New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2033-2050
  • ISSN: 2391-5455

Abstract

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We obtain Hardy type inequalities ∫ 0 ∞ M ω r u r ρ r d r ⩽ C 1 ∫ 0 ∞ M u r ρ r d r + C 2 ∫ 0 ∞ M u ' r ρ r d r , and their Orlicz-norm counterparts ω u L M ( ℝ + , ρ ) ⩽ C ˜ 1 u L M ( ℝ + , ρ ) + C ˜ 2 u ' L M ( ℝ + , ρ ) , with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

How to cite

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Agnieszka Kałamajska, and Katarzyna Pietruska-Pałuba. "New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights." Open Mathematics 10.6 (2012): 2033-2050. <http://eudml.org/doc/268960>.

@article{AgnieszkaKałamajska2012,
abstract = {We obtain Hardy type inequalities \[\int \_0^\infty \{M\left( \{\omega \left( r \right)\left| \{u\left( r \right)\} \right|\} \right)\rho \left( r \right)dr\} \leqslant C\_1 \int \_0^\infty \{M\left( \{\left| \{u\left( r \right)\} \right|\} \right)\rho \left( r \right)dr + C\_2 \int \_0^\infty \{M\left( \{\left| \{u^\{\prime \}\left( r \right)\} \right|\} \right)\rho \left( r \right)dr,\} \}\] and their Orlicz-norm counterparts \[\left\Vert \{\omega u\} \right\Vert \_\{L^M (\mathbb \{R\}\_ + ,\rho )\} \leqslant \tilde\{C\}\_1 \left\Vert u \right\Vert \_\{L^M (\mathbb \{R\}\_ + ,\rho )\} + \tilde\{C\}\_2 \left\Vert \{u^\{\prime \}\} \right\Vert \_\{L^M (\mathbb \{R\}\_ + ,\rho )\} ,\] with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.},
author = {Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba},
journal = {Open Mathematics},
keywords = {Hardy inequalities; Orlicz-Sobolev spaces; Nondoubling measures; nondoubling measures},
language = {eng},
number = {6},
pages = {2033-2050},
title = {New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights},
url = {http://eudml.org/doc/268960},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Agnieszka Kałamajska
AU - Katarzyna Pietruska-Pałuba
TI - New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2033
EP - 2050
AB - We obtain Hardy type inequalities \[\int _0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int _0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int _0^\infty {M\left( {\left| {u^{\prime }\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }\] and their Orlicz-norm counterparts \[\left\Vert {\omega u} \right\Vert _{L^M (\mathbb {R}_ + ,\rho )} \leqslant \tilde{C}_1 \left\Vert u \right\Vert _{L^M (\mathbb {R}_ + ,\rho )} + \tilde{C}_2 \left\Vert {u^{\prime }} \right\Vert _{L^M (\mathbb {R}_ + ,\rho )} ,\] with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.
LA - eng
KW - Hardy inequalities; Orlicz-Sobolev spaces; Nondoubling measures; nondoubling measures
UR - http://eudml.org/doc/268960
ER -

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