This paper deals with the fractional Sobolev spaces W s,p . We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains. These notes grew out of a few lectures given in an undergraduate class held at the Università di Roma "Tor Vergata". It is a pleasure to thank the students for their warm interest, their sharp observations and their precious feedback.

Abstract. We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

Abstract. We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.