{'fraction_non_alphanumeric': 0.019464720194647202, 'fraction_numerical': 0.004866180048661801, 'mean_word_length': 4.755244755244755, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Microlocal regularity on step two nilpotent Lie groups

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.