Sull'insieme nodale delle autofunzioni del Laplaciano frazionario.

We prove an estimate on the Hausdorff measure of the nodal set of eigenfunctions of the fractional Laplacian, which are intended in a local sense. The estimate is given in terms of the Almgren frequency of the Caffarelli-Silvestre extension of the eigenfunction. In the literature, such an estimate was previously obtained for s-harmonic functions, which corresponds to the special case of the zero eigenvalue. In our case, we changed the definition of Almgren frequency taking the presence of the eigenvalue into account, and we show that the most crucial properties (interior regularity, compactness of blow-up sequences and doubling-type properties) are preserved. Thus, the same techniques employed in the s-harmonic case can be adapted to our more general case, and a similar estimate on the Hausdorff measure of the nodal set is obtained.​