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Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as $$ \mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\mathbb{R}^2_+)\,|\quad \|f\|_{klmn}<\infty,\quad k,l,m,n=0,1,\ldots\right\}, $$ with seminorms $$ \|f\|_{klmn}=\sup\limits_{(x,y)\in\mathbb{R}^2_+}\left|x^m\left(y+\frac1y\right)^n\partial_x^k\partial_y^lf(x,y)\right|. $$ Through extension by zero this space is continuously embedded in the usual Schwarz space on the plane, but obviously has a stronger topology.

Question: What would be a good reference for the definition and basic properties of this space? Frechet, nuclear, tempered distributions etc.

Note that I am not looking for answers containing spelled out proofs but only references. Thank you.

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    $\begingroup$ Do you know if the composition with the diffeomorphism $\mathbb R\times (0,\infty) \to \mathbb R^2$, $(x,y)\mapsto (x, y-1/y)$ makes your space isomorphic to $\mathscr S(\mathbb R^2)$? $\endgroup$
    – Jochen Wengenroth
    Commented Sep 5, 2022 at 6:41
  • $\begingroup$ Thanks. That was exactly the first thing that I tried, and it looks like $\mathcal{S}(\mathbb{R}^2)\to\mathcal{S}(\mathbb{R}^2_+)$ is indeed continuous. The inverse diffeo looks like $(x,y)\to(x,(y+\sqrt{y^2+1})/2)$, and taking higher derivatives is more tedious. $\endgroup$
    – Bedovlat
    Commented Sep 5, 2022 at 6:50

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