Consider the following unbounded curve in $\Bbb C$ $$ \Gamma=\{x+ih(x),\;\;x\in\Bbb R\} $$ where $h\colon\Bbb R\to\Bbb R$ is a bounded $\mathscr C^1$ real function, such that $h'(x)\to0$ as $|x|\to\infty$; hence $\Gamma$ is roughly a perturbation of $\Bbb R$ inside $\Bbb C$.
My case deals with the gaussian kernel $$ K_t(\zeta,z):=\frac{1}{t\sqrt \pi}e^{-\frac{(\zeta-z)^2}{t^2}} $$ where $t>0$ and $z$ is a fixed point in $\Gamma$ and $\zeta\in\widetilde\Gamma:=\{\zeta\in\Gamma\;:\;\Re(\zeta-z)\le\eta\}$ for some $\eta\ge0$. Then for any $\epsilon>0$ there exists $t_0>0$ such that $$ \left|\int_{\widetilde\Gamma}K_t(\zeta,z)\,d\zeta-1\right|<\epsilon $$ for all $0<t<t_0$. This tells us that the complex number $$ w:=\int_{\widetilde\Gamma}K_t(\zeta,z)\,d\zeta $$ stays in a $\epsilon$-ball around $1$. Fine, so far.
I wonder: what can we say about $$ \int_{\widetilde\Gamma}|K_t(\zeta,z)|\,d|\zeta|\;\;? $$ Clearly one gets $$ \epsilon >\left|\int_{\widetilde\Gamma}K_t(\zeta,z)\,d\zeta-1\right| \ge 1- \left|\int_{\widetilde\Gamma}K_t(\zeta,z)\,d\zeta\right| \ge 1-\int_{\widetilde\Gamma}|K_t(\zeta,z)|\,d|\zeta| $$ from which $$ \int_{\widetilde\Gamma}|K_t(\zeta,z)|\,d|\zeta|>1-\epsilon\;. $$ Would it be possible to prove that $$ \int_{\widetilde\Gamma}|K_t(\zeta,z)|\,d|\zeta|<1+\epsilon\;? $$