Kernels of Toeplitz operators, smooth functions, and Bernstein type inequalities

Abstract

Let ϕ be a unimodular function on the unit circle\(\mathbb{T}\) and let Kp(ϕ) denote the kernel of the Toeplitz operator Tϕ in the Hardy space Hp, p≥1;\(K_p (\varphi )\mathop = \limits^{def} \{ f \in H^p :T_\varphi f = 0\} \). Suppose Kp(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in Kp(ϕ). One of the main results is as follows.Theorem 1Let 1<p, q<+∞, 1<r≤+∞, q−1=p−1+r−1. Suppose |ϕ|≡1 on\(\mathbb{T}\) and ϕ∈W 1r (i.e.,\(\varphi ' \in L^r (\mathbb{T})\)). Then Kp(ϕ)⊂W 1q . Moreover, for any f∈Kp(ϕ) we have ‖f′‖q≤c(p, r)‖ϕ′‖r ‖f‖. Bibliography: 19 titles.