Pointwise estimates to the modified Riesz potential

Petteri Harjulehto, Ritva Hurri-Syrjänen

Research output: Contribution to journalArticleScientificpeer-review

Abstract

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.

Original languageEnglish
JournalManuscripta Mathematica
Volume156
Issue number3-4
Pages (from-to)521-543
Number of pages23
ISSN0025-2611
DOIs
Publication statusPublished - Jul 2018
MoE publication typeA1 Journal article-refereed

Fields of Science

  • 111 Mathematics
  • IRREGULAR DOMAINS
  • ORLICZ SPACES
  • INEQUALITY
  • EXTENSION
  • OPERATORS

Cite this

Harjulehto, Petteri ; Hurri-Syrjänen, Ritva. / Pointwise estimates to the modified Riesz potential. In: Manuscripta Mathematica. 2018 ; Vol. 156, No. 3-4. pp. 521-543.
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Pointwise estimates to the modified Riesz potential. / Harjulehto, Petteri; Hurri-Syrjänen, Ritva.

In: Manuscripta Mathematica, Vol. 156, No. 3-4, 07.2018, p. 521-543.

Research output: Contribution to journalArticleScientificpeer-review

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T1 - Pointwise estimates to the modified Riesz potential

AU - Harjulehto, Petteri

AU - Hurri-Syrjänen, Ritva

PY - 2018/7

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N2 - In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.

AB - In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.

KW - 111 Mathematics

KW - IRREGULAR DOMAINS

KW - ORLICZ SPACES

KW - INEQUALITY

KW - EXTENSION

KW - OPERATORS

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