Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point.
Define the integral
$$ Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta $$ and $\beta \in \mathbb{N}^d$ a multiindex.
Question: Do we have the inequality \begin{equation} \left\|\frac{\partial}{\partial x_i } Iu \right\| \le C \|u\| \end{equation} for some constant $C>0$ and $\|\cdot \|$ meaning the $L^2(\Delta)$-norm?
The integral $I$ arises for instance as remainder term in a Taylor expansion. I found (1) easy to prove for $d=1$ (simply substitute $\tau = tx$), for higher dimensions I tried polar coordinates but to no avail.
Edit My guess is that the answer is no, since $I$ only smoothes along rays emanating from zero...
