Abstract:

Let $1 < p < \infty $, $\alpha>0$ and $\beta > -1$. Let $A^p_{\beta,\log^\alpha}$ denote the logarithmic weighted Bergman space of those functions $f$ which are analytic in the unit disk D such that

$\|f\|_{A_{\beta,\log^\alpha}^p}\overset{\rm def}{=} \left(\int_\mathbb{D}|f(z)|^p(1-|z|^2)^{\beta} \left(\log\frac{2}{1-|z|^2} \right)^\alpha{\rm d}A(z) \right)^{1/p}< \infty.$

This paper computes the lower and upper bounds for the norm of the Hilbert matrix operator $\mathcal{H}$ acting from the logarithmically weighted Bergman space $A_{p-2,\log^{\alpha}}^p$ to the Bergman space $A^p$ when $\alpha > p$ and $1 < p < 2$. We also compute norm estimates for the Hilbert matrix operator acting from the logarithmically weighted Bergman space $A^p_{\beta,\log^\alpha}$ to the weighted Bergman space $A^p_{\beta}$.

Key words: operator norm, Hilbert matrix operator, logarithmically weighted Bergman space