Transfert Formula in the Real Hyperbolic Space B n

Let s ∈ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq1783.tif"/> such that − 1 < Re   s   <   1 n − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq1784.tif"/> . Then we show that every eigenfunction f of the Lobatchevsky Laplacian D in the real unit ball Bⁿ , with (s ² – l)(n – l)² as eigenvalue can be written as an integral transform (transfert formula) of a unique Euclidean harmonic function F in Bⁿ : f ( x ) = Γ ( n 2 ) ( 1 − | x | 2 ) n − 2 2 Γ ( ( 1 + s ) ( n − 1 2 ) ) Γ ( 1 − s ( n − 1 ) 2 ) ∫ 0 1 [ t ( 1 − | x | 2 ) ( 1 − t ) ( 1 − t | x | 2 ) ] s ( n − 1 ) + 1 2 F ( t x ) t 4 − n 2 d t . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq1785.tif"/>