Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes − k $\mathbb {R}^{n-k}$ where the weight is Muckenhoupt type. concentrate example α x = | $w_\alpha (x) {\big\vert x_n\big\vert }^\alpha$ when ≤ 1 $\big\vert \le 1$ ∈ $x\in \mathbb {R}^n$ and (x)=1$ otherwise, > $\alpha >-1$ . Here we use some refined atomic decomposition argument as well an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome description real interpolation space B 2 θ r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big )\big )}_{\theta ,r}$ 0 < ∞ $00$ sufficiently large, $0<\theta <1$ $0<r\le \infty$ Apart from case $1/r= (1-\theta )/{p_1}+ {\theta }/{p_2}$ question seems to be open for many years. Based our first result can now quickly solve this long-standing problem. benefit very recent finding Besoy, Cobos Triebel.