Gagliardo-Nirenberg inequalities in logarithmic spaces

• Autorzy: Agnieszka Kałamajska , Katarzyna Pietruska-Pałuba
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 26D10: Inequalities involving derivatives and differential and integral operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2006
• Tom: 106
• Numer: 1
• Strony: 93-107

Daty

wydano

2006

Twórcy

autor

Agnieszka Kałamajska

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

autor

Katarzyna Pietruska-Pałuba

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Identyfikatory

DOI

10.4064/cm106-1-8