I don't understand why you have the $\delta$. One has the following estimate (see, for instance, Theorem 4.18 in the Clay lecture notes "Nonlinear Schrodinger Equations at Critical Regularity" by Rowan Killip and Monica Visan):

**Bilinear Strichartz.** Let $j,k$ be integers such that $j\leq k$. Then
$$\|(e^{it\Delta}P_j f)(e^{it\Delta}P_k g)\|_{L_{t,x}^2(\mathbb{R}\times\mathbb{R}^d)} \lesssim 2^{\frac{j(d-1)}{2} -\frac{k}{2}} \|f\|_{L^2(\mathbb{R}^d)}\|g\|_{L^2(\mathbb{R}^d)}.$$

Let$\tilde{P}_j$ and $\tilde{P}_k$ be "fattened" Littlewood-Paley projectors such that their symbols are identically one on the support of the symbol of $P_j$ and $P_k$, respectively, for all $j,k\in\mathbb{Z}$, so that in particular, $P_j\tilde{P}_j = P_j$ and $P_k\tilde{P}_k=P_k$. We then have as a consequence of Bilinear Strichartz that

$$\|(e^{it\Delta}P_j f)(e^{it\Delta}P_k g)\|_{L_{t,x}^2(\mathbb{R}\times\mathbb{R}^d)} \lesssim 2^{\frac{j(d-1)}{2}-\frac{k}{2}} \|\tilde{P}_j f\|_{L^2(\mathbb{R}^d)} \|\tilde{P}_k g\|_{L^2(\mathbb{R}^d)}.$$
By Bernstein's lemma,
\begin{align}
2^{\frac{j(d-1)}{2}}\|\tilde{P}_j f\|_{L^2(\mathbb{R}^d)} &\sim \|\tilde{P}_j f\|_{\dot{H}^{\frac{d-1}{2}}(\mathbb{R}^d)} \lesssim\|f\|_{\dot{H}^{\frac{d-1}{2}}(\mathbb{R}^d)},\\
2^{-\frac{k}{2}}\|\tilde{P}_k g\|_{L^2(\mathbb{R}^d)} &\sim \|\tilde{P}_kg\|_{\dot{H}^{-1/2}(\mathbb{R}^d)} \lesssim \|g\|_{\dot{H}^{-1/2}(\mathbb{R}^d)}.
\end{align}