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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Marco Fraccaroli (University of Bonn)

On distributions with full $GL_{2}(\mathbb{R})$ dilation symmetry

The tempered distribution in $\mathbb{R}^{4}$ \[ \Lambda(\varphi) := \text{p.v.} \int_{\mathbb{R}^{4}} \frac{1}{\text{det}(x \ y)} \varphi(x,y) \ dx dy, \] arises in a recent paper by Gressman, He, Kovač, Street, Thiele, Yung, where $x,y \in \mathbb{R}^{2}$ and the principal value is taken as $\text{det}(x \ y)$ goes to 0. It satisfies the following invariance property:

For a matrix $A \in GL_{2}(\mathbb{R})$ define the transform \[ D_{A}^{1} \varphi (x,y) := \frac{1}{\text{det } A} \varphi (A^{-1}x,A^{-1}y) \] for every $\varphi \in \mathcal{S}(\mathbb{R}^{4})$. Then for every $A$ and $\varphi$ as above \[ \Lambda(D_{A}^{1}\varphi)=\Lambda(\varphi). \] Motivated by this we want to classify all the tempered distributions in $\mathbb{R}^{4}$ satisfying the $D_{A}^{\alpha}$-invariance property for matrices with positive determinant (where $\alpha$ identify the exponent of $({\text{det } A}^{-1})$ in the definition of $D_{A}^{\alpha} \varphi$) and distinguish them according to their behaviour against a matrix with determinant -1, namely one of the two possible invariances of the form \[ \Lambda (\varphi(x_{1},-x_{2},y_{1},-y_{2}))=\pm \Lambda(\varphi(x,y)). \]