Let $\phi$ be a function on $(0,\infty)$ continuous except on a null set, and $\phi_{\epsilon}(\xi)=\phi(\epsilon\xi)\ (\epsilon>0).$ Also $\tilde{T}_{\epsilon}$ be the operator on Jacobi series such that $(\tilde{T}_{\epsilon}f)^{\wedge}(n)= \phi_{\epsilon}(n)\hat{f}(n)\ (n\in{\bf Z})$, where $\hat{f}(n)$ is the coefficient of Jacobi expanstion of $f$, and ${\cal H}_{\alpha}(Tf)(\xi)=\phi(\xi){\cal H}_{\alpha}f(\xi)\ (\xi\in(0,\infty))$, where ${\cal H}_{\alpha}f$ is the modified Hankel transform of $f$ with order $\alpha$. Then Igari [4] proved that if the operator norm of $\tilde{T}_{\epsilon}$ is uniformly bounded for all $\epsilon>0$, $T$ is an operator on Hankel transforms (the details in $\S 1,\S 2$). After that, Connett-Schwartz[2] and Kanjin[5] proved the weak version and the maximal version by using [4], respectively. In this paper, we prove the analogy of Igari[4] in the Lorentz space, in the same way. Also in $\S 3$, as an application of this result, we show a result with respect to the partial sum operator of the Jacobi series.