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 


$$ \int\_{\mathbb{R}^n} \left\lvert f(x) \right\rvert dx \le \alpha\_1 + ...+ \alpha\_n?$$



Interpolation between $L^1$ and $L^2$ spaces
Yes for $n = 1$, and no for $n \geq 2$. Indeed, if we had an inequality $||f||\_1 \leq C(\alpha\_1 + \dots + \alpha\_n)$ then we could set $f = \mathbb{1}\_{[0,K]^n}$ and get


$$
K^n \leq C n K^{\frac{n+1}{2}},
$$
and we get $n \leq 1$ by letting $K$ tend to infinity.


