Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba

Agnieszka Kałamajska ; Katarzyna Pietruska-Pałuba

Colloquium Mathematicae, Tome 106 (2006), p. 93-107 / Harvested from The Polish Digital Mathematics Library

Résumé

We obtain interpolation inequalities for derivatives: $\int M_{q, α} (| \nabla f (x) |) d x \leq C [\int M_{p, β} (Φ ₁ (x, | f |, | \nabla^{(2)} f |)) d x + \int M_{r, γ} (Φ ₂ (x, | f |, | \nabla^{(2)} f |)) d x]$ , 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 ${| | \cdot | |}_{(s, κ)}$ is the Orlicz norm relative to the function $M_{s, κ} (t) = t^{s} {(l n (2 + t))}^{κ}$ . 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.

Publié le : 2006-01-01

Zbl 1091.26013

EUDML-ID : urn:eudml:doc:283948

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     title = {Gagliardo-Nirenberg inequalities in logarithmic spaces},
     journal = {Colloquium Mathematicae},
     volume = {106},
     year = {2006},
     pages = {93-107},
     zbl = {1091.26013},
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Agnieszka Kałamajska; Katarzyna Pietruska-Pałuba. Gagliardo-Nirenberg inequalities in logarithmic spaces. Colloquium Mathematicae, Tome 106 (2006) pp. 93-107. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-1-8/