Geodesic
Kernels of Toeplitz operators, smooth functions, and Bernstein type inequalities
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 20, Tome 201 (1992), pp. 5-21 
K. M. D'yakonov. Kernels of Toeplitz operators, smooth functions, and Bernstein type inequalities. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 20, Tome 201 (1992), pp. 5-21. http://geodesic.mathdoc.fr/item/ZNSL_1992_201_a0/
@article{ZNSL_1992_201_a0,
     author = {K. M. D'yakonov},
     title = {Kernels of {Toeplitz} operators, smooth functions, and {Bernstein} type inequalities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--21},
     year = {1992},
     volume = {201},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1992_201_a0/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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, $q<+\infty$, $1, $q^{-1}=p^{-1}+r^{-1}$. 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 $||f'||_q\leqslant c(p,r)||\varphi'||_r||f||_p$.