Remarks on the critical Besov space and its embedding into weighted Besov–Orlicz spaces
Volume 201 / 2010
Abstract
We present several continuous embeddings of the critical Besov space $B^{{n}/{p},\rho}_p(\Bbb R^n)$. We first establish a Gagliardo-Nirenberg type estimate $$ \|u\|_{\dot B^{0,\nu}_{q,w_r}}\leq C_n\left(\frac{1}{n-r}\right)^{\frac{1}{q}+\frac{1}{\nu}-\frac{1}{\rho}} \left(\frac{q}{r}\right)^{\frac{1}{\nu}-\frac{1}{\rho}} \|u\|_{\dot B^{0,\rho}_p}^{\frac{(n-r)p}{nq}}\|u\|_{\dot B^{{n}/{p},\rho}_p}^{1-\frac{(n-r)p}{nq}}, $$ for $1< p\leq q< \infty$, $1\leq \nu< \rho\leq\infty$ 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 $B^{{n}/{p},\rho}_p(\Bbb R^n)\hookrightarrow B^{0,\nu}_{{\mit\Phi}_0,w_r}(\Bbb R^n)$, where the function ${\mit\Phi}_0$ of the weighted Besov–Orlicz space $B^{0,\nu}_{{\mit\Phi}_0,w_r}(\Bbb R^n)$ is a Young function of the exponential type. Another point of interest is to embed $B^{{n}/{p},\rho}_p(\Bbb R^n)$ into the weighted Besov space $B^{0,\rho}_{p,w_n}(\Bbb R^n)$ with the critical weight $w_n(x)={1}/{|x|^n}$; more precisely, we prove $B^{{n}/{p},\rho}_p(\Bbb R^n)\hookrightarrow B^{0,\rho}_{p,W_s}(\Bbb R^n)$ with the weight $W_s(x)=\frac{1}{|x|^n\left[\log(e+{1}/{|x|})\right]^s}$ for any $s>1$.
