COLLOQUIUM MATHEMATICUM VOL. 119 2010 NO. 1 ON FOURIER ASYMPTOTICS OF A GENERALIZED CANTOR MEASURE BY BÉRENGER AKON KPATA, IBRAHIM FOFANA and KONIN KOUA (Abidjan) Abstract. Let d be a positive integer and μ a generalized Cantor measure satisfying μ = P m j=1 aj μ ◦ S −1 j , where 0 <aj < 1, P m j=1 aj =1, Sj = ρR + bj with 0 <ρ< 1 and R an orthogonal transformation of R d . Then 8 > > > > < > > > > : 1 <p ≤ 2 ⇒ sup r>0 r d(1/α ′ −1/p ′ ) “  J r x | b μ(y)| p ′ dy ” 1/p ′ ≤ D1ρ −d/α ′ ,x ∈ R d , p =2 ⇒ inf r≥1 r d(1/α ′ −1/2) “  J r 0 | b μ(y)| 2 dy ” 1/2 ≥ D2ρ d/α ′ , where J r x = Q d i=1 (xi − r/2,xi + r/2), α ′ is defined by ρ d/α ′ =( P m j=1 a p j ) 1/p and the constants D1 and D2 depend only on d and p. 1. Introduction. Let us start with some notations. We will denote by χ N the characteristic function of the subset N of R d . Given 1 ≤ q ≤∞, ‖·‖ q will denote the usual Lebesgue norm and q ′ the conjugate exponent of q: 1/q +1/q ′ =1 with the convention 1/∞ =0. Let μ be a non-negative Radon measure on R d . For 1 ≤ q< ∞, 0 <β ≤ d and r> 0, we define the average H (q,β,r)= 1 r d−β  J r 0 |  μ(y)| q dy, where  μ is the Fourier transform of the measure μ and J r x =  d i=1 (x i − r/2, x i + r/2) for x =(x 1 ,...,x d ) ∈ R d . We are interested in lower and upper bounds of H (q,β,r) when r varies in (0, ∞). This topic has been studied by K. S. Lau and J. Wang [L-W], and R. S. Strichartz ([St1]–[St3]) in the setting of self-similar measures. We recall some results obtained by R. S. Strichartz. 2010 Mathematics Subject Classification : Primary 42B10; Secondary 28A80. Key words and phrases : Fourier transform, generalized Cantor measure, self-similar mea- sure. DOI: 10.4064/cm119-1-6 [109] c  Instytut Matematyczny PAN, 2010