I've come across Holder inequality used in the below case where $\lambda_i's$ are over some integers and we apply Holder inequality with exponents $2k$ and $2k/(2k-1)$ over the variables $\lambda_1,\dots , \lambda_r$. However, I cannot find a source that states Holder inequality applied over multiple sums. How does this work out here? $J_{k,r}$ is just some counting function and $e(\theta)$ is $\exp(2\pi i \theta)$.
After some rewriting, we can just apply the usual Hölder inequality. We can indeed just take a single sum over the $r$-dimensional variable $\lambda=(\lambda_1,\ldots,\lambda_r)$ taking the values of all possible combinations of $\lambda_1$, ..., $\lambda_r$. Then, to simplify things, we can write: $$ f(\lambda) := J_{k,r}(\lambda_1,\ldots,\lambda_r;Z), \ \ \ \ \ \ \ \ \ \ g(\lambda) := \left|\sum_{x\leq Z} e(\alpha_1x\lambda_1+\ldots+\alpha_rx^r\lambda_r)\right|, $$ and if we take as you said $p=\frac{2k}{2k-1}$ and $q=2k$, this all just amounts to applying the regular Hölder inequality: $$ \sum_{\lambda} f(\lambda) g(\lambda) \leq \left(\sum_{\lambda} f(\lambda)^p\right)^{1/p} \, \left(\sum_{\lambda} g(\lambda)^q\right)^{1/q}. $$