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Choice D is correct. If the graph of $y=g\left(x\right)$ is the result of shifting the graph of $y=f\left(x\right)$ down $k$ units in the xy-plane, the function $g$ can be defined by an equation of the form $g\left(x\right)=f\left(x\right)-k$. It’s given that $f\left(x\right)=7x^3$ and the graph of $y=g\left(x\right)$ is the result of shifting the graph of $y=f\left(x\right)$ down $2$ units. Substituting $7x^3$ for $f\left(x\right)$ and $2$ for $k$ in the equation $g\left(x\right)=f\left(x\right)-k$ yields $g\left(x\right)=7x^3-2$. 

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This equation defines a function $g$ for which the graph of $y=g\left(x\right)$ is the result of shifting the graph of $y=f\left(x\right)$ up, not down, $2$ units.