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Sobolev Spaces in Mathematics II pp 103–121Cite as

Conductor Inequalities and Criteria for Sobolev-Lorentz Two-Weight Inequalities

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Part of the book series: International Mathematical Series ((IMAT,volume 9))

Abstract

We present integral conductor inequalities connecting the Lorentz p, q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p, q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures.

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Author information

Authors and Affiliations

  1. McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada

    Serban Costea

  2. Ohio State University, Columbus, OH 43210, USA

    Prof. Vladimir Maz'ya

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  1. Serban Costea
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  2. Prof. Vladimir Maz'ya
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Correspondence to Serban Costea .

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Editors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, USA

    Vladimir Maz'ya

  2. Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

    Vladimir Maz'ya

  3. Department of Mathematics, Linköping University, Linköping, Sweden

    Vladimir Maz'ya

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Costea, S., Maz'ya, V. (2009). Conductor Inequalities and Criteria for Sobolev-Lorentz Two-Weight Inequalities. In: Maz'ya, V. (eds) Sobolev Spaces in Mathematics II. International Mathematical Series, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85650-6_6

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