This cannot hold true, at least not without further assumptions on $l$ (more precisely, on the decay of $h'(x)$ at infinity). Indeed, let $h_\epsilon:=\Gamma_\epsilon \ast h$ be the $\epsilon$-mollification as in your statement (here $\Gamma_\epsilon$ is the heat kernel at time $\epsilon$). You are asking whether you can contol $|h_\epsilon'(x)|\leq \frac{C}{x}$ for large $x$, uniformly in $\epsilon$. But it is well-known that, if for example $h$ is smooth enough (say $C^1$) then $h_{\epsilon}'=\Gamma_\epsilon\ast (h')$, which converges at least pointwise to $h'$ as $\epsilon\to 0$. So, roughly speaking, if $h'$ does not decay at infinity at least as $\frac{1}{|x|}$ you cannot expect this estimate to be true.

This cannot hold true, at least not without further assumptions on $l$ (more precisely, on the decay of $l'(x)$ at infinity). Indeed, let $l_\epsilon:=\Gamma_\epsilon \ast l$ be the $\epsilon$-mollification as in your statement (here $\Gamma_\epsilon$ is the heat kernel at time $\epsilon$). You are asking whether you can contol $|l_\epsilon'(x)|\leq \frac{C}{x}$ for large $x$, uniformly in $\epsilon$. But it is well-known that, if for example $l$ is smooth enough (say $C^1$) then $l_{\epsilon}'=\Gamma_\epsilon\ast (l')$, which converges at least pointwise to $l'$ as $\epsilon\to 0$. So, roughly speaking, if $l'$ does not decay at infinity at least as $\frac{1}{|x|}$ you cannot expect this estimate to be true.

This cannot hold true, at least not without further assumptions on $l$. Indeed, let $h_\epsilon:=\Gamma_\epsilon \ast h$ be the $\epsilon$-mollification as in your statement (here $\Gamma_\epsilon$ is the heat kernel at time $\epsilon$). You are asking whether you can contol $|h_\epsilon'(x)|\leq \frac{C}{x}$ for large $x$, uniformly in $\epsilon$. But it is well-known that, if for example $h$ is smooth enough (say $C^1$) then $h_{\epsilon}'=\Gamma_\epsilon\ast (h')$, which converges at least pointwise to $h'$ as $\epsilon\to 0$. So, roughly speaking, if $h'$ does not decay at infinity at least as $\frac{1}{|x|}$ you cannot expect this estimate to be true.

This cannot hold true, at least not without further assumptions on $l$ (more precisely, on the decay of $h'(x)$ at infinity). Indeed, let $h_\epsilon:=\Gamma_\epsilon \ast h$ be the $\epsilon$-mollification as in your statement (here $\Gamma_\epsilon$ is the heat kernel at time $\epsilon$). You are asking whether you can contol $|h_\epsilon'(x)|\leq \frac{C}{x}$ for large $x$, uniformly in $\epsilon$. But it is well-known that, if for example $h$ is smooth enough (say $C^1$) then $h_{\epsilon}'=\Gamma_\epsilon\ast (h')$, which converges at least pointwise to $h'$ as $\epsilon\to 0$. So, roughly speaking, if $h'$ does not decay at infinity at least as $\frac{1}{|x|}$ you cannot expect this estimate to be true.