How to write a joint kernel density of two random variables with known individual densities? Consider two random variables $X$ and $Y$ with densities
$${f}_1(x) = \frac{1}{n_1h_1} \sum\limits_{i=1}^{n_1}K\left(\frac{x-u_i}{h_1}\right) ~~~~\text{and}  ~~~~ {f}_2(y) = \frac{1}{n_2h_2} \sum\limits_{j=1}^{n_2}K\left(\frac{y-v_j}{h_2}\right),$$respectively, where $K$ is a kernel and $h_1$ and $h_2$ are corresponding bandwidth parameters. Then, how to write the joint density of $X$ and $Y$?

It depends on what assumptions you are willing to make. Basically, you need a multivariate kernel. If you can assume that the variables are independent, you can use the product kernel
$$
\hat{f_h}(\mathbf{x}) = \frac{1}{n} \sum_{i=1}^n \prod_{j=1}^d \frac{1}{h_j} K\Big(\frac{x_i-\mathbf{X}_{ij}}{h_j}\Big)
$$
It won't do anything about handling dependence or correlation between variables. If you can't assume independence, you need a proper multivariate kernel density estimator defined in terms of multivariate kernels.
