We investigate the \(L^p\) decay estimates of a class of non-localized oscillatory fractional integral operators of the form \(\begin{aligned} T_{\lambda ,-\alpha }f(x) = \int _{\mathbb {R}} e^{i\lambda P(x - y)} |x - y|^{-\alpha } f(y)\,dy, \end{aligned}\) where \(0 \le \alpha < 1\) , \(\lambda \ge 1\) , and P is a real polynomial of degree \(n \ge 2\) . Our goal is to characterize the sharp decay rate \(\delta \) in the estimate \(\Vert T_{\lambda ,-\alpha }\Vert _{L^p \rightarrow L^p} \lesssim \lambda ^{-\delta }\) in terms of the parameters \(\alpha \) and p. To this end, we introduce a hexagonal region \(\mathcal {H}(\gamma , \frac{1 - \alpha }{n})\) in the \((\frac{1}{p}, \delta )\) -plane that precisely describes the admissible range for decay, where \(\begin{aligned} \gamma = \min \left\{ \frac{1 - \alpha }{\ell + 2}, \frac{1}{2 + m} \right\} . \end{aligned}\) Here, \(\ell \) is the order of vanishing of \(P''\) at the origin, and m is the maximum multiplicity among nonzero real roots of \(P''(t) = 0\) . We prove that the decay estimate holds if and only if \((\frac{1}{p}, \delta )\) lies inside this hexagon. Our approach decomposes the operator into localized parts near singular points and a non-localized remainder. We apply the van der Corput-type estimates to handle \(L^2\) decay near singularities and derive Hardy space bounds to interpolate to the full \(L^p\) range. For the non-localized parts, we establish decay estimates using multiplier techniques adapted to translation-invariant kernels. This yields a complete description of the \(L^p\) behavior of \(T_{\lambda ,-\alpha }\) in terms of both singular structure and kernel decay.
