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Theorems of imbedding Besov spaces into ideal spaces. (Russian) Zbl 0636.46036
We say a continuous and increasing function \(\phi(t)\), \(0\leq t<\infty\), \(\phi (0)=0\), belongs to the class S (resp. \(S_ k)\) if \(\phi (t)/t^{\epsilon}\) (resp. \(t^{k-\epsilon}/\phi(t))\) is increasing a.e.. \(\Omega_ k\) is the class of continuous and strictly increasing functions \(\omega\) (t), \(0\leq t\leq 1\), such that \(\omega(0)=0\), \(\omega(1)=1\) and \(t^ k/\omega(t)\) is increasing. We write \(T_{[a,b],\phi,\theta}(g)= (\int^{b}_{a}(g(u)/\phi (u))^{\theta} d\phi (u)/\phi (\) \(u))^{1/\theta}\), \(1\leq \theta \leq \infty\), in particular, \(T_{[0,1],\phi,\theta}=T_{\phi,\theta}\). Given a certain symmetric Banach space E of functions and given a function \(\omega \in S_ k\), \(B^{\omega}_{E,\theta}\) is the space of functions f(x), \(x\in R^ n\), with norm \(\| f\|_{B^{\omega}_{E,\theta}}=\| f\|_ E+T_{\omega,\theta}(\omega^ k_ E(u)f)\), \(\omega^ k_ E(u)f\) denoting the k-th order modulus of continuity of f in the metric of E. \(S(B^{\omega}_{E,\theta})\) (resp. \(S^{\#}(B^{\omega}_{E,\theta}))\) is the set of functions g(t) such that \(g^*(t)\leq f^*(t)\) (resp. \(g^*(t)\leq (f^{\#})^*(t))\) for some f in the unit ball of \(B^{\omega}_{E,\theta}\), where \(f^*\) is the non-increasing rearrangement of \(| f|\), \(f^{\#}(x)=\sup_{x\in Q}\int_{Q}| f(y)-f_ Q| dy/| Q|\) and \(f_ Q\) is the mean of f on the cube Q. Let \(\psi_{\theta}(t)= 1/(1+T_{[t,1],\psi,\theta'} (\omega(^ n\sqrt{u})))\), \(1/\theta +1/\theta'=1\), \(\{t_ i\}\) with \(\omega(t_ i)=2^{-i+1}\) and let \(\{\nu_{\theta}(i)\}\), \(\nu_{\theta}(1)=1\), be inductively defined ad \(\nu_{\theta}(i+1)=t_{r-1}\) (resp. \(=t_ r)\) if \(t_{r-1}<\nu_{\theta}(i)\) (resp. \(t_{r-1}=\nu_{\theta}(i))\), r being the least number \(\psi_{\theta}(t^ n_ r)<\psi_{\theta}((\nu_{\theta}(i))\) n). \(E(\psi_{\theta}, \{\nu_{\theta}(i)\},\theta)\) is the symmetric space of functions f(x) with norm \(\sum_{i}(\sup \{(f_ E^{**}(t)\psi_{\theta}(t))^{\theta}\); \(\nu^ n_{\theta}(i+1)\leq t\leq \nu\) \(n_{\theta}(i)\})^{1/\theta}\), where \(f_ E^{**}(t)=\| \chi_{[0,t]}(u)f\) \(*(u)\|_ E/\psi_ E(t)\), \(\psi_ E(t)=\| \chi_{[0,t]}(u)\|_ E\), \(\chi_{[0,t]}(u)\) being the characteristic function of the interval [0,t]. Now the author says \(S(B^{\omega}_{E,\theta})\) is equivalent to the unit ball of \(E(\psi_{\theta},\{\nu_{\theta}(i)\},\theta)\) and if \(\omega\in S\) or if the maximal operator of Hardy-Littlewood for cubes is allowed in E, then \(S^{\#}(B^{\omega}_{E,\theta})\) is equivalent to the unit ball of \(E(\psi_ 1,\{\nu_ 1(i)\},\theta)\) (Theorem 1). There are given numbers of propositions concerning the imbedding problems. But the present reviewer can hardly follow such involved details further.
Reviewer: K.Yoshinaga

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: EuDML