Damped oscillatory integrals and Weierstrass polynomials

We consider the Sogge-Stein problem related to the damped oscillatory integrals. We show that in three-dimensional Euclidean spaces minimal exponent, which guarantees optimal decaying of the Fourier transform of the surfaces-carried measures with mitigating factor is bounded by $\frac{3}{2}$. A proof of the main theorem is based on Weierstrass type results.