Suppose $f\in C^\infty(\mathbb{R})$ is compactly supported on $[-N,N]$, and $1\leq p<\infty$, and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)\textrm dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right).$$
It follows easily that $\int_\mathbb{R}K_t(x)\textrm dx=\int_\mathbb{R}K(x)\textrm dx=1$.
I'm in the process of proving some convergence bound, and I am left to show that $$\lim_{t\rightarrow 0}\int_{|x|>N+1}\left|\int_{\mathbb{R}}f(x-y)K_t(y)\textrm dy\right|^p \textrm dx=0.$$
How can I show that?
I assume that $K$ is non-negative and $N=1$. Using Jensen's inequality, it's enough to prove that $$I_t:=\int_{\{|x|\gt 2\}}\int_\mathbb R |f(x-y)|^pK_t(y)\mathrm dy\mathrm dx\to 0\quad \mbox{as }t\to 0.$$ The substitution $s=y/t$ yields $$I_t=\int_{\{|x|\gt 2\}}\int_\mathbb R |f(x-ts)|^pK(s)\mathrm ds\mathrm dx=\int_2^\infty+\int_{-\infty}^{-2}.$$
Since the support of $f$ is contained in $[-1,1]$, we get $$I_t=\int_{|y|\geqslant t^{-1}}K(y)\mathrm dy.$$