An estimates of a function : $L^2$-norm

Question

if we have the following estimates: if we take $\Omega$ is a bounded open set of $\mathbb{R}$, and $f,g\in L^2(\Omega)$ with $C_1,C_2>0$ constants: $$\int_{\Omega}fgdx\leq C_1 + C_2 \|f\|_2 $$ My question is: Can I have an estimates of $\|g\|_2$?

Answer 1

First, I believe the answer is no, because your integral on the left hand side can be equal to zero or even to arbitrary negative number (for example).

Second, the question is a bit vague, so in order to make my answer a little more substantial I will make some assumptions by my own. Namely, let $H$ be a Hilbert space with the inner product $\langle\cdot,\cdot\rangle$ and let $g \in H$ be such that the inequality $$ \langle f,g \rangle \leq C_1 + C_2 \lVert f \rVert $$ holds for some $C_1,C_2 >0$ and for any $f \in H$. I claim that $g = 0$.

Indeed, our inequality obviously implies $$ \langle f,g \rangle \leq C_1 + C_2 \lVert f \rVert^2, $$ which is equivalent to $$ \langle f,g \rangle - C_2 \lVert f \rVert^2 \leq C_1. $$ On the other hand, $$ \langle f,g \rangle - C_2 \lVert f \rVert^2 = \langle f,g \rangle - C_2 \langle f,f \rangle = \langle f-C_2f,g \rangle = \langle (1-C_2)f,g \rangle. $$ Since $f$ is arbitrary, it follows that $$ \langle f,g \rangle \leq C_1. $$ Now take $f = \frac{C_1}{\varepsilon} g$ where $\varepsilon>0$. We get the estimate $$ \langle g,g \rangle = \lVert g \rVert^2 \leq \varepsilon. $$ This holds for any $\varepsilon>0$, in particular, for $\varepsilon$ arbitrary small. But then $\lVert g \rVert$ must be equal to zero, and the result follows.