_V\|K\|_2h_{1,n}^{1/2}$ and the characteristics $A=R$ and $v=22$, and the lemma will follow by application of Talagrand’s inequality. We just need to estimate $Eg^2(t,X)$. We have, making several natural changes of variables, $$\begin{aligned}
 Eg^2(t,X_1)&=&
E\Big\{\int f^{1/2}(x)K\Big(\frac{x-X_1}{h_{1,n}}\Big)L\Big(\frac{t-x}{h_{2,n}}f^{1/2}(x)\Big)
I(|t-x|<h_{2,n}B)dx\nonumber\\
&&~~~~~~~~~~\times \int f^{1/2}(y)K\Big(\frac{y-X_1}{h_{1,n}}\Big)L\Big(\frac{t-y}{h_{2,n}}f^{1/2}(y)
\Big)I(|t-y|<h_{2,n}B)
dy\Big\}\nonumber\\
&=&h_{1,n}^2\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
f^{1/2}(t-h_{1,n}v_2)\nonumber\\
&&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)
K\Big(\frac{t-u}{h_{1,n}}-v_1\Big)K
\Big(\frac{t-u}{h_{1,n}}-v_2\Big)\nonumber\\
&&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(u)dudv_1dv_2\nonumber\\
&=&h_{1,n}^3\int\int\int f^{1/2}(t-h_{1,n}v_1)L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
f^{1/2}(t-h_{1,n}v_2)\nonumber\\
&&~~~~~~~~~~\times L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)K(v)K(v+v_1-v_2)\nonumber\\
&&~~~~~~~~~~~~~~~\times I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)f(t-h_{1,n}v_1-h_{1,n}v)dvdv_1dv_2\nonumber\\
&\le& h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}v_1f^{1/2}(t-h_{1,n}v_1)\Big)
 L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\notag\\
 &&~~~~~~~~~~\times K(v)K(v+v_1-v_2)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_1|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdv_1dv_2\nonumber\\
&= &h_{1,n}^3||f||_\infty^2\int\int\int \left|L\Big(\frac{h_{1,n}}{h_{2,n}}(w+v_2)f^{1/2}(t-h_{1,n}w-h_{1,n}v_2)\Big)
L\Big(\frac{h_{1,n}}{h_{2,n}}v_2f^{1/2}(t-h_{1,n}v_2)\Big)\right|\nonumber\\
&&~~~~~~~~~~\times K(v)K(v+w)I\Big(\frac{h_{1,n}}{h_{2,n}}|w+v_2|<B\Big)
I\Big(\frac{h_{1,n}}{h_{2,n}}|v_2|<B\Big)dvdwdv_2\nonumber\\
&=&h_{1,n}^2h_{2,n}||f||_\infty^2\in