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

Hidemitsu Wadade. "Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces." Studia Mathematica 201.3 (2010): 227-251. <http://eudml.org/doc/285857>.

@article{HidemitsuWadade2010,
abstract = {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.},
author = {Hidemitsu Wadade},
journal = {Studia Mathematica},
keywords = {embeddings; Besov space; Gagliardo-Nirenberg type estimate; Trudinger type estimate; weighted Besov-Orlicz space},
language = {eng},
number = {3},
pages = {227-251},
title = {Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces},
url = {http://eudml.org/doc/285857},
volume = {201},
year = {2010},
}

TY - JOUR
AU - Hidemitsu Wadade
TI - Remarks on the critical Besov space and its embedding into weighted Besov-Orlicz spaces
JO - Studia Mathematica
PY - 2010
VL - 201
IS - 3
SP - 227
EP - 251
AB - 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.
LA - eng
KW - embeddings; Besov space; Gagliardo-Nirenberg type estimate; Trudinger type estimate; weighted Besov-Orlicz space
UR - http://eudml.org/doc/285857
ER -