# Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba

Colloquium Mathematicae (2006)

- Volume: 106, Issue: 1, page 93-107
- ISSN: 0010-1354

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topAgnieszka Kałamajska, and Katarzyna Pietruska-Pałuba. "Gagliardo-Nirenberg inequalities in logarithmic spaces." Colloquium Mathematicae 106.1 (2006): 93-107. <http://eudml.org/doc/283948>.

@article{AgnieszkaKałamajska2006,

abstract = {We obtain interpolation inequalities for derivatives:
$∫ M_\{q,α\}(|∇f(x)|)dx ≤ C[∫M_\{p,β\}(Φ₁(x,|f|,|∇^\{(2)\}f|))dx + ∫M_\{r,γ\}(Φ₂(x,|f|,|∇^\{(2)\}f|))dx]$,
and their counterparts expressed in Orlicz norms:
||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)$,
$where $||·||_\{(s,κ)\}$ is the Orlicz norm relative to the function $M_\{s,κ\}(t) = t^\{s\}(ln(2+t))^\{κ\}$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher order gradients are also considered.},

author = {Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba},

journal = {Colloquium Mathematicae},

keywords = {Gagliardo-Nirenberg inequalities; logarithmic Orlicz spaces; Carathéodory functions},

language = {eng},

number = {1},

pages = {93-107},

title = {Gagliardo-Nirenberg inequalities in logarithmic spaces},

url = {http://eudml.org/doc/283948},

volume = {106},

year = {2006},

}

TY - JOUR

AU - Agnieszka Kałamajska

AU - Katarzyna Pietruska-Pałuba

TI - Gagliardo-Nirenberg inequalities in logarithmic spaces

JO - Colloquium Mathematicae

PY - 2006

VL - 106

IS - 1

SP - 93

EP - 107

AB - We obtain interpolation inequalities for derivatives:
$∫ M_{q,α}(|∇f(x)|)dx ≤ C[∫M_{p,β}(Φ₁(x,|f|,|∇^{(2)}f|))dx + ∫M_{r,γ}(Φ₂(x,|f|,|∇^{(2)}f|))dx]$,
and their counterparts expressed in Orlicz norms:
||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ)$,
$where $||·||_{(s,κ)}$ is the Orlicz norm relative to the function $M_{s,κ}(t) = t^{s}(ln(2+t))^{κ}$. The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher order gradients are also considered.

LA - eng

KW - Gagliardo-Nirenberg inequalities; logarithmic Orlicz spaces; Carathéodory functions

UR - http://eudml.org/doc/283948

ER -

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