Geodesic
Bilinear embedding theorems for differential operators in~$\mathbb R^2$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 210-234

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We prove bilinear inequalities for differential operators in $\mathbb R^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However, here we study the phenomenon in itself. We consider elliptic case, where our analysis is complete, and non-elliptic, where it is not. The latter case is related to Strichartz estimates in a very easy case of two dimensions. 
@article{ZNSL_2014_424_a8,
     author = {D. M. Stolyarov},
     title = {Bilinear embedding theorems for differential operators in~$\mathbb R^2$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {210--234},
     publisher = {mathdoc},
     volume = {424},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a8/}
}
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D. M. Stolyarov. Bilinear embedding theorems for differential operators in~$\mathbb R^2$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 210-234. http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a8/