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

• Autorzy: Hidemitsu Wadade
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26D15: Inequalities for sums, series and integrals
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 201
• Numer: 3
• Strony: 227-251

Daty

wydano

2010

Twórcy

autor

Hidemitsu Wadade

• Advanced Mathematical Institute, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, 558-8585 Osaka, Japan

Identyfikatory

DOI

10.4064/sm201-3-2