In this paper we consider translation invariant operators with additional symmetry coming from group actions. As the classic Hilbert and Riesz transforms can be characterized up to scalar by means of relative invariance of conformal transformation groups, certain multiplier operators are characterized by relative invariance of some other affine subgroups. In this article, we formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar, and provide several examples of multiplier operators having 'large symmetry'. Finally, we classify which of these examples are Lp-bounded.
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© Toshiyuki Kobayashi