We study the maximal cone multiplier operator associated with truncated Fourier multipliers adapted to conic regions in frequency space. In dimension three, we establish sharp \(L^p\) bounds for the associated maximal operator, thereby completely resolving the conjectured range for all \(1< p < \infty \) . Our approach combines multi-scale analysis with wave packet decomposition adapted to annular and conic geometry. The same method can also be applied in the two-dimensional spherical setting to establish new \(\ell ^p\) annular inequalities, leading to a new proof of the sharp \(L^4\) boundedness of the maximal Bochner–Riesz operator in the plane. In higher dimensions \(n \ge 4\) , we solve the problem completely for \(p < 2\) and obtain sharp partial results for certain values of \(p > 2\) , away from the conjectured critical exponent \(p_c = \frac{2(n-1)}{n-2}\) . While the full conjectured range \(p > p_c\) remains open, our results improve upon previously known bounds and go below the general barrier \(p = \frac{2(n+1)}{n-1}\) for maximal operators associated with general Fourier integral operators satisfying the cinematic curvature condition. In particular, we identify a new dimension-dependent threshold p(n) above which we establish sharp \(L^p\) estimates for the maximal cone multiplier operator. This work extends and refines earlier results on cone multiplier problems and demonstrates how geometric and frequency-localized methods can yield sharp maximal bounds across different settings.
