Hello everyone.

I am trying to establish a fractional integration lemma of this form.

For $\alpha\geq 0$, and $1\leq p,q<\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alpha}{d}$ or $1\leq p,q\leq\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}<\frac{\alpha}{d}$, for $f$ a function of $t\in\mathbb{R}$ and $x\in\mathbb{R}^3$, such that $f(t,\cdot)$ is in $L^q(\mathbb{R}^3)$, we have

$\Vert\Lambda_{t}^{-a}f(t,\cdot)\Vert_{p}\leq C t^{\frac{a}{2}-\frac{3}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}\Vert f\Vert_{q}$ where $\Lambda^{-\alpha}=\frac{1}{|D|^{\alpha}}$ and $\Lambda_{t}^{-\alpha}=\sqrt{t}^{\alpha}Z^{\alpha}\left(\sqrt{t}|D|\right)$ where $Z$ is a smooth function equal to $|\xi|^{-1}$ for $|\xi|\geq 2$ and equal to $1$ for $|\xi|\leq 1$. ($D$ is the Fourier multiplier, i>e $f(D)u=\mathcal{F}^{-1}(f(\xi))\hat{u}$ It must be linked to Riemann Liouville integral operators but I do not know how and how to consider the rescaling in $t$. Normally Fourier should be a simpler way to understand fractionnal integration but here I am quite stuck.

Thanks in advance to anyone who has ever done fractional integration!