From Sobolev inequality to doubling (original) (raw)

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.

On the Hardy–Sobolev–Maz'ya Inequality and Its Generalizations

International Mathematical Series, 2009

The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.

Markov-type inequalities and duality in weighted Sobolev spaces

The aim of this paper is to provide Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, some basic facts about Sobolev spaces with respect to certain vector measures are stated.

Improved Hardy-Sobolev inequalities

2007

The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux's technique is applied to the operator L := x • ∇ and the analysis requires the determination of the operator semigroup {e −tL * L } t>0 and its properties.

A product property of Sobolev spaces with application to elliptic estimates

Rendiconti del Seminario Matematico della Università di Padova, 2014

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p P [1Y I), nY m P Z , and m ! 2 that satisfy m b nap, kfck mYpYV K sup Vs jfj 2 3 k ck m Y p Y V k ck m À 1 Y q Y V kck mÀ1YpYV kfk mYpYV 4 5 for all fY c P W mYp (V) that satisfy spt c & V s & V and domains V & R n that are nonempty, open, and satisfy the cone condition. Here q p if p b n, q P (naÇY pna(n À p)] if n b p, q P (naÇY I) if p n, K K(nYpYmYqYg), where g is the cone from the cone condition, and Ç X [[ nap ]], the largest integer less than or equal to nap.

The weighted Sobolev and mean value inequalities

Proceedings of the American Mathematical Society, 2014

In this paper we prove a Michael-Simon inequality in the weighted setting and using this inequality we obtain a diameter control depending of the f-mean curvature, which is based in the work of Topping.

A Note about Young’s Inequality with Different Measures

International Journal of Mathematics and Mathematical Sciences

The key purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators defined with doubling measures in Lebesgue spaces with different measures. Relating Bessel decaying the kernel of the operators is satisfying some elementary properties. Doubling measure, Young's inequality, and Minköwski’s inequality will be used in proofs of boundedness of integral operators. In addition, we also explore the relation between the parameters of the kernel and generalized integral operators and see the norm of these generalized operators which will also be bounded by the norm of their kernel with different measures.

Sobolev inequalities in disguise

Indiana University Mathematics Journal, 1995

We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or sub-elliptic geometry, as well as on graphs and to certain non-local Sobolev norms. It only uses elementary cut-off arguments. This method has interesting consequences concerning Trudinger type inequalities.