Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian

We study the local well-posedness of nonlinear Schrödinger equation associated to Grushin operator with random initial data. To best our knowledge, no result is known in Sobolev spaces Hk when k⩽32. In this article, we prove that there exists a large family data such that, respect suitable randomization Hk, k∈(1,32], almost-sure holds. The proof relies on bilinear and trilinear estimates.