L^p Space - Weak Lp

Weak Lp

Let (S, Σ, μ) be a measure space, and f a measurable function with real or complex values on S. The distribution function of f is defined for t > 0 by

If f is in Lp(S, μ) for some p with 1 ≤ p < ∞, then by Markov's inequality,

A function f is said to be in the space weak Lp(S, μ), or Lp,w(S, μ), if there is a constant C > 0 such that, for all t > 0,

The best constant C for this inequality is the Lp,w-norm of f, and is denoted by

The weak Lp coincide with the Lorentz spaces Lp,∞, so this notation is also used to denote them.

The Lp,w-norm is not a true norm, since the triangle inequality fails to hold. Nevertheless, for f in Lp(S, μ),

and in particular Lp(S, μ) ⊂ Lp,w(S, μ). Under the convention that two functions are equal if they are equal μ almost everywhere, then the spaces Lp,w are complete (Grafakos 2004).

For any 0 < r < p the expression

is comparable to the Lp,w-norm. Further in the case p > 1, this expression defines a norm if r = 1. Hence for p > 1 the weak Lp spaces are Banach spaces (Grafakos 2004).

A major result that uses the Lp,w-spaces is the Marcinkiewicz interpolation theorem, which has broad applications to harmonic analysis and the study of singular integrals.