variable weak Hardyspace. As an application, the variable weak Hardy space$W\!H^{p(\cdot)}(\mathbb R^n)$ with$p_-:=\mathop\mathrm{ess\,inf}_{x\in\rn}p(x)\in(1,\infty)$ is proved tocoincide with the variable Lebesgue space $W\!L^{p(\cdot)}(\mathbb R^n)$.

Interpolation between $H^{p(\cdot)}(\mathbb R^n)$ and $L^\infty(\mathbb R^n)$: Real Method