Two theorems on oscillatory integrals are proved. Suppose \(N\) and \(n\) are positive integers, and \({\mathcal P}_ N\) denotes the real polynomials in one variable of degree at most \(N.\) Then there exists a constant \(C(N,n)\) such that for all intervals \([a,b]\) and all \(p\) in \({\mathcal P}_ N,\) \[ |\int_{[a,b]} e^{ip(x)}| p^{(n)}(x)|^{1/ n}dx|\leq C(N,n), \] and further for all \(s\) in \({\mathbf R},\) \[ |\int_{[a,b]}e^{ip(x)}| p^{(n)}(x)|^{{1/ n} + is}dx| \leq C(N,n)(1 + | s|)^{1/ n}, \] provided that \(n=1\) or \(2\).