Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces

Hidemitsu Wadade

Studia Mathematica, Tome 196 (2010), p. 227-251 / Harvested from The Polish Digital Mathematics Library

Résumé

We present several continuous embeddings of the critical Besov space $B^{n/p,ρ}_{p}(ℝⁿ)$. We first establish a Gagliardo-Nirenberg type estimate $||u||_{Ḃ^{0,ν}_{q,w_r}} ≤ Cₙ(1/(n-r))^{1/q + 1/ν - 1/ρ} (q/r)^{1/ν - 1/ρ} ||u||_{Ḃ^{0,ρ}_{p}}^{(n-r)p/nq}| |u||_{Ḃ^{n/p,ρ}_{p}}^{1-(n-r)p/nq}$, for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function $w_{r}(x) = 1/(|x|^{r})$ with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding $B^{n/p,ρ}_{p}(ℝⁿ) ↪ B^{0,ν}_{Φ₀,w_{r}}(ℝⁿ)$, where the function Φ₀ of the weighted Besov-Orlicz space $B^{0,ν}_{Φ₀,w_{r}}(ℝⁿ)$ is a Young function of the exponential type. Another point of interest is to embed $B^{n/p,ρ}_{p}(ℝⁿ)$ into the weighted Besov space $B^{0,ρ}_{p,wₙ}(ℝⁿ)$ with the critical weight wₙ(x) = 1/|x|ⁿ; more precisely, we prove $B^{n/p,ρ}_{p}(ℝⁿ) ↪ B^{0,ρ}_{p,W_{s}}(ℝⁿ)$ with the weight $W_{s}(x) = 1/(|x|ⁿ[log(e+1/|x|)]^{s})$ for any s > 1.

Détail

Publié le : 2010-01-01

Zbl 1216.46034

EUDML-ID : urn:eudml:doc:285857