Asymptotic crossing relations for the Regge amplitudes are proposed as a model-independent expression of local duality. They allow local duality to be formulated solely in terms of Regge trajectories without any violation of unitarity or neglect of resonance widths. The crossing relations connect the direct- and crossed-channel Regge-pole amplitudes for the elastic scattering of two equal-mass particles as $s\to \infty$ in a given domain of the real $\mathrm{st}$ plane. The main purpose of this paper is to determine the type of domain in which such crossing relations will possess nontrivial solutions that are both self-consistent and in keeping with the known properties of the Regge trajectories and residues. An extensive and systematic study of all domains involving large, positive $s$ and values of $t$ in some interval [$t_1, t_2$] along the positive $t$ axis is presented. We consider both intervals of fixed length and intervals whose length is increasing with $s$. Only one type of interval is found that is satisfactory, namely, intervals consisting of values of $t$ that are increasing in proportion to $s$, i.e., for any given $s$, $t\in [\frac{s}{r_1}, s]$ for some $r_1>1\mathrm{}$. The reasons are the following. It is the only type of interval (1) in which we can provide an existence proof for the solutions to the crossing relations, the proof being valid for asymptotically parallel trajectories; (2) for which the generalization to unequal-mass scattering encounters no obvious inconsistencies; and (3) for which the residues calculated from the crossing relations have an asymptotic form of the type found in the Veneziano model. Furthermore, we can prove the validity of the proposed crossing relations over this type of interval if the modified $s$-channel background integral $B_s^{\prime }$ satisfies the bound $\ln \left|B_s^{\prime }\right|<\mathrm{}\frac{\surd 2a{s}^p}{p}$, where $\mathrm{Re}\alpha \mathrm{}\left(s\right)\mathrm{}\to a{s}^p$ as $s\to \infty$ for $p>\mathrm{}0.802\mathrm{}$. The analogous proof for any of the other types of domains considered requires the bound $\ln \left|B_s^{\prime }\right|<N\ln s$ for some $N$, which is a much stronger bound. The proof to which we refer assumes only that the contributions of Regge branch cuts to the scattering amplitude are negligible compared to the contributions of Regge poles, ${\alpha }_n\mathrm{}\left(s\right)\mathrm{}$, for $s$ positive and sufficiently large. In the indicated domain, the crossing relations imply certain homogeneous integral equations that the residues must satisfy. Although a complete solution is not given, we show that as $s\to \infty$ the residues behave as $\exp (-\gamma a{s}^p)$, where $\gamma$ is a logarithmic function of $r_1$ and $p$. From the expression for $\gamma$ we obtain the upper bound $\gamma <\ln (3+2\mathrm{}\surd 2)-\frac{\surd 2}{p}$ (for $p=1\mathrm{}$, $\gamma <0.348\mathrm{}$). The constant analogous to $\gamma$ in the Veneziano model is 0.38. We add that if the crossing relations proposed here are not valid, then either the background integrals must generate the crossed-channel Regge terms or the background integrals in concert with the direct-channel Regge terms must do so.

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