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

Let $\varphi$ be a unimodular function on the unit circle $\mathbb{T}$ and let $K_p(\varphi)$ denote the kernel of the Toeplitz operator $T_\varphi$ in the Hardy space $H^p$, $p\geqslant1: K_p(\varphi)\stackrel{\mathrm{def}}{=}\{f\in H^p: T_\varphi f=0\}$. Suppose $K_p(\varphi)\ne\{0\}$. The problem is to find out how the smoothness of the symbol $\varphi$ influences the boundary smoothness of functions in $K_p(\varphi)$. One of the main results is as follows.
THEOREM 1. Let $1<p$, $q<+\infty$, $1<r\leqslant+\infty$, $q^{-1}=p^{-1}+r^{-1}$. Suppose $||\varphi||\equiv1$ on $\mathbb{T}$ and $\varphi\in W_r^1$ (i.e. $\varphi'\in L^r(\mathbb{T})$). Then $K_p(\varphi)\subset W_q^1$. Moreover, for any $\varphi\in K_p(\varphi)$ we have $||f'||_q\leqslant c(p,r)||\varphi'||_r||f||_p$.