+ 2 \int_{0}^{1} \int_{0}^{1}dudv (a_{m}^{x}(u,v))^{2} (uv) ^{m+1}\right)\end{aligned}$$ converges to zero as $N,M\to \infty$. This is true because $$\sum_{i=0}^{N-1} \sum_{j=0}^{M-1}(a_{m}^{x}(s_{i}, t_{j}))^{2} s_{i}^{m+1} (s_{i+1}-s_{i}) t_{j}^{m+1} (t_{j+1}-t_{j}) \to _{N,M\to \infty} \int_{0}^{1} \int_{0}^{1}dudv (a_{m}^{x}(u,v))^{2} (uv) ^{m+1}$$ and $$\sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j})s_{i}^{m+1}  t_{j}^{m+1}\int_{s_{i}}^{s_{i+1}} \int_{t_{j}}^{t_{j+1}}dudv a_{m}^{x}(u,v)\to _{N,M\to \infty} \int_{0}^{1} \int_{0}^{1}dudv (a_{m}^{x}(u,v))^{2} (uv) ^{m+1}$$ using Riemann sums convergence. To conclude the convergence of $\Vert T_{N,M}(x)-\xi (x) \Vert ^{2} _{\alpha , 2}$ to zero for every $\alpha <\frac{-1}{2}$ it suffices to check that $$\sum_{i=0}^{N-1} \sum_{j=0}^{M-1}(a_{m}^{x}(s_{i}, t_{j}))^{2} s_{i}^{m+1} (s_{i+1}-s_{i}) t_{j}^{m+1} (t_{j+1}-t_{j})\leq cm^{-\frac{1}{2}},$$

$$\sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j})s_{i}^{m+1}  t_{j}^{m+1}\int_{s_{i}}^{s_{i+1}} \int_{t_{j}}^{t_{j+1}}dudv a_{m}^{x} (u,v) \leq cm^{-\frac{1}{2}},$$

and $$\int_{0}^{1} \int_{0}^{1}dudv (a_{m}^{x}(u,v))^{2} (uv) ^{m+1}\leq cm^{-\frac{1}{2}}.$$ with $c>0$ a constant not depending on $m$. The last bound has been proved before (see the proof of Theorem \[th1\]) and the first two bounds can be obtained analogously using relations (\[id\]) and (\[cn\]).

2D-stochastic currents in deterministic Sobolev spaces
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The purpose is this paragraph in to study the mapping (\[xi\]) as a function of the spatial variable $x\in \mathbb{R}$.

Regularity in the deterministic Sobolev spaces
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We study in this part the regularity with respect to the variable $x  \in \mathbb{R}^{d} $ of the mapping given by (\[xi\]) in the (deterministic) Sobolev spaces $H^{-r}
(\mathbb{R}^{d}; \mathbb{R}^{d})$. This is the 