Characterizations of Besov and Triebel-Lizorkin spaces on metric measure spaces (Q2844273)

This paper considers Besov and Triebel-Lizorkin spaces on metric measure spaces \((X,d,\mu)\) with the doubling property. Let \(s\in(0,\infty)\) and \(p,q\in(0,\infty]\). Recall that the homogeneous Besov space \(\dot B^s_{p,q}(X)\) is defined as the set of all \(p\)-locally integrable functions \(u\) on \(X\) such that NEWLINE\[NEWLINE\|u\|_{\dot B^s_{p,q}(X)}=\left(\int_0^\infty \left(\int_X \frac{t^{-sp}}{\mu(B(x,t))}\int_{B(x,t)}|u(x)-u(y)|^p\,d\mu(y)d\mu(x)\right)^{\frac qp} \frac{dt}{t}\right)^{\frac 1q}NEWLINE\]NEWLINE is finite. The authors prove that the space \(\dot B^s_{p,q}(X)\) can be characterized by fractional Hajłasz gradients, that is, \(u\in \dot B^s_{p,q}(X)\) if and only if there exists a fractional \(s\)-Hajłasz gradient \(\vec{g}=\{g_k\}_{k\in\mathbb{Z}}\) of \(u\) such that \( \|\vec{g}\|_{\ell^q(L^p(X))}\) is finite. Here \(\vec{g}=\{g_k\}_{k\in\mathbb{Z}}\) is called a fractional \(s\)-Hajłasz gradient of \(u\) if there exists \(E\subset X\) with measure zero such that for all \(k\in\mathbb{Z}\) and \(x,y\in X\setminus E\) with \(2^{-k-1}\leq d(x,y) <2^{-k}\), NEWLINE\[NEWLINE|u(x)-u(y)|\leq d(x,y)^s [g_k(x)+g_k(y)].NEWLINE\]NEWLINE Applying this characterization, the authors further prove that, under different parameters, the quantity NEWLINE\[NEWLINE\left(\frac{t^{-sp}}{\mu(B(x,t))}\int_{B(x,t)}|u(x)-u(y)|^p\,d\mu(y)\right)^{1/p}NEWLINE\]NEWLINE in the definition of \(\|u\|_{\dot B^s_{p,q}(X)}\) can be equivalently replaced by one of the following four kinds of functions NEWLINE\[NEWLINEC^{s,\sigma}_t(u)(x)=t^{-s} \left(\frac1{\mu(B(x,t))}\int_{B(x,t)}|u(x)-u(y)|^\sigma\,d\mu(y)\right)^{1/\sigma},NEWLINE\]NEWLINE NEWLINE\[NEWLINEA^{s,\sigma}_t(u)(x)=t^{-s} \left(\frac1{\mu(B(x,t))}\int_{B(x,t)}|u(y)-u_{B(x,t)}|^\sigma\,d\mu(y)\right)^{1/\sigma},NEWLINE\]NEWLINE NEWLINE\[NEWLINEI^{s,\sigma}_t(u)(x)=t^{-s} \left(\inf_{c\in\mathbb{R}}\frac1{\mu(B(x,t))}\int_{B(x,t)}|u(y)-c|^\sigma \,d\mu(y)\right)^{1/\sigma}NEWLINE\]NEWLINE and NEWLINE\[NEWLINES^{s,\epsilon,\sigma}_t(u)(x)=t^{\epsilon-s} \sup_{r\in(0,t]}r^{-\epsilon} \left(\inf_{c\in\mathbb{R}}\frac1{\mu(B(x,r))}\int_{B(x,r)}|u(y)-c|^\sigma \,d\mu(y)\right)^{1/\sigma},NEWLINE\]NEWLINE where \(s\in[0,\infty)\), \(\epsilon\in[0,s]\), \(\sigma\in(0,\infty)\), and NEWLINE\[NEWLINEu_{B(x,t)}:=\frac1{\mu(B(x,t))}\int_{B(x,t)}u(y)\,d\mu(y).NEWLINE\]NEWLINE A similar result for Triebel-Lizorkin spaces is also obtained. As further applications of these characterizations, under a suitable Poincaré inequality condition, the authors also prove the triviality of Besov and Triebel-Lizorkin spaces for certain parameters. Some examples of nontrivial Besov and Triebel-Lizorkin spaces are also presented to show the necessity of the Poincaré inequality condition.