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## Sobolev Norms on $SL(d,\mathbb Z)\backslash SL(d,\mathbb R)$

Assume $\theta: \mathbb R^d \rightarrow \mathbb R$ is a smooth function with compact support and assume you know bounds for all $(2,l)$-Sobolev norms $\Vert f \Vert_{2,l}$ and bounds for all $C^l$-norms $\Vert f \Vert_{C^l}$ of $f$, $l=0,1,2,\dots$ (by $\Vert f \Vert_{2,0}$ we shall denote the $L^2$ norm.). Basically assume you know a lot about the function $\theta$.

Define the function $F: SL(d,\mathbb Z)\backslash SL(d,\mathbb R)\rightarrow \mathbb R$ to be given by $$M\mapsto \sum_{\mathbf k\in \mathbb Z^d} \theta(\mathbf k \cdot M).$$

Then I was wondering, what can you say about the $(2,l)$-Sobolev norms of $F$ for $l\geq 0$? What can I do to better understand Sobolev-norms specifically on $SL(d,\mathbb Z)\backslash SL(d,\mathbb R)$.

Thanks!

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 In your first paragraph do you mean $f$ to be $\theta$? Presuming so, it is nevertheless quite unclear what you intend by "Sobolev norms" of a function on $SL(d,\mathbb Z)\backslash SL(d,\mathbb R)$. Perhaps a clear and appropriate characterization of those Sobolev spaces is an underlying issue? Clarify? – paul garrett Nov 15 at 19:13