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Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with $H^q(X,V)=0$, then $dim{H}^q(X,V\otimes E)\<\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\<\infty$.

We study the $\bar{\partial }$-equation with Hölder estimates in $q$-convex wedges of ${ℂ}^n$ by means of integral formulas. If $D\subset {ℂ}^n$ is defined by some inequalities $\{{\rho }_i\le 0\}$, where the real hypersurfaces $\{{\rho }_i=0\}$ are transversal and any nonzero linear combination with nonnegative coefficients of the Levi form of the ${\rho }_i$’s have at least $(q+1)$ positive eigenvalues, we solve the equation $\bar{\partial }f=g$ for each continuous $(n,r)$-closed form $g$ in $D$, $n-q\le r\le n$, with the following estimates: if $d$ denotes the distance to the boundary of $D$ and if $d^{\beta }g$ is bounded, then for all $ϵ\>0$,...