Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some constant $c \geq 0$, where $\left| \! \left| \cdot \right| \! \right|$ is the operator norm, \begin{align*} \left| \! \left| \sqrt{A+B} C \right| \! \right| \leq c \left| \! \left| \sqrt{A} C \right| \! \right| + c \left| \! \left| \sqrt{A} C \right| \! \right| ? \end{align*}

Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some constant $c \geq 0$, where $\left| \! \left| \cdot \right| \! \right|$ is the operator norm, \begin{align*} \left| \! \left| \sqrt{A+B} C \right| \! \right| \leq c \left| \! \left| \sqrt{A} C \right| \! \right| + c \left| \! \left| \sqrt{B} C \right| \! \right| ? \end{align*}