Compute of two norms of a function of three variables

Question

Let $f$ be a function defined on $\mathbb R^3$ by $$f(x,y,z)=\exp(-2\mathbb i\pi (x+y+z)) |x|^{1-k} |y|^{k-1} \operatorname{sign}(x) \operatorname{sign}(z),$$ where $sign(x)$ means the sign of $x$ and $k \geq 2$ is an even integer. Suppose that the product $x y z \in [a,b]$ $(0<a<b).$ What are bounds of norms of spaces $L^{1}$ and $L^2$ of $f$ in terms of $a$ and $b$? Recall that $$\parallel f \parallel_{1}=\int_{\mathbb R^3} |f(x,y,z)|dxdydz$$ and $$\parallel f \parallel_{2}=\left[\int_{\mathbb R^3} |f(x,y,z)|^2 dxdydz\right]^{1/2}.$$ Thanks in advance.