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.

We establish the monotonicity property for the mass of non-pluripolar products on compact Kähler manifolds, and we initiate the study of complex Monge-Ampère type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kähler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for nonpluripolar products with small unbounded locus.Without the non-zero mass condition X θ n φ > 0 this characterization cannot hold (see Remark 3.3). The equivalence between (i) and (iii) in the above theorem shows that P θ [u] is the same potential for any u ∈ E(X, θ, φ), and equals to P θ [φ]. Given this and the inclusion E(X, θ, φ) ⊂ E(X, θ, P θ [φ]), one is tempted to consider only potentials φ in the image of the operator ψ → P θ [ψ], when studying the classes of relative full mass E(X, θ, φ). These potentials seemingly play the same role as V θ , the potential with minimal V * K,θ is supported on K, we thus haveand the desired inequality holds in this case. If M := M φ (K) ≥ 1, then by (12) we have φ ≤ M −1 V * K,φ + (1 − M −1 )φ ≤ φ + 1, and by definition of the relative capacity we can write:

Let X be a compact Kähler manifold and {θ} be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of θ-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class {η}, we show the inclusion E(X, η) ∩ PSH(X, θ) ⊂ E(X, θ). Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.