1. The problem states that the function $f$ is defined by $f(x) = 7x^3$ and the graph of $y = g(x)$ is obtained by shifting the graph of $y = f(x)$ down 2 units. 2. When a graph is shifted down by 2 units, the new function $g(x)$ is given by subtracting 2 from the original function $f(x)$: $$g(x) = f(x) - 2$$ 3. Substitute $f(x) = 7x^3$ into the equation: $$g(x) = 7x^3 - 2$$ 4. This matches option D. Therefore, the equation defining $g$ is: $$\boxed{g(x) = 7x^3 - 2}$$