Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|\_{L^p}$ into an $L^2$ sum of $\|f\_\theta\|\_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where $\theta$ ranges over $\delta^{1/2} \times \cdots \times \delta^{1/2} \times \delta$-sized slab neighborhoods of $S$, and where $f\_\theta$ is the restriction of $f$ to frequences in $\theta$. I'm wondering if we can replace the $L^p$ norm here with the $L^p\_x L^q\_t$ norm for $p\neq q$. Better still, if we can let $p=\infty>q$ and get any bound that beats the trivial triangle inequality bound?



Decoupling in mixed norm spaces
For hypersurfaces there is a decent chance that the methods can be adapted to handle $L^q\_t L^p\_x$ type norms, but not $L^p\_x L^q\_t$ type norms. This is because the parabolic rescalings used in the Bourgain-Demeter argument (see Proposition 4.1 of <http://arxiv.org/pdf/1403.5335.pdf> ) react well with the former type of norm (after writing the parabolic rescaling in physical space rather than frequency space, where they become something like a generalised Galilean transform) but not the latter.


For higher codimension surfaces, such as curves in three and higher dimensions, the situation looks to be more complicated, as the affine change of variables used (see e.g. Lemma 7.2 of <http://arxiv.org/pdf/1512.01565.pdf>) will not interact well with either of the above norms. However, this may only be a technical obstacle rather than a genuine one, as there may be a way to run the Bourgain-Demeter type arguments without appealing to any of these change of variables. For instance, Lee and Vargas in 


*Sanghyuk Lee and Ana Vargas*, [**Sharp null form estimates for the wave equation**](http://dx.doi.org/10.1353/ajm.0.0024), *Amer. J. Math.* **130** (2008), no. 5, 1279--1326. 


were able to get good bilinear null form estimates in mixed norm spaces despite the fact that Lorentz transforms did not interact well with these norms.


