Sobolev type spaces based on Lorentz-Karamata spaces

On Trudinger-Moser type inequalities involving Sobolev-Lorentz spaces

Generalizations of the Trudinger-Moser inequality to Sobolev-Lorentz spaces with weights are considered. The weights in these spaces allow for the addition of certain lower order terms in the exponential integral. We prove an explicit relation between the weights and the lower order terms; furthermore, we show that the resulting inequalities are sharp, and that there are related phenomena of co...

Valuations on Sobolev Spaces

All affinely covariant convex-body-valued valuations on the Sobolev space W (R) are completely classified. It is shown that there is a unique such valuation for Blaschke addition. This valuation turns out to be the operator which associates with each function f ∈W (R) the unit ball of its optimal Sobolev norm. 2000 AMS subject classification: 46B20 (46E35, 52A21,52B45) Let ‖ ·‖ denote a norm on...

Sobolev spaces on graphs

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...

Sobolev Spaces

Poincaré–type Inequalities for Broken Sobolev Spaces

We present two versions of general Poincaré–type inequalities for functions in broken Sobolev spaces, providing bounds for the Lq–norm of a function in terms of its broken H1–norm.