Characterizations of \(B^s_{p,q}(G),L^s_{p,q}(G),W^s_p(G)\) and certain other function spaces. Applications (Q2714726)

The author proves two fundamental theorems.NEWLINENEWLINENEWLINETheorem A. Let \(G\subset \mathbb{R}^m\) be a set satisfying the following conditions: for some \(\delta_0\in (0,1]\), \(T\in (0,\infty)\) and for any \(x\in G\) there exists a path NEWLINE\[NEWLINE\rho(t^\lambda):= (\rho_1(t^{\lambda_1}),\dots, \rho_n(t^{\lambda_n}))= \rho(t, x),NEWLINE\]NEWLINE where \(t\in [0,T]\) and \(\lambda\in (0,\infty)^n\), \(|\lambda|= n\); moreover, the functions \(\rho\) and \(\rho_i\) satisfying two conditions:NEWLINENEWLINENEWLINEa) \(\rho_i\) is absolutely continuous in \([0, T^{\lambda_i}]\), \(i\in \{1,\dots, n\}\), \(|\rho_i'|\leq 1\), a.s. in \([0, T^{\lambda_i}]\);NEWLINENEWLINENEWLINEb) \(\rho(0)= 0\), NEWLINE\[NEWLINEx+ V(\lambda, x,\delta_0)= x+\bigcup_{t\in (0,T]} [\rho(t^\lambda)+ t^\lambda \delta^\lambda_0 Q_0],\;x+ V(\lambda, x,\rho_0)\subset G,NEWLINE\]NEWLINE \(s_*> 0\), \(s= s_*/\lambda\in (0,\infty)^n\), \(a\in [1,\infty]\), \(p\in [1,\infty]^n\), \(q\in [1,\infty]\).NEWLINENEWLINENEWLINEThen the following holdNEWLINENEWLINENEWLINE1) \(\widetilde B^s_{p,q,a}(G)= B^s_{p,q}(G)=\widetilde B^s_{p,q,a}(G)^*\) for \((1/s,1/a')< 1((\lambda, 1/a')< s_*)\);NEWLINENEWLINENEWLINE2) \(B^s_{p,q,a}(G)= B^s_{p,q}(G)\) for \(1/a'< s\).NEWLINENEWLINENEWLINETheorem B. Let \(G\subset \mathbb{R}^n\) be a set satisfying the assumption of Theorem A, \(s_*> 0\), \(s= s_*/2\in (0,\infty)^n\), \(a\in [1,\infty]\), \(p\in (1,\infty)^n\), \(q\in (1,\infty)\) or \(p\in [1,\infty]\), \(q=\infty\). Then the following holdNEWLINENEWLINENEWLINE1) \(\widetilde L^s_{p,q,a}(G)= L^s_{p,q}(G)= \widetilde L^s_{p,q,a}(G)\) for \((1/s, 1/a')< 1((\lambda, 1/a')< s_*)\);NEWLINENEWLINENEWLINE2) \(L^s_{p,q,a}(G)= L^s_{p,q}(G)\) for \(1/a'< s\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].