# zbMATH — the first resource for mathematics

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

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Full Text: