In this paper we prove norm inflation in new regions of Sobolev regularities for the scalar Zakharov system in the spatial domain R d for arbitrary d ∈ N. To this end, we apply abstract considerations of Bejenaru and Tao from [BT06] and modify arguments of Iwabuchi and Ogawa [IO15].

We prove a weak maximum principle for nonlocal symmetric stable operators including the fractional Laplacian. The main focus of this work is on minimal regularity assumptions of the functions under consideration.