Quantitative maximal ${\mathrm{L}}^p$-regularity for generators of real contraction semigroups

Let $-A$ be the generator of a real, symmetric contraction semigroup on a $\sigma$-finite measure space. We prove a universal bound for maximal ${\mathrm{L}}^p$-regularity: In space-time norms $$\mathrm{\parallel }Au{\mathrm{\parallel }}_p\le (p^{\ast }+\frac{p^{\ast }}{2(p^{\ast }-1)})\mathrm{\parallel }f{\mathrm{\parallel }}_p,$$ where $p^{\ast }=\max (p,p\mathrm{/}(p-1))$ and $u$ is the mild solution to the Cauchy problem with right-hand side $f$ starting at $u(0)=0$. Linear growth of the constant in $p^{\ast }$ is optimal. Our proof introduces an elementary approach to maximal regularity that avoids both vector-valued harmonic analysis and dilation theory. We also obtain quantitative maximal regularity estimates for real, non-symmetric semigroups that are contractive on a sector.