Constrained differential operators, fractional integrals, and Sobolev inequalities
A talk in the Nonlocal Equations: Analysis and Numerics series by
Andrea Cianchi from Florence
| Abstract: | Inequalities for Riesz potentials are well-known to be equivalent to Sobolev
inequalities of the same order for domain norms ”far” from $L^1$, but to be
weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Raita and Spector, and by Stolyarov proved that this gap can be filled
in Riesz potential inequalities for vector-valued functions in $L^1$ fulfilling
a co-canceling differential condition. This work demonstrates that such
a property is not just peculiar to the space $L^1$. Indeed, under the same
differential constraint, a Riesz potential inequality is shown to hold for any
domain and target rearrangement-invariant norms that render a Sobolev
inequality of the same order true. This is based on a new interpolation
inequality, which, via a kind of duality argument, yields a parallel property
of Sobolev inequalities for any linear homogeneous elliptic canceling differential operator. Specifically, Sobolev inequalities involving the full gradient
of a certain order share the same rearrangement-invariant domain and target spaces as their analogs for any other homogeneous elliptic canceling
differential operator of equal order. As a consequence, Riesz potential
inequalities under the co-canceling constraint and Sobolev inequalities for
homogeneous elliptic canceling differential operators are offered for general
families of rearrangement-invariant spaces, such as the Orlicz spaces and
the Lorentz-Zygmund spaces. Especially relevant instances of inequalities
for domain spaces neighboring $L^1$ are singled out. This is joint work with
D.Breit and D.Spector. Within the CRC this talk is associated to the project(s): A7 |