We know the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach Space. We say $A$ has maximal $L^p$-regularity if all $f\in L^p([0,T),X)$, $v_t$ and $Av$ belong to $L^p([0,T),X)$, and there exists $C_p>0$ such that $$\|v_t\|_{L^p([0,T),X)}+\|Av\|_{L^p([0,T),X)}\le C_p \|f\|_{L^p([0,T),X)}.$$ My question is how $C_p$ depends on $p$. Could anyone give an upper estimate of $C_p$?
In fact, for the heat semigroup on the whole space, $C_p\le c_1 p^\tau$ with $c_1,\tau>0$ known constants. I anticipate the same for a general operator, at least for the Neumann heat semigroup on a bounded domain.
