Spectrophotometric study of dehalogenation of benzyl halides with nickel in aprotic dipolar solvents in the presence of oxygen

Oxidative dissolution of nickel in the benzyl halogenide – dipolar aprotic solvent system in the presence of air oxygen proceeds with the formation of benzaldehyde, benzyl alcohol, 1,2-diphenylethane, and trace amounts of 4,4'-ditolyl. Spectrophotometry studies have revealed that the oxidation products are formed in the solution rather than at the nickel surface. It has been shown that the oxygen adsorbed at the nickel surface practically does not pass into the solution.     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

On imbedding of certain function classes in symmetric spaces

2— - , () — - (), (, ) (0< <)— - . , >1; ; > ; ₁(), ₂() — -;f(x) — —. , () — ( , X, L , , ( ) -.

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The author thanks Prof. M. K. Potapov for having proposed the problem and for the help he has given in the work.     

Об операторе Кальдерона

The Calderon type operator $$Sx(t) = \int\limits_0^\infty {x(s)d\mathop {\min }\limits_{i = 0,1} \{ \varphi _i (s)/\Psi _\iota (t)\} } $$ is investigated from the point of view of its bounded action in symmetric spaces of measurable functions on [0, ∞), whereφ _i(t) andΨ _i(t) are concave positive functions on [0, ∞). The following assertions are proved. Theorem 1. Let 1) \(\alpha _{\varphi _1 } > \beta (E) \geqq \alpha (E) > \beta _{\varphi _0 } ,\) , 2)either \(\beta _{\psi _0 }< 1\) or \(\beta _{\psi 1}< 1\) ,where \(\alpha _{\varphi _1 } \) and \(\beta _{\psi _1 } \) denote exponents of the functions φ _i and Ψ_i, i=0, 1,and α(E),β(E) are indices of the space E. Then the Calderon operator acts boundedly from E into E _δ,where δ(t) and χ(t) stand for measurable solutions of the equations $$\psi _i (\delta (t)) = \chi (t)\varphi _i (t),i = 0,1,$$ and $$||x||_{E_{\delta ,\chi } } = ||x^{**} (\delta (t))\chi (t)||_E .$$ Theorem 2.If the ratio φ ₀/φ(t)/φ₁(t) is non-increasing, Ψ_i(t) are semimultiplicative and \(\alpha _{\psi _i } \) (i=0, 1),then a necessary condition for the Calderon operator to act boundedly from E into E _{δ, χ} $$\beta _{\varphi _1 } \geqq \beta (E) \geqq \alpha (E) \geqq \alpha _{\varphi _0 } .$$