Estimates of capacities associated with Besov spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 20, Tome 201 (1992), pp. 124-156
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Let $X$ be the Besov space $BL_{p,\theta}^l(\mathrm{R}^n)$, $0, $0<\theta\leqslant\infty$, $0. Let $\overline{\mathrm{cap}}(\cdot,X)$ be the capacity associated with the space $X$ (defined on subsets of $\mathrm{R}^n$) and $\varphi$ be a function defined on $[0,1]$ such that $\varphi(0)=0$, $\varphi(1)=1$ and for some $\varepsilon>0$ the functions $\varphi(t)t^{-\varepsilon}$, $t^{n-\varepsilon}/\varphi(t)$ increase. DEFINITON. Let $A\subset\mathrm{R}^n$, $0<\beta<\infty$. Define $$ h_{\varphi,\beta}(A)=\inf\left(\sum_{i=0}^{+\infty}(m_i\omega(2^{-i}))^\beta\right)^{1/\beta}, $$ where the infimum is taken over all coverings of $A$ by a countable number of balls, whose radii $r_j$ do not exceed 1, while $m_i$ is the number of balls from this covering whose radii $r_j$ belong the set $(2^{-i-1},2^{-i}]$, $i\in N_0$. THEOREM 1. Let $p\leqslant1$, $\theta=\infty$, and let the function $\varphi(t)t^{lp-n}$ increase. Then the following condition are equivalent: a) for any compact set $K$, $K\subset\mathrm{R}^n$, if $\overline{\mathrm{cap}}(K, X)=0$ then $h_{\varphi,\infty}(K)=0$. b) $\sum\limits_{i=0}^\infty(\varphi(1/x_i)x_i^{n-lp})^{1/p}<+\infty$, $x_0=1$, $x_{i+1}=2^{x_i}$, $i\in N_0$. THEOREM 2. Let $\theta<1$. Then for any set $A$ the inequalities $c_1\overline{\mathrm{cap}}(A,X)\leqslant h_{t^{n-lp},\theta/p}(A)\leqslant c_2\overline{\mathrm{cap}}(A,X)$ hold.