mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Strichartz estimates are a family of inequalities for linear

dispersive partial differential equation In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. E ...

s. These inequalities establish size and decay of solutions in mixed norm Lebesgue spaces. They were first noted by Robert Strichartz and arose out of connections to the Fourier restriction problem.

Examples

Consider the linear

Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

\mathbb^d

with ''h'' = ''m'' = 1. Then the solution for initial data

u_0

is given by

e^u_0

. Let ''q'' and ''r'' be real numbers satisfying

2\leq q, r \leq \infty

;

\frac+\frac=\frac

; and

(q,r,d)\neq(2,\infty,2)

. In this case the homogeneous Strichartz estimates take the form: :

\, e^ u_0\, _\leq C_ \, u_0\, _.

Further suppose that

\tilde q, \tilde r

satisfy the same restrictions as

q, r

and

\tilde q', \tilde r'

are their dual exponents, then the dual homogeneous Strichartz estimates take the form: :

\left\,  \int_\mathbb e^F(s)\,ds\right\, _\leq C_\, F\, _.

The inhomogeneous Strichartz estimates are: :

\left\,  \int_ e^F(s)\,ds\right\, _\leq C_\, F\, _.

References

Theorems in analysis Inequalities {{Mathanalysis-stub