On shifted convolution of half-integral weight cusp forms (Q625848)

Let \[ f(z)=\sum_{n\geq 1}a_f(n)n^{\frac{l-1}{2}}e^{2\pi nz}\in S_l(\Gamma_0(1)) \] and \[ g(z)=\sum_{n\geq 1}a_g(n)n^{\frac{k-1}{2}}e^{2\pi nz}\in S_k(\Gamma_0(4N)) \] be two cusp forms, where \(l\in{\mathbb N}\) and \(k\geq \frac 52\) is a half-integer, \(\phi(x)\) a smooth function with support in \([\frac{X}{2},\frac{5X}{2}]\) satisfying \(\phi^{(p)}(x)\ll (\frac{X}{P})^{-p}\) for all integers \(p\geq 0\), where \(1\leq P\leq X\). The author proves that for a fixed integer \(b>0\) \[ \sum_{n\geq 1}a_g(n+b)a_f(n)\phi(n)\ll_{f,g,b,\epsilon}X^{\frac 34 +\epsilon} P^{\frac 32}. \] If \[ h(z)=y^{\frac 12}\sum_{n\in{\mathbb Z}}\rho_h(n) K_{it}(2\pi|n|y)e^{2\pi inx} \] is a Maass cusp form for \(SL(2,{\mathbb Z})\) with the Laplacian eigenvalue \( \lambda=\frac 14+t^2,t>0\) then \[ \sum_{n\geq 1}a_g(n+b)\rho_h(n)\phi(n)\ll_{f,h,b,\epsilon}X^{\frac 34 +\epsilon} P^{\frac 32}. \] It is expected that the bounds would hold with \(X^{\frac 12+\epsilon}\) in place of \(X^{\frac 34+\epsilon}\), which can be proved in the case \(b=0\).