On Choquet integrals and Poincar\'e-Sobolev inequalities

Abstract: We consider integral inequalities in the sense of Choquet with respect to the Hausdorff content

m a t h c a l H_{i} n f t y^{d e l t a}

. In particular, if

O m e g a

is a bounded John domain in

m a t h b b R^{n}

,

n g e q 2

, and

0 < d e l t a l e n

, we prove that the corresponding

(d e l t a p / (d e l t a - p), p)

-Poincar'e-Sobolev inequalities hold for all continuously differentiable functions defined on

O m e g a

whenever

d e l t a / n < p < d e l t a

. We prove also that the

(p, p)

-Poincar'e inequality is valid for all

p > d e l t a / n

.

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