# Strichartz estimate

In applied mathematics, **Strichartz estimates** are a family of inequalities for linear dispersive partial differential equations. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by R. Strichartz and arose out of contentions to the Fourier restriction problem.^{[1]}

## Examples

Consider the linear Schrödinger equation in with *h* = *m* = 1. Then the solution for initial data is given by . Let *q* and *r* be real numbers satisfying ; ; and .

In this case the homogeneous Strichartz estimates take the form:^{[2]}

Further suppose that satisfy the same restrictions as and are their dual exponents, then the dual homogeneous Strichartz estimates take the form:^{[2]}

The inhomogeneous Strichartz estimates are:^{[2]}

## References

- ↑ R.S. Strichartz (1977), "Restriction of Fourier Transform to Quadratic Surfaces and Decay of Solutions of Wave Equations",
*Duke Math. J*,**44**: 705–713 - 1 2 3 Tao, Terence (2006),
*Nonlinear dispersive equations: Local and global analysis*, CBMS Regional Conference Series in Mathematics,**106**, ISBN 0-8218-4143-2