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

100%

Wadade H.

Studia Mathematica

2010

tom 201

nr 3

227-251

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.

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

100%

Wadade H.

Studia Mathematica

2014

tom 223

nr 1

77-95

We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H^{n/p}_{p,q,λ₁,...,λₘ}(ℝⁿ)$ into the generalized Morrey space $ℳ_{Φ,r}(ℝⁿ)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H^{n/p + 1}_{p,q,λ₁,...,λₘ}(ℝⁿ)$. O'Neil's inequality and its reverse play an essential role in the proofs of the main theorems.

Ograniczanie wyników

2 Studia Mathematica

2 Wadade H.

1 2014

1 2010

Wyniki wyszukiwania

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

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces