Statement

Remarks and examples

• The equality for p = ∞ fails whenever there exists a set A in the σ-field Σ with μ(A) = ∞ that has no subset B ∈ Σ with 0 < μ(B) < ∞ (the simplest example is the σ-field Σ containing just the empty set and S, and the measure μ with μ(S) = ∞). Then the indicator function 1_{A satisfies ||1A||∞ = 1, but every g ∈ L¹(μ) has to be μ-almost everywhere constant on A, because it is Σ-measurable, and this constant has to be zero, because g is μ-integrable. Therefore, the above supremum for the indicator function 1A is zero and the extremal equality fails.}
• For p = ∞, the supremum is in general not attained. As an example, let S denote the natural numbers (without zero), Σ the power set of S, and μ the counting measure. Define ƒ(n) = (n − 1)/n for every natural number n. Then ||ƒ ||_∞ = 1. For g ∈ L¹(μ) with 0 < ||g ||₁ ≤ 1, let m denote the smallest natural number with g(m) ≠ 0. Then

Applications

• The extremal equality is one of the ways for proving the triangle inequality ||ƒ₁ + ƒ₂||_p ≤ ||ƒ₁||_p + ||ƒ₂||_p  for all ƒ₁ and ƒ₂ in L^p(μ), see Minkowski inequality
  Minkowski inequality
  In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp...
  
  .

• Hölder's inequality implies that every ƒ ∈ L^p(μ) defines a bounded (or continuous) linear functional κ_ƒ on L^q(μ) by the formula

The extremal equality (when true) shows that the norm of this functional κ_ƒ as element of the continuous dual space L^q(μ)^∗ coincides with the norm of ƒ in L^p(μ) (see also the L^p-space article).

Generalization of Hölder's inequality

1, …, p_n ∈ (0, ∞] such that


Then, for all measurable real- or complex-valued functions f₁, …, f_n defined on S,


In particular,


Note:

• For r ∈ (0, 1), contrary to the notation, ||.||_r is in general not a norm, because it doesn't satisfy the triangle inequality
  Triangle inequality
  In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....
  
  .

Interpolation

Let p1, …, p_n ∈ (0, ∞] and let θ1, …, θn ∈ (0, 1) denote weights with θ1+ … + θn = 1. Define p as the weighted harmonic mean

Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

, i.e.,


Given a measurable real- or complex-valued functions f on S, define


Then by the above generalization of Hölder's inequality,


As an interpolation

Riesz-Thorin theorem

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.This theorem bounds the norms of linear maps...

 result for n = 2,


where θ ∈ (0, 1) and

Reverse Hölder inequality

Assume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then, for all measurable real- or complex-valued functions f and g on S such that g(s) ≠ 0 for μ-almost all s ∈ S,


If ||fg ||₁ < ∞ and ||g ||_{−1/(p −1)} > 0, then the reverse Hölder inequality is an equality if and only if there exists an α ≥ 0 such that
    μ-almost everywhere.

Note: ||f ||_1/p and ||g ||_{−1/(p −1)} are not norms, these expressions are just compact notation for
   and   

Conditional Hölder inequality

Let be a probability space, a sub-σ-algebra, and p, q ∈ (1, ∞) Hölder conjugates, meaning that 1/p + 1/q = 1. Then, for all real- or complex-valued random variables X and Y on Ω,


Remarks:

• If a non-negative random variable Z has infinite expected value
  Expected value
  In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
  
  , then its conditional expectation
  Conditional expectation
  In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution....
  
  is defined by

• On the right-hand side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying a > 0 with ∞ gives ∞.

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.