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<p>We prove the well-posedness of a Cauchy problem of the kind:</p> <p class="disp_formula"> $ \left\{\begin{array}{@{}l@{}c} \mathcal{L}u = f, &amp; \text{ in }D'(\mathbb{R}^N\times(0,+\infty)),\\ u(x,0) = g(x),&amp;\forall x\in\mathbb{R}^N, \end{array}\right. $ </p> <p>where $ f $ is Dini continuous in space and measurable in time and $ g $ satisfies suitable regularity properties. The operator $ \mathcal{L} $ is the degenerate Kolmogorov-Fokker-Planck operator</p> <p class="disp_formula"> $ \mathcal{L} = \sum\limits_{i,j = 1}^q a_{ij}(t)\partial_{x_ix_j}^2+ \sum\limits_{k,j = 1}^N b_{kj}x_k\partial_{x_j}-\partial_t $ </p> <p>where $ \{a_{ij}\}_{ij = 1}^q $ is measurable in time, uniformly positive definite and bounded while $ \{b_{ij}\}_{ij = 1}^N $ have the block structure:</p> <p class="disp_formula">$ \{b_{ij}\}_{ij = 1}^N = \left( \begin{matrix}{} \mathbb{O} &amp; \dots &amp; \mathbb{O} &amp; \mathbb{O} \\ \mathbb{B}_1 &amp; \dots &amp; \mathbb{O} &amp; \mathbb{O} \\ \vdots &amp; \ddots&amp; \vdots &amp; \vdots \\ \mathbb{O} &amp; \dots &amp; \mathbb{B}_\kappa &amp; \mathbb{O} \end{matrix} \right) $</p> <p>which makes the operator with constant coefficients hypoelliptic, 2-homogeneous with respect to a family of dilations and traslation invariant with respect to a Lie group.</p>