Redirection with nested commands

I am running a $command, in a instance of a screen, as a different user.

It's run by the equivalent of this command (more detailed at bottom):

su $USERNAME -c "screen -dmS $SCREENNAME $COMMAND"

and that works great. The problem is I want to redirect the output of $COMMAND to some logfile (e.g $LOGFILE).

Just adding > $LOGFILE to the end seems to read from screen instead of $COMMAND, so I am stumped as to what I should do. I've tried grouping it with parenthesis and braces, but that caused errors so I probably wasn't doing it correctly.

So how can I redirect the output of $COMMAND to $LOGFILE?

---

The code I am running is "equivalent" to this:

execute() {
  su $USERNAME -c "$*"
}
COMMAND="$SOMEDIR/$SOMEPROG $SOMEARGS"
execute screen -dmS $SCREENNAME $COMMAND