Biorthogonal bases and multiple interpolation in weighted Hardy spaces

Abstract

Let {z_k=x_k+iy_k} be a sequence on upper half plane$\mathbb{R}_ + ^2$ and {s_i} be the number of appearence of z_k in {z₁,z₂,...,z_k}. Suppose sup s_i<+∞. Let ω(x) be a weight belonging to A^∞ and$w_j = \smallint _{x_i }^{x_i + y_i } w(x)dx$. We Consider the weighted Hardy space$H_{ + w}^p H_{ + w}^p (\mathbb{R}_ + ^2 )$ and operator T_p mapping f(z)∈H ^p_+w into a sequence defined by$(T_p f)_j = w_j^{\tfrac{1}{p}} y_j^{s_j - 1} f^{(s_j - 1)} (z_k ),0< p \leqslant + \infty ,j = 1,2, \cdots$, 0<p≤+∞, j=1,2,.... Then T_p(H ^p_+w )=l^p if and only if {z_k} is uniformly separated. Besides the effective solution for interpolation is obtained.