This paper investigates the Cesàro summability method in the setting of $\mathcal{L}-$fuzzy normed spaces, extending the classical theory of summability to the framework of $\mathcal{L}-$fuzzy norms. We also establish a Tauberian theorem characterizing the conditions under which Cesàro summability implies ordinary convergence in this setting. Furthermore, we derive an analogue of Hardy's classical two-sided Tauberian theorem by employing the notion of q-boundedness in $\mathcal{L}-$fuzzy normed spaces, thereby providing new criteria for analyzing the convergence behavior of sequences and series in this context.