$(x_{k}, y_{k})$ is missing.

Using (\[d12\]), the $\mathbb{D} ^{\alpha ,2}$ norm of the random distribution $\xi (x)$ can be written as $$\begin{aligned}
&&\Vert \xi (x) \Vert _{\alpha , 2} ^{2} \\
&=& \sum_{m\geq 0} (1+m) ^{\alpha } (m+2)! \Vert   \left( \frac{1}{m+1}\frac{1}{m+2} \sum_{k,l=1; k\not= l} ^{m+2} a_{m} ^{x} (x_{k}, y_{l}) 1_{[0, x_{k}]\times [0, y_{l} ] } (x_{l}, y_{k}) \right.\\
  &&\left. \times 1_{[0, x_{k} ] \times [0, y_{l}]} ^{\otimes m}\left( (x_{1}, y_{1}), (x_{2}, y_{2}),\ldots , \overline{ (x_{k}, y_{k}) }, \overline{ (x_{l}, y_{l}) },  \ldots ,(x_{m+2}, y_{m+2}) \right)\right) \Vert _{L^{2} (([0,1] ^{2}) ^{m+2} )}\\
&=& \sum_{m\geq 0} (1+m) ^{\alpha } (m+2)! \left( \frac{1}{m+1}\frac{1}{m+2}\right) ^{2} \sum_{k, l=1; k\not= l}^{m+2} \sum_{k', l'=1; k'\not= l'}^{m+2} \int_{([0, 1] ^{2} ) ^{m+2} } dx_{1}dy_{1}..dx_{m+2} dy_{m+2} \\
&&\sum_{k,l=1; k\not= l} ^{m+2} a_{m} ^{x} (x_{k}, y_{l})1_{[0,x_{k}]}^{\otimes m+1} (x_{1}, .., \overline{x_{k}},.., x_{m+2} )1_{[0,y_{l}] }^{\otimes m+1} (y_{1},.., \overline{y_{l}},.., y_{m+2} )  \\
&&\sum_{k',l'=0; k'\not= l'} ^{m+2} a_{m} ^{x} (x_{k'}, y_{l'}) 1_{[0,x_{k'}]}^{\otimes m+1} (x_{1}, .., \overline{x_{k'}},.., x_{m+2} )1_{[0,y_{l'}] }^{\otimes m+1} (y_{1},.., \overline{y_{l'}},.., y_{m+2} ).\end{aligned}$$ Notice that if $k\not= k'$, the terms $1_{[0, x_{k} ] } (x_{k'}) $ and $1_{[0, x_{k'} ] } (x_{k}) $ appear in the above expression. Therefore, all the summand with $k\not= k'$ will vanish. Similarly, the summands with $l\not= l'$ 