The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function $$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$ where as usual $\widehat{x_j}$ stands for the point $(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_d)$. Then the GI is $$\int_{{\mathbb R}^d}f(x)\,dx\le\prod_{j=1}^d\|f_j\|_{L^{d-1}({\mathbb R}^{d-1})}.$$ (Remark the nice constant $\bf1$).
I suspect that the following natural extension is true. Let $k$ be a integer, $1\le k\le d-1$. For every subset $I\subset[\![1,d]\!]$ of cardinal $k$, let $f_I$ be a non-negative measurable function over ${\mathbb R}^k$. Denote $x_I=(x_i)_{i\in I}$ and form the function $$f(x)=\prod_{|I|=k}f_I(x_I).$$
Is it true that $$\int_{{\mathbb R}^d}f(x)\,dx\le\prod_{|I|=k}\|f_I\|_{L^q({\mathbb R}^k)},$$ where $q=\binom{d-1}{k-1}$ ?
The exponent $q$ above is determined by scaling invariance. The case $k=d-1$ is GI. The case $k=1$ is just Fubini, and then the equality holds true. The case $k=d$ is a tautology.
