Analog of an inequality of Bohr for integrals of functions from ${L^{p}}(R^{n})$. I

Let $p\in(2,+\infty],$ $n\ge1$ and $\Delta=(\Delta_1,\ldots,\Delta_n),$ $\Delta_k>0,$ $1\le k\le n.$ It is proved that for functions $\gamma(t)\in L^p(R^n)$ spectrum of which is separated from each of $n$ the coordinate hyperplanes on the distance not less than $\Delta_k,$ $1\le k\le n$ respectively, the inequality is valid:
$$ \left\|\int\limits_{E_t}\gamma(\tau)\,d\tau\right\| _{L^{\infty}(R^n)}\le C^n(q)\left[\prod_{k=1}^n\frac{1} {\Delta_k^{1/q}}\right]\left\|\gamma(\tau)\right\|_{L^p(R^n)}, $$
where $t=(t_1,\ldots,t_n)\in R^n,$ $E_t=\{\tau\,|\,\tau=(\tau_1,\ldots,\tau_n)\in R^n,$ $\tau_j\in[0,t_j],$ if $t_j\ge0,$ and $\tau_j\in[t_j,0],$ if $t_j<0,\ 1\le j\le n\},$ and the constant $C(q)>0,$ $\displaystyle\frac{1}{p}+ \frac{1}{q}=1$ does not depend on $\gamma(\tau)$ and vector $\Delta.$