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For functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from$L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belongingof their Fourier transform $\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well asof a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geqt^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$ to $L_{1}(\mathbb{R}^{1}_{+})$. Asfor the positivity of $\widehat{f}$ on $\mathbb{R}^{2}$, it is completelyreduced to the same question on $\mathbb{R}^{1}$ for a function$f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.

On the Fourier transform of function of two variables which depend only on the maximum of these variables