In this note we introduce the notion of the local product on a sheet and associated space. As an application we prove under some special conditions the following inequalities \begin{align} 2\pi \frac{|\log(\langle \vec{a},\vec{b}\rangle)|}{(||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4})|\langle \vec{a},\vec{b}\rangle|}\bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s+3]{\sum \limits_{i=1}^{n}x^{4s+3}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq \bigg|\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\mathbf{e}\bigg(-i\frac{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}\bigg)dx_1dx_2\cdots dx_n\bigg|\nonumber \end{align} and \begin{align} \bigg|\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\mathbf{e}\bigg(i\frac{\sqrt[4s+3]{\sum \limits_{j=1}^{n}x^{4s+3}_j}}{||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4}}\bigg)dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq 2\pi \frac{|\langle \vec{a},\vec{b}\rangle|\times |\log(\langle \vec{a},\vec{b}\rangle)|}{(||\vec{a}||^{4s+4}+||\vec{b}||^{4s+4})}\bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s+3]{\sum \limits_{i=1}^{n}x^{4s+3}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \end{align} and \begin{align} \bigg |\int \limits_{|a_n|}^{|b_n|} \int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[4s]{\sum \limits_{i=1}^{n}x^{4s}_i}dx_1dx_2\cdots dx_n\bigg|\nonumber \\ \leq \frac{|\langle \vec{a},\vec{b}\rangle|}{2\pi |\log(\langle \vec{a},\vec{b}\rangle)|}\times (||\vec{a}||^{4s+1}+||\vec{b}||^{4s+1}) \times \bigg|\prod_{i=1}^{n}|b_i|-|a_i|\bigg|\nonumber \end{align}for all $s\in \mathbb{N}$, where $\langle,\rangle$ denotes the inner product and where $\mathbf{e}(q)=e^{2\pi iq}$.