On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) $p$-K\"ahler structures

Inspired by a recent work of D. Wei--S. Zhu on the extension of closed complex differential forms and Voisin's usage of the $\partial\bar{\partial}$-lemma, we obtain several new theorems of deformation invariance of Hodge numbers and reprove the local stabilities of $p$-K\"ahler structures with the $\partial\bar{\partial}$-property. Our approach is more concerned with the $d$-closed extension by means of the exponential operator $e^{\iota_\varphi}$. Furthermore, we prove the local stabilities of transversely $p$-K\"ahler structures with mild $\partial\bar{\partial}$-property by adapting the power series method to the foliated case, which strengthens the works of A. El Kacimi Alaoui--B. Gmira and P. Ra\'zny on that of the transversely K\"ahler foliations with homologically orientability. We observe that a transversely K\"ahler foliation, even without homologically orientability, also satisfies the $\partial\bar{\partial}$-property. So even when $p=1$ (transversely K\"ahler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott--Chern numbers with mild $\partial\bar{\partial}$-properties are also presented.