On Choquet Integrals and Sobolev Type Inequalities

Abstract We consider integrals in the sense of Choquet with respect to the $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean n -space, $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , $$0&lt;\delta \le n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . In particular, for these functions we prove Sobolev inequalities in the limiting case $$p=\delta /n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>/</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> and in the case $$p&gt;\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> , here p is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré–Sobolev and Morrey inequalities.