Abstract Let G be a second countable locally compact group with type-I left regular representation, G ˆ its dual and K = ( K π ) π ∈ G ˆ a specific measurable field of operators. In this paper, we investigate an inversion formula for L p ( G ) . Let 1 p , r ≤ 2 , 1 p + 1 q = 1 s + 1 r = 1 , and F p : L p ∩ L 1 ( G ) ⟶ L q ( G ˆ ) be defined by F p ( f ) π = π ( f ) K π 1 q . The map F p extends uniquely to a linear map of L p ( G ) into L q ( G ˆ ) , denoted by F p . Let F ¯ p be the transpose of F p and f ∈ L p ( G ) . We prove that f ¯ ∈ F ¯ r [ L r ( G ˆ ) ] if and only if F p ( f ) K 1 p − 1 s ∈ L r ( G ˆ ) , and, if that is the case, we have f ¯ = F ¯ r ( K 1 p − 1 s [ F p ( f ) ] ⁎ ) .

中文翻译：

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某些局部紧群的 L−Fourier 反演公式

摘要 令 G 是第二个具有 I 型左正则表示的可数局部紧群，G ˆ 其对偶和 K = ( K π ) π ∈ G ˆ 是一个特定的可测算子域。在本文中，我们研究了 L p ( G ) 的反演公式。设 1 p , r ≤ 2 , 1 p + 1 q = 1 s + 1 r = 1 , 且 F p : L p ∩ L 1 ( G ) ⟶ L q ( G ˆ ) 由 F p ( f ) π 定义= π ( f ) K π 1 q 。映射 F p 唯一地扩展到 L p ( G ) 到 L q ( G ˆ ) 的线性映射，用 F p 表示。令 F¯ p 是 F p 和 f ∈ L p ( G ) 的转置。我们证明 f¯ ∈ F¯ r [ L r ( G ˆ ) ] 当且仅当 F p ( f ) K 1 p − 1 s ∈ L r ( G ˆ ) ，如果是这样，我们有f¯ = F¯ r (K 1 p − 1 s [F p ( f ) ] ⁎ ) 。