"*On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel*" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):

A transmitter with $t$ antennas broadcasts to $r$ independent receivers through a flat-fading channel (modeled by a matrix $H\in \mathbb{C}^{r\times t}$) with iid additive white gaussian noise. i.e. each reception looks like: $$\vec{y}=H\vec{x}+\vec{z}; \quad \vec{z}\sim \mathcal{N}(\vec{0}, I_{r\times r})$$ Each receiver seeks to recieve a different message.

The **sum capacity** of such a channel is: $\displaystyle \underset{\vec{r}\in R}{\sup} \textstyle\sum_{i}r_i$ where $R$ is the achievable-rate region. It's just the maximum overall rate.

Given $H$, what is the **sum capacity** of a channel where a single transmit antenna is talking to $r$ independent (non-communicating) receive antennas?

Note that finding the actual region $R$ is difficult, but the sum capacity is a known result.

The paper says that the capacity region of the system in question is well-known. It cites Cover/Thomas' information theory textbook. However, Cover/Thomas only characterizes the $r=2$ case (relevant details start at pg515), and doesn't focus on the *sum capacity*

"*Sum Capacity of the Vector Gaussian Broadcast Channel and Uplink–Downlink Duality*" by Viswanath and David finds the sum capacity for the more complicated case where $t>1$, and says that the $t=1$ case is well-known. (but also alludes to $t=1$ systems being 'degenerate')