If \(\mathbf{X}\) is a Banach space, \(\boldsymbol{\Gamma}: \mathbb{R}^k\rightarrow \mathbb{R}^n\) is a \(C^\infty\)-mapping and \(K\) is a Calderón-Zygmund kernel in \(\mathbb{R}^k\), then a vector-valued singular integral \(T_\Gamma\) is defined on \(\mathscr{S} (\mathbb{R}^n, \mathbf{X})\) by \[ T_\Gamma f(x)= p.v. \int_{\mathbb{R}^k} f(x-\Gamma(t)) K(t) \, dt. \] Let the class \(\mathcal{I}\) be formed by those UMD spaces \(\mathbf{X}\) which are isomorphic to a closed subspace of a complex interpolation space \([\mathbf{H}, \mathbf{Y}]_\theta, 0<\theta<1\), between a Hilbert space \(\mathbf{H}\) and another UMD space \(\mathbf{Y}\). The main result of the paper reads as follows:
Theorem. If \(\mathbf{X}\in\mathcal{I}\) and \(\mathcal{P}(t)=(\mathcal{P}_1(t),\mathcal{P}_2(t),\dots,\mathcal{P}_n(t))\), where \(\mathcal{P}_j, j=1,\dots,n,\) are polynomials in \(t\in \mathbb{R}^k\), then, for any \(p\in (1,\infty)\), there exists a constant \(C_p>0\) such that \[ \|T_{\mathcal{P}}f\|_{L^p(\mathbb{R}^n; \mathbf{X})}\le C_p \|f\|_{L^p(\mathbb{R}^n;\mathbf7X}). \] The constant \(C_p\) may depend on \(\mathbf{X}, k, n\) and the total degree of \(\mathcal{P}\), but it is independent on the coefficients of \(\mathcal{P}\).