$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy \leq ||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $

Question

We are given the following, $$ b:\mathbb R^d \times \mathbb R^d \rightarrow \mathbb R,\;\; f:\mathbb R^d\rightarrow \mathbb R $$ and $$ f\in L^2(\mathbb R^d)\; ,\;b\in L^2(\mathbb R^d\times \mathbb R^d) $$

Show that

$$ \int_{\mathbb R^d}\int_{\mathbb R^d}b(x,y)\,f(x)\,f(y)dx\,dy\;\leq\;||b_+||_{L^2(\mathbb R^d\times \mathbb R^d)}||f||_{L^2(\mathbb R^d)}^2 $$

Here $b_+(x,y)= max \{b(x,y),0\}$ i.e. positive part of function $b$.

Sorry for this trivial question, I see it should not be too difficult to ask here. But I cannot do it.

Answer 1

Since $b = b_{+} - b_{-}$ and $b_{-}$ is nonnegative, $b \le b_{+}$. Therefore

\begin{align}\int_{\Bbb R^d} \int_{\Bbb R^d} b(x,y)f(x)f(y)\, dx\, dy &\le \int_{\Bbb R^d} \int_{\Bbb R^d} b_{+}(x,y) |f(x)||f(y)|\, dx\, dy\\ & \le \|b_{+}\|_{L^2(\Bbb R^d\times \Bbb R^d)}\|f(x)f(y)\|_{L^2(\Bbb R^d\times \Bbb R^d)}\\ &= \|b_{+}\|_{L^2(\Bbb R^d \times \Bbb R^d)}\|f\|_{L^2(\Bbb R^d)}^2. \end{align}

Answer 2

It suffices to show that if b is a.e.positive, then the left handside integral is nonnegative. This is trivial since b is a positive kernel of a Hilbert-Schimict operator, which is positive improving.