Width of the Hardy classes \(H^ p\) in weighted \(L_ q\) spaces
Let \(X,Y\) be normed linear spaces, \(x \in X \Rightarrow x \in Y(\| x \|_ X \neq \| x \|_ Y\), generally), \(B\) be the unit ball in \(X\), \(A \in L(X \to Y)\), \[ \begin{aligned} \lambda_ n(X,Y) & =\inf_ A \sup_{z \in B} \| z-Az \|_ Y,\dim AX \leq n,\\ d^ n(X,Y) & = \inf_{Y_ n} \sup_{z \in B} \inf_{W \in Y_ n} \| z-W \|_ Y,\;Y_ n \subset Y,\dim Y_ n \leq n,\\ d_ n(X,Y) & =\inf_{Y_{- n}} \sup_{z \in B \cap Y_{-n}} \| W \|_ Y,Y_{-n} \subset Y,\text{ codim } Y_{-n} \leq n.\end{aligned} \] The values \(\lambda_ n(X,Y)\), \(d^ n(X,Y)\), \(d_ n(X,Y)\) are called linear, Kolmogorov, Gel'fand widths, respectively, of the ball \(B\) in the metric of the space \(Y\). The widths have applications in the theory of approximation, theory of interpolation of linear operators, another questions. The width was calculated for some pairs of spaces. Let \(H^ p\) be the Hardy space in the unit circle, \(T_ r=\{z:| z |=r\}\), \(0<r<1\), \(L_ p (T_ r, \mu)\) be the Lebesgue space respecting finite Borel measure \(\mu\). The author proves that \[ \begin{multlined} \lambda_ n (H^ p,L_ q \bigl( T_ r,\mu) \bigr)=d^ n \bigl( H^ p,L_ q (T_ r,\mu) \bigr)=d_ n \bigl( H^ p,L_ q (T_ r,\mu) \bigr)=\\ G^{1/q} (\mu)r^ n,G (\mu)=\exp S \left( \ln {d \mu \over dm} \right) dm,\end{multlined} \] where \(m\) is the normalized Lebesgue measure on \(T_ r\), \(1 \leq q \leq p \leq \infty\), \[ \begin{multlined} \varlimsup_{n \to \infty} \lambda_ k \bigl( H^ p,L_ q (T_ r,\mu) \bigr) r^{-n} \leq G^{1/q} (\mu) (1-r^ s)^{- {1 \over s}},\;1 \leq p \leq 2 \leq q \leq \infty, \\ {1 \over p}={1 \over q} +{1 \over s},\;\mu=\varphi dm,\;\varphi \in C(T_ r).\end{multlined} \] {}.
scientific article; zbMATH DE number 484294