A new approach to function spaces on quasi-metric spaces (Q558709)

Let \(n\in\mathbb N\) and \(0<d<n\). A compact set \(\Gamma\subset\mathbb R^n\) is called a \(d\)-set if \(\mathcal H_\Gamma^d(B(\gamma,r))\sim r^d\), \(\gamma\in\Gamma\), \(0<r<\text{diam}\Gamma\), where \(B(\gamma,r)\) is the ball in \(\mathbb R^n\) centered at \(\gamma\) and of radius \(r\), and \(\mathcal H_\Gamma^d\) is the restriction of the Hausdorff measure \(\mathcal H^d\) in \(\mathbb R^n\) to \(\Gamma\). An abstract \(d\)-space \((X,\varrho,\mu)\) is a set \(X\) equipped with a quasi-metric \(\varrho\) and a Borel measure \(\mu\) such that \((X,\varrho)\) is complete and \(\mu(B(x,r))\sim r^d\), \(x\in X\), \(0<r\leq\text{diam}X\). There exists \(\varepsilon_0\in(0,1]\) such that, for any \(\varepsilon\in(0,\varepsilon_0)\), \(\varrho^\varepsilon\) is equivalent to a metric. The space \((X,\varrho^\varepsilon,\mu)\) is called the snowflaked version of \((X,\varrho,\mu)\). The author starts with adapting some known results about the Besov spaces on \(d\)-sets in \(\mathbb R^n\) and proving new ones about intrinsic atomic characterisations. Then he shows that for any \(d\)-space there is a number \(\varepsilon_0\in(0,1]\) such that for any \(\varepsilon\in(0,\varepsilon_0)\) there is a bi-Lipschitzian map \(H\) of \((X,\varrho^\varepsilon,\mu)\) onto \((\Gamma,\varrho_n,\mathcal H_\Gamma^{d/\varepsilon})\) where \(\varrho_n(x,y)\) is the usual Euclidean distance in \(\mathbb R^n\) and \((\Gamma,\varrho_n,\mathcal H_\Gamma^{d/\varepsilon})\) is a compact \((d/\varepsilon)\)-set in \(\mathbb R^n\). The map \(H\) is called a Euclidean chart of \((X,\varrho,\mu)\). The maps \(H\) are illustrated by an example when \(\Gamma\) is the Koch snowflake curve in \(\mathbb R^2\), \((X,\varrho,\mu)\) is the \(d\)-space with \(X=[0,1]\), \(\varrho(x,y)=| x-y| ^\varepsilon\) with \(\varepsilon=d^{-1}=\log3/\log4\), and \(\mu\) is the Lebesgue measure in \(\mathbb R^1\). The snowflaked transforms \(H\) are used to define Besov spaces \(B^s_p(X,\varrho,\mu;H)\), \(1<p<\infty\), \(s>0\), as \(B^{s/\varepsilon}_p(\Gamma,\varrho_n,\mathcal H^{d/\varepsilon}_\Gamma)\circ H\). The spaces \(B^s_p(X,\varrho,\mu;H)\) are characterized intrinsically by quarkonial and by atomic decompositions. The scale of the Besov spaces is extended by duality to \(s<0\) and by interpolation to \(s=0\). The author then describes a method (called the standard approach) to introduce the spaces \(B^s_p(X)\), \(1<p<\infty\), \(| s|<\varepsilon_0\), as the class of all \(f\in(\text{Lip}^\varepsilon(X))'\) with the norm \(\| f\mid B^s_p(X)\|=\left(\sum_{j=0}^\infty 2^{jsp}\| E_j f\mid L_p(X)\|^p\right)^{1/p}<\infty\), where \((E_jf)(x)=\int_X E_j(x,y)f(y)\mu(dy)\) with properly defined kernels \(E_j(x,y)\). He proves that \(B^s_p(X;H)=B^s_p(X)\), i.e., the spaces \(B^s_p(X;H)\) for \(| s|<s_0\), \(1<p<\infty\) do not depend on the Euclidean charts. Two applications concern compact embeddings \(B^{s_1}_{p_1}(X;H)\hookrightarrow B^{s_2}_{p_2}(X;H)\) and the corresponding entropy numbers, and the Riesz potentials on \(d\)-sets \(\Gamma\) and its eigenvalues. It should be noted that the paper contains an extensive list of references and the presented theory is contextualized in the state-of-the-art of the recent results.