On differentiability properties of the symbol of a multidimensional singular integral operator

Let $f$ be the characteristic and $\Phi$ the symbol of $n$-dimensional singular integral operator, let $\delta$ be the Beltrami operator on the sphere $S^{n-1}$ of the space $\mathbf R^n$, and let $H^l_p(S^{n-1})$ be the space of Bessel potentials on this sphere with norm
$$ \|g\|_{H^l_p(S^{n-1})}=\|(E+\delta)^{l/2}g\|_{L_p(S^{n-1})}, $$
where $E$ is the identity operator.
The differentiability properties of the symbol in the spaces $H^l_p(S^{n-1})$ were studied earlier in the case $p=2$.
In this paper it is proved that in the case $p\in(1, \infty)$, $p\ne2$, the following assertions hold:
a) If $f\in L_p(S^{n-1})$, then $\Phi\in H^\alpha_p(S^{n-1})$, $\alpha<\frac n2-|\frac 1p-\frac 12|(n-2)$, while this assertion fails to hold for any $\alpha>\frac n2-|\frac 1p-\frac 12|(n-2)$.
b) If $\Phi\in H^\nu_p(S^{n-1})$, where $\nu>\frac n2+|\frac 1p-\frac 12|(n-2)$, then $f\in L_p(S^{n-1})$, while this assertion fails to hold for any $\nu<\frac n2+|\frac 1p-\frac 12|(n-2)$.
From these results it follows that for the range $R(\Phi)$ of the symbol $\Phi$ with characteristic $f\in L_p(S^{n-1})$ the inclusions $H^\nu_p\subset R(\Phi)\subset H^\alpha_p$ hold, and, in contrast to the case $p=2$, a more precise description of $R(\Phi)$ in terms of the spaces $H^l_p(S^{n-1})$ is not possible.
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