The extrapolation of low-energy theorems for certain weak quantities from the soft point to the $\rho$ mass is studied. A parabola in the plane of variables, essentially connected with the mass and energy of the particles involved, is taken for the path of extrapolation, as proposed by Fubini and Furlan. As a consequence, sum rules involving the $\pi \rho$ and $\rho N$ elastic scattering matrices at threshold are obtained, in which the most important contributions are explicitly exhibited. An important term arises from the right-hand side of the equal-time commutation relation of space components of vector currents, and thus the resulting sum rules are particularly suitable for discussing models which predict such commutators. The same results are obtained in the saturation approach, starting from commutators of electric dipoles and space-space commutators of vector currents. Combining the $\rho \pi$ scattering matrix sum rule with the Weinberg formula, an expression for $\frac{1}{6}〈r_{\pi }^{}{}_{}{}^2〉$ is found. In this particular case and under the simplest assumptions, both field algebra and quark model are compatible with the data within present experimental errors. The investigation is extended to the algebra of tensor currents and together with a hypothesis of partial conservation of tensor current and local $\mathrm{SU}\left(2\right)\mathrm{}$ algebra, this leads to a value for the difference of the mean-square electromagnetic pion and nucleon radius as an exact consequence.

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