Properties of a bent function construction by a subspace of an arbitrary dimension

Let $f$ be a bent function in $2k$ variables, $L$ be an affine subspace of $\mathbb F_2^{2k}$, and $\mathrm{Ind}_L$ be a Boolean function with values $1$ on $L$. Here, we study the properties of the function $f\oplus\mathrm{Ind}_L$. Particularly, we give some necessary and sufficient conditions under which the increase or decrease of the dimension of $L$ by $1$ doesn't change the property bent of $f\oplus\mathrm{Ind}_L$. We prove that if the function $f(x_1,\dots,x_{2k})\oplus x_{2k+1}x_{2k+2}\oplus\mathrm{Ind}_U$ is a bent function and $U$ is an affine subspace, then the function $f\oplus\mathrm{Ind}_L$ is a bent function for some affine subspace $L$ of dimension $\operatorname{dim}U-1$ or $\operatorname{dim}U-2$. An example of bent function $f$ in $10$ variables for which $f\oplus\mathrm{Ind}_L$ is a bent function for only $\operatorname{dim}L\in\{9,10\}$ is provided.