A Note on the Sup Norm of Eisenstein Series

The sup norm problem has been an active area and now there exist non-trivial estimates for cusp forms of large level, on higher rank groups, and for half-integral weight forms [BH] [T1] [HT1] [HT2] [T2] [HRR] [BP] [BM] [M] [K]. Nevertheless, the basic estimate for Γ = PSL2(Z) in the eigenvalue aspect has not been improved. The case of Eisenstein series seems to have been largely neglected up to now, at least for the sup norm problem, but not for some other norms: Luo and Sarnak proved QUE for Eisenstein series [LS], Spinu estimated the L norm [Sp], and the author has investigated QUE for geodesic restrictions [Y]. The Eisenstein series case is similar in some ways to the cuspidal case, but has some technical problems because of the constant term in the Fourier expansion. Actually, the main impetus for this note was the realization that one can choose an efficient amplifier for the Eisenstein series, which leads to the improved exponent compared to the cusp form case (see [IS, Remark 1.6]).