Let a sequence $ ( f_{X,Y}^{(K)} )_{K=1}^\infty $ of continuous joint probability density functions (PDFs) $ f_{X,Y}^{(K)}: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R} $ be given and assume it converges uniformly to another continuous PDF $ f_{X, Y}: \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R} $, i.e., it holds $$ \lim_{K\to\infty} \| f_{X,Y}^{(K)} - f_{X,Y} \|_\infty = 0. $$ Let us define the marginals via $$ f_X^{(K)}(x) = \int_{\mathbb{R}^n} f_{X,Y}^{(K)}(x, y) dy $$ and correspondingly also $$ f_X(x) = \int_{\mathbb{R}^n} f_{X,Y}(x,y) dy. $$ Under what conditions do we have $$ \lim_{K\to\infty} \| f_X^{(K)} - f_X \|_\infty = 0, $$ i.e., uniform convergence of $ (f_X^{(K)})_{K=1}^\infty $ to $ f_X $?
I know that there are some statements if we look at the convergence of the cumulative distribution functions (convergence in distribution) but I'm interested in the PDFs and I don't know the right keywords to find the right literature.