Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka 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 -