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In this paper, we mainly focus on the Cauchy problem of an integrablenonlocal Hirota equation with initial value in weighted Sobolev space. Throughthe spectral analysis of Lax pairs, we successfully transform the Cauchyproblem of the nonlocal Hirota equation into a solvable Riemann-Hilbertproblem. Furthermore, in the absence of discrete spectrum, the long-timeasymptotic behavior of the solution for the nonlocal Hirota equation isobtained through the Dbar steepest descent method. Different from the localHirota equation, the leading order term on the continuous spectrum and residualerror term of $q(x,t)$ are affected by the function $Im\nu(z_j)$.

Long time asymptotic analysis for a nonlocal Hirota equation via the Dbar steepest descent method