On a generalization of an inequality of Bohr

Let $p\in (1, 2], n\ge 1, S\subseteq R^{n}$ and $\Gamma(S, p)$— the set of all functions, $\gamma(t)\in L ^{p}(R ^{n})$ the support of the Fourier transform of which lies in $S$. We obtain the inequality conditions $||\int \limits_{E_t}\gamma(\tau)d\tau|| _{L ^{\infty}(R^n)}\le C||\gamma(\tau)|| _{L ^{p}(R^n)}$, where $t=(t _{1}, t _{2}, \dots , t _{n})\in R^{n}, E _{t} = \{\tau|\tau=(\tau _{1},\tau _{2},\dots ,\tau _{n})\in R^{n}, \tau_j\in [0,t_j]$, if $t_j\ge 0$ and $\tau_{j}\in (t_j,0]$, if $\tau_{j}< 0, 1\le j\le n\}, \gamma(\tau)\in \Gamma(S,p)$ and constant $C$ does not depend on $\gamma(t)$. Also were considered some validity conditions on the inequality on non-trivial subsets $\Gamma(S, p)$ in cases, where they were not satisfied on the whole $\Gamma(S, p)$.