I've recently come across the Riemann-Liouville operator in $L^p(a,b)$ (with $1\leq p \leq +\infty$), whose definition is :
$V^\alpha > f(t)=\dfrac{1}{\Gamma(\alpha)}\int_a^t(t-x)^{\alpha-1}f(x)dx$
for $\alpha\in\mathbb{C}$ with real part $\Re(\alpha)>0$ and $f\in L^p(a,b)$. The bound provided by Wikipedia is : $$\lVert V^\alpha f\rVert_p \leq \dfrac{(b-a)^{\Re(\alpha)/p}}{|\Gamma(\alpha)|\Re(\alpha)} \lVert f \rVert_p.$$
Unfortunately, I can't reach that bound. Here's my attempt : let $q$ be such that $1/p+1/q=1$. Then : \begin{align}|V^\alpha f(t)| &= \left|\dfrac{1}{\Gamma(\alpha)}\int_a^t (t-x)^{\alpha-1}f(x)dx\right| \\\\ &\leq \dfrac{1}{|\Gamma(\alpha)|}\int_a^t (t-x)^{\Re(\alpha)-1}|f(x)|dx \\\\ &= \dfrac{1}{|\Gamma(\alpha)|}\int_a^t \Bigl((t-x)^{(\Re(\alpha)-1)/p}|f(x)|\Bigr)(t-x)^{(\Re(\alpha)-1)/q}dx \quad(\text{preparing Hölder inequality}) \\\\ &\leq \dfrac{1}{|\Gamma(\alpha)|} \Bigl(\int_a^t \Bigl[(t-x)^{(\Re(\alpha)-1)/p}|f(x)|\Bigr]^pdx \Bigr)^{1/p} \Bigl(\int_a^t \Bigl[(t-x)^{(\Re(\alpha)-1)/q}\Bigr]^q dx\Bigr)^{1/q} \\\\ &= \dfrac{1}{|\Gamma(\alpha)|} \Bigl(\int_a^t (t-x)^{\Re(\alpha)-1}|f(x)|^pdx \Bigr)^{1/p} \Bigl(\int_a^t (t-x)^{(\Re(\alpha)-1)} dx\Bigr)^{1/q} \\\\ &= \dfrac{1}{|\Gamma(\alpha)|} \Bigl(\int_a^t (t-x)^{\Re(\alpha)-1}|f(x)|^pdx \Bigr)^{1/p} \dfrac{(t-a)^{\Re(\alpha)/q}}{\Re(\alpha)^{1/q}} \\\\ &\leq \dfrac{(b-a)^{\Re(\alpha)/q}}{|\Gamma(\alpha)|\Re(\alpha)^{1/q}} \int_a^t (t-x)^{\Re(\alpha)-1}|f(x)|^p dx \quad(t-a\leq b-a) \end{align} Elevating this final inequality to the power $p$, by noticing that $\frac{p}{q}=p-1$, then integrating to the power $p$ : \begin{align} \lVert V^\alpha f \rVert_p^p &= \int_a^b |V^\alpha f(t)|^pdt \\\\ &\leq \dfrac{(b-a)^{\Re(\alpha)(p-1)}}{|\Gamma(\alpha)|^p\Re(\alpha)^{p-1}} \int_a^b \Bigl( \int_a^t (t-x)^{\Re(\alpha)-1}|f(x)|^pdx \Bigr)dt \\\\ &= \dfrac{(b-a)^{\Re(\alpha)(p-1)}}{|\Gamma(\alpha)|^p\Re(\alpha)^{p-1}} \int_a^b |f(x)|^p \Bigl( \int_x^b (t-x)^{\Re(\alpha)-1}dt \Bigr)dx \quad(\text{Fubini theorem})\\\\ &= \dfrac{(b-a)^{\Re(\alpha)(p-1)}}{|\Gamma(\alpha)|^p\Re(\alpha)^{p-1}} \int_a^b |f(x)|^p \dfrac{(b-x)^{\Re(\alpha)}}{\Re(\alpha)}dx \\\\ &\leq \Bigl( \dfrac{(b-a)^{\Re(\alpha)}}{|\Gamma(\alpha)|\Re(\alpha)} \lVert f \rVert_p\Bigr)^p \quad(b-x\leq b-a) \end{align} Which provides :$$ \lVert V^\alpha f \rVert_p \leq \dfrac{(b-a)^{\Re(\alpha)}}{|\Gamma(\alpha)|\Re(\alpha)} \lVert f \rVert_p. $$ (Observe the lack of exponent $frac{1}{p}$ at the numerator)