A multilinear version of Schur’s test is obtained for products of L spaces and is used to derive boundedness for multilinear multiplier operators acting on Sobolev and Besov spaces.

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.

We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces.

1 Some Basic Analysis 3 1.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Fourier inversion on S and L . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Sobolev spa...

The function spaces Dk(R) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp = n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div–curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539–543 (see also J. Van Schaftingen, Estimates for L1-vector fields...

We denote by Lloc(IR) the space of locally integrable functions f : IR 7→ IR. These are the Lebesgue measurable functions which are integrable over every bounded interval. The support of a function φ, denoted by Supp(φ), is the closure of the set {x ; φ(x) 6= 0} where φ does not vanish. By C∞ c (IR) we denote the space of continuous functions with compact support, having continuous derivatives ...

Fixed point theorems for generalized Lipschitzian semigroups are proved in puniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in Lp spaces, in Hardy space Hp , and in Sobolev spaces Hk,p , for 1<p <∞ and k≥ 0.