Geodesic
On translation and dilation invariant subspaces of $L^p(\mathbb R^n)$, $0$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 35, Tome 345 (2007), pp. 5-24

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We prove that each translation and dilation invariant subspace $X\subset L^p(\mathbb R^n)$, $X\ne L^p(\mathbb R^n)$, is contained in a maximal translation and dilation invariant subspace of $L^p(\mathbb R^n)$. Moreover, we prove that the set of all maximal translation and dilation invariant subspaces of $L^p(\mathbb R^n)$ has the power of the continuum. 
@article{ZNSL_2007_345_a0,
     author = {A. B. Aleksandrov},
     title = {On translation and dilation invariant subspaces of $L^p(\mathbb R^n)$, $0<p<1$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--24},
     publisher = {mathdoc},
     volume = {345},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_345_a0/}
}
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A. B. Aleksandrov. On translation and dilation invariant subspaces of $L^p(\mathbb R^n)$, $0