Stein's Singular Integrals book mentions a number of results concerned with characterising operators that commute with translations. Some of those results are mentioned in this other StackExchange post. Are there any examples of questions/problems where one can see the need for such operators? How would one motivate the problem of characterising such operators (for someone with an understanding of analysis at the level of Folland/Royden)?
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1The reason for considering things commutative up to an action is that such things can often be passed to the quotient by the action. It's a bit weaker than being invariant under such actions, but often suffices. In this case one might care about either operators on affine space as supposed to choosing a specific origin, or one might want to define a mollifier that doesn't depend on the point in space chosen, etc – Brevan Ellefsen Jul 22 '23 at 19:00
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As an example of that perspective, note the group of Euclidian isometries contains translations as a normal subgroup. Modding out by them yields the orthogonal group, which is the group of isometries of the sphere (after restricting to the group to the sphere). You could thus be interested in proving results about spherical operators by working with operators commuting with translation, in the hopes there could be an induced spherical operator. – Brevan Ellefsen Jul 22 '23 at 19:09