Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $\eta^\epsilon$ a mollifier, and $B_3^{\alpha, \infty}$ the Besov space. Define $$I_\epsilon(x) = \int \eta^\epsilon(y)\big(f(x-y)-f(x)\big)^2 dy$$ Using the inequality $$\|f(\cdot + y) - f(\cdot)\|_{L^3} \leq C |y|^\alpha \|f\|_{B_3^{\alpha, \infty}} \tag{1}$$ I would like to show that $$\|I_\epsilon\|_{L^{3/2}} \leq C \epsilon^{2\alpha}\|f\|^2_{B_3^{\alpha,\infty}}$$
Unravelling the definitions gives $$\|I_\epsilon\|_{L^{3/2}} = \Bigg[ \int \Bigg( \int \eta^\epsilon(y) \big(f(x-y) - f(x)\big)^2 dy\Bigg)^{3/2} dx \Bigg]^{2/3}.$$
My idea is to somehow get the $\frac32$ inside the integral and then use Hölder's Inequality to get two separate integrals. From there I can use the fact that $\eta^\epsilon$ is a mollifier so its integral is one, and I can use the inequality $(1)$ to take care of the second integral. However I am having trouble with the exponent being outside the integral in the first step.
How can I proceed?
The ingredient you're missing is Minkowski's integral inequality, which lets you move the $\frac32$ inside the integral and interchanges the order of integration. This gives \begin{align*} \lVert I_{\varepsilon} \rVert_{L^{\frac32}(\mathbb R^3)} &\leq \int_{\mathbb R^3} \left( \int_{\mathbb R^3} \lvert\eta^{\varepsilon}(y)\rvert^{\frac32} \lvert f(x-y) - f(x)\rvert^3 \,\mathrm{d}x\right)^{\frac23} \,\mathrm{d}y\\ & =\int_{\mathbb R^3} \eta^{\varepsilon}(y) \lVert f(\cdot - y ) - f \rVert_{L^3}^2\,\mathrm{d}y\\ & \leq C\int_{\mathbb R^3} \eta^{\varepsilon}(y) \lvert y \rvert^{2\alpha} \lVert f \rVert_{B_3^{\alpha,\infty}}^2\,\mathrm{d}y \\ &\leq C \varepsilon^{2\alpha} \lVert f \rVert_{B_3^{\alpha,\infty}}^2 \end{align*} as required, where we have applied said inequality in the first line.
Edited to add: Note that in the last line, we used the fact that $\eta^{\varepsilon}$ is supported on a ball of radius $\varepsilon$, so on said support we have $\lvert y \rvert^{2\alpha} \leq \varepsilon^{2\alpha}$. Then we use the fact that $\eta^{\varepsilon}$ integrates to one, as it is a mollifier.