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

Hidemitsu Wadade¹

¹ Advanced Mathematical Institute Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku 558-8585 Osaka, Japan

Studia Mathematica, Tome 201 (2010) no. 3, pp. 227-251

Résumé

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 $s>1$.

Détail

DOI : 10.4064/sm201-3-2

Keywords: present several continuous embeddings critical besov space rho bbb first establish gagliardo nirenberg type estimate dot leq frac n r right frac frac frac rho frac right frac frac rho dot rho frac n r dot rho frac n r leq infty leq rho leq infty weight function prove corresponding trudinger type estimate obtain terms embedding rho bbb hookrightarrow mit phi bbb where function mit phi weighted besov orlicz space mit phi bbb young function exponential type another point interest embed rho bbb weighted besov space rho bbb critical weight precisely prove rho bbb hookrightarrow rho bbb weight frac log right