$|\mathcal M\nu(w)-1|\ll\|w\|$ on the strip, for a mass-one kernel

Let $\nu$ be a $C^{1}$ kernel supported in $[1/2,2]$ with multiplicative mass one,

Then there is a constant $C>0$, depending only on $\nu$, such that for every $w$ in the strip $0<\operatorname{Re}w\le1$,

where $\mathcal M\nu(w)=\int_0^{\infty}\nu(x)x^{w-1}dx$.

The mass-one hypothesis is exactly the statement $\mathcal M\nu(0)=1$, so the estimate is a Lipschitz bound for $\mathcal M\nu$ measured from the origin — but valid on the whole strip, with $\operatorname{Im}w$ unrestricted, rather than only in a neighbourhood of $0$. One may take $C=3A+1$ with $A=\int_0^\infty|\nu(x)|\,dx/x$.

This is the analytic engine behind the estimate $\bigl|\mathcal M\widetilde{1_\varepsilon}(p)-p^{-1}\bigr|\le C\varepsilon$ for the smoothed cutoff: by the factorisation $\mathcal M\widetilde{1_\varepsilon}(p)=p^{-1}\mathcal M\nu(\varepsilon p)$, that estimate is the present one applied at $w=\varepsilon p$, and the uniformity in $\operatorname{Im}w$ is what allows $p$ to range over a full vertical line.