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

Blanca F. Besoy, Dorothee D. Haroske and Hans Triebel

Mathematische Nachrichten, 2022, vol. 295, issue 9, 1669-1689

Abstract: We study traces of weighted Triebel–Lizorkin spaces Fp,qs(Rn,w)$F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes Rn−k$\mathbb {R}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight wα(x)=|xn|α$w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when |xn|≤1$\big\vert x_n\big\vert \le 1$, x∈Rn$x\in \mathbb {R}^n$, and wα(x)=1$w_\alpha (x)=1$ otherwise, where α>−1$\alpha >-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space (Bp1,p1s1(Rn−k),Bp2,p2s2(Rn−k))θ,r$\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$, 0 0$s>0$ sufficiently large, 0

Date: 2022
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