We improve the estimates in the restriction problem in dimension n⩾4. To do so, we establish a weak version of a k-linear restriction estimate for any k. The exponents in this weak k-linear estimate are sharp for all k and n.
In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field.