Estimates of capacities associated with Besov spaces

Abstract

Let X be the Besov space BL ^l_{p, θ} (ℝⁿ), 0<p<∞, 0<θ≤∞, 0<lp<n. Let$\overline {cap} ( \cdot , X)$ be the capacity associated with the space X (defined on subsets of ℝⁿ), and ϕ be a function defined on [0,1] such that ϕ(0)=0, ϕ(1)=1 and for some ε>0 the functions ϕ(t)t^−ε,$\tfrac{{t^{m - \varepsilon } }}{{\varphi (t)}}$ increase.

Definition

Let A⊂ℝⁿ, 0<β≤∞. Define

$$h_{\varphi ,\beta } (A) = \inf \left( {\sum\limits_{i = 0}^{ + \infty } {\left( {m_j \varphi (2^{ - i} } \right)^\beta } } \right)^{1/\beta } $$

where the infinum 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 to the set (2⁻ⁱ⁻¹, 2⁻ⁱ], i∈N₀.

Theorem 1

Let p≤1, θ=∞, and let the function ϕ(t)t^lp−n increase. Then the following conditions are 2quivalent;

• a) for any compact set K, K⊂ℝⁿ, if$\overline {cap} (K, X) = 0$, then h_ϕ,∞(K)=0;

• b)

  

Theorem 2

Let θ<1. Then for any set A the inequalities$c_1 \overline {cap} (A,X) \leqslant h_{t^{n - lp} ,\theta /p} (A) \leqslant c_2 \overline {cap} (A,X)$ hold. Bibliography:6 titles.