Global $L^p$ estimates for the heat equation by approximation

Question

Consider the heat equation $\partial_t u = \Delta u + f $ on $\mathbb{R}^N$ with $u(0) = 0$. Let $u \in C_c^\infty(\mathbb{R}^{N+1})$ be a solution of the heat equation to some $f \in C_c^\infty(\mathbb{R}^{N+1})$. Then by multiplying with $f$ and integrating it follows that $$ \frac{1}{4} \| u \|_2 + \| \nabla u\|_2 \le \|f\|_2 ,$$ where $\| \cdot\|_2$ denotes the $L^2(\mathbb{R}^{N+1})$ norm. Is this inequality true for all $f \in L^2$? Is it possible to prove it by approximation, i.e. reducing the result to compactly supported smooth functions? Do you know any reference on this ?+