I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:

Let $f \in \mathbb{R}^n$ be such that

$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \right\rVert_{L^2(\mathbb{R}^{n-1})} dx_1$$

up to

$$ \alpha_n:= \int_{\mathbb{R}} \left\lVert f(\cdot,\cdot,....,x_n) \right\rVert_{L^2(\mathbb{R}^{n-1})} dx_n$$ are finite.

Does this give us an upper bound for the $L^1$ norm of $f$ in the sense that

$$ \int_{\mathbb{R}^n} \left\lvert f(x) \right\rvert dx \le \alpha_1 + ...+ \alpha_n?$$