Hölder's inequality - Alchetron, The Free Social Encyclopedia

In mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of L^p spaces.

Theorem (Hölder's inequality). Let (S, Σ, μ) be a measure space and let p, q ∈ [1, ∞] with 1/p + 1/q = 1. Then, for all measurable real- or complex-valued functions f and g on S, If, in addition, p, q ∈ (1, ∞) and f ∈ L^p(μ) and g ∈ L^q(μ), then Hölder's inequality becomes an equality if and only if | f |^p and |g|^q are linearly dependent in L¹(μ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α|f |^p = β |g|^q μ-almost everywhere.

The numbers p and q above are said to be Hölder conjugates of each other. The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if || fg ||₁ is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L^p(μ) and g is in L^q(μ), then the pointwise product fg is in L¹(μ).

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L^p(μ), and also to establish that L^q(μ) is the dual space of L^p(μ) for p ∈ [1, ∞).

Hölder's inequality was first found by Rogers (1888), and discovered independently by Hölder (1889).

Conventions

The brief statement of Hölder's inequality uses some conventions.

In the definition of Hölder conjugates, 1/ ∞ means zero.

If p, q ∈ [1, ∞), then || f ||_p and || g ||_q stand for the (possibly infinite) expressions

If p = ∞, then || f ||_∞ stands for the essential supremum of | f |, similarly for || g ||_∞.

The notation || f ||_p with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if || f ||_p is finite and f is considered as equivalence class of μ-almost everywhere equal functions. If f ∈ L^p(μ) and g ∈ L^q(μ), then the notation is adequate.

On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying a > 0 with ∞ gives ∞.

Estimates for integrable products

As above, let f and g denote measurable real- or complex-valued functions defined on S. If || fg ||₁ is finite, then the pointwise products of f with g and its complex conjugate function are μ-integrable, the estimate

| ∫ S f g ¯ d μ | ≤ ∫ S | f g | d μ = ∥ f g ∥ 1

and the similar one for fg hold, and Hölder's inequality can be applied to the right-hand side. In particular, if f and g are in the Hilbert space L²(μ), then Hölder's inequality for p = q = 2 implies

| ⟨ f , g ⟩ | ≤ ∥ f ∥ 2 ∥ g ∥ 2 ,

where the angle brackets refer to the inner product of L²(μ). This is also called Cauchy–Schwarz inequality, but requires for its statement that || f ||₂ and || g ||₂ are finite to make sure that the inner product of f and g is well defined. We may recover the original inequality (for the case p = 2) by using the functions | f | and | g | in place of f and g.

Generalization for probability measures

If (S, Σ, μ) is a probability space, then p, q ∈ [1, ∞] just need to satisfy 1/p + 1/q ≤ 1, rather than being Hölder conjugates. A combination of Hölder's inequality and Jensen's inequality implies that

∥ f g ∥ 1 ≤ ∥ f ∥ p ∥ g ∥ q

for all measurable real- or complex-valued functions f and g on S.

Notable special cases

For the following cases assume that p and q are in the open interval (1,∞) with 1/p + 1/q = 1.

Counting measure

For the n-dimensional Euclidean space, when the set S is {1, ..., n} with the counting measure, we have

∑ k = 1 n | x k y k | ≤ ( ∑ k = 1 n | x k | p ) 1 p ( ∑ k = 1 n | y k | q ) 1 q for all ( x 1 , … , x n ) , ( y 1 , … , y n ) ∈ R n or C n .

If S = N with the counting measure, then we get Hölder's inequality for sequence spaces:

∑ k = 1 ∞ | x k y k | ≤ ( ∑ k = 1 ∞ | x k | p ) 1 p ( ∑ k = 1 ∞ | y k | q ) 1 q for all ( x k ) k ∈ N , ( y k ) k ∈ N ∈ R N or C N .

These Hölder inequalities for the counting measure still hold when 1/p + 1/q ≤ 1.

Lebesgue measure

If S is a measurable subset of Rⁿ with the Lebesgue measure, and f and g are measurable real- or complex-valued functions on S, then Hölder inequality is

∫ S | f ( x ) g ( x ) | d x ≤ ( ∫ S | f ( x ) | p d x ) 1 p ( ∫ S | g ( x ) | q d x ) 1 q .

Probability measure

For the probability space ( Ω , F , P ) , let E denote the expectation operator. For real- or complex-valued random variables X and Y on Ω , Hölder's inequality reads

E [ | X Y | ] ⩽ ( E [ | X | p ] ) 1 p ( E [ | Y | q ] ) 1 q .

Let 0 < r < s and define p = s r . Then q = p p − 1 is the Hölder conjugate of p . Applying Hölder's inequality to the random variables | X | r and 1 Ω we obtain

E [ | X | r ] ⩽ ( E [ | X | s ] ) r s .

In particular, if the s^th absolute moment is finite, then the r^th absolute moment is finite, too. (This also follows from Jensen's inequality.)

These Hölder inequalities for a probability measure still hold when 1/p + 1/q ≤ 1.

Product measure

For two σ-finite measure spaces (S₁, Σ₁, μ₁) and (S₂, Σ₂, μ₂) define the product measure space by

S = S 1 × S 2 , Σ = Σ 1 ⊗ Σ 2 , μ = μ 1 ⊗ μ 2 ,

where S is the Cartesian product of S₁ and S₂, the σ-algebra Σ arises as product σ-algebra of Σ₁ and Σ₂, and μ denotes the product measure of μ₁ and μ₂. Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If f and g are Σ-measurable real- or complex-valued functions on the Cartesian product S, then

∫ S 1 ∫ S 2 | f ( x , y ) g ( x , y ) | μ 2 ( d y ) μ 1 ( d x ) ≤ ( ∫ S 1 ∫ S 2 | f ( x , y ) | p μ 2 ( d y ) μ 1 ( d x ) ) 1 p ( ∫ S 1 ∫ S 2 | g ( x , y ) | q μ 2 ( d y ) μ 1 ( d x ) ) 1 q .

This can be generalized to more than two σ-finite measure spaces.

Vector-valued functions

Let (S, Σ, μ) denote a σ-finite measure space and suppose that f = (f₁, ..., f_n) and g = (g₁, ..., g_n) are Σ-measurable functions on S, taking values in the n-dimensional real- or complex Euclidean space. By taking the product with the counting measure on {1, ..., n}, we can rewrite the above product measure version of Hölder's inequality in the form

∫ S ∑ k = 1 n | f k ( x ) g k ( x ) | μ ( d x ) ≤ ( ∫ S ∑ k = 1 n | f k ( x ) | p μ ( d x ) ) 1 p ( ∫ S ∑ k = 1 n | g k ( x ) | q μ ( d x ) ) 1 q .

If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbers α, β ≥ 0, not both of them zero, such that

α ( | f 1 ( x ) | p , … , | f n ( x ) | p ) = β ( | g 1 ( x ) | q , … , | g n ( x ) | q ) ,

for μ-almost all x in S.

This finite-dimensional version generalizes to functions f and g taking values in a normed space which could be for example a sequence space or an inner product space.

Proof of Hölder's inequality

There are several proofs of Hölder's inequality; the main idea in the following is Young's inequality.

If || f ||_p = 0, then f is zero μ-almost everywhere, and the product fg is zero μ-almost everywhere, hence the left-hand side of Hölder's inequality is zero. The same is true if || g ||_q = 0. Therefore, we may assume || f ||_p > 0 and || g ||_q > 0 in the following.

If || f ||_p = ∞ or || g ||_q = ∞, then the right-hand side of Hölder's inequality is infinite. Therefore, we may assume that || f ||_p and || g ||_q are in (0, ∞).

If p = ∞ and q = 1, then | fg | ≤ || f ||_∞ | g | almost everywhere and Hölder's inequality follows from the monotonicity of the Lebesgue integral. Similarly for p = 1 and q = ∞. Therefore, we may also assume p, q ∈ (1, ∞).

Dividing f and g by || f ||_p and || g ||_q, respectively, we can assume that

∥ f ∥ p = ∥ g ∥ q = 1.

We now use Young's inequality, which states that

a b ≤ a p p + b q q

for all nonnegative a and b, where equality is achieved if and only if a^p = b^q. Hence

| f ( s ) g ( s ) | ≤ | f ( s ) | p p + | g ( s ) | q q , s ∈ S .

Integrating both sides gives

∥ f g ∥ 1 ≤ 1 p + 1 q = 1 ,

which proves the claim.

Under the assumptions p ∈ (1, ∞) and || f ||_p = || g ||_q, equality holds if and only if | f |^p = | g |^q almost everywhere. More generally, if || f ||_p and || g ||_q are in (0, ∞), then Hölder's inequality becomes an equality if and only if there exist real numbers α, β > 0, namely

α = ∥ g ∥ q q , β = ∥ f ∥ p p ,

such that

α | f | p = β | g | q μ-almost everywhere (*).

The case || f ||_p = 0 corresponds to β = 0 in (*). The case || g ||_q = 0 corresponds to α = 0 in (*).

Statement

Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then, for every f ∈ L^p(μ),

∥ f ∥ p = max { | ∫ S f g d μ | : g ∈ L q ( μ ) , ∥ g ∥ q ≤ 1 } ,

where max indicates that there actually is a g maximizing the right-hand side. When p = ∞ and if each set A in the σ-field Σ with μ(A) = ∞ contains a subset B ∈ Σ with 0 < μ(B) < ∞ (which is true in particular when μ is σ-finite), then

∥ f ∥ ∞ = sup { | ∫ S f g d μ | : g ∈ L 1 ( μ ) , ∥ g ∥ 1 ≤ 1 } .

Remarks and examples

The equality for p = ∞ fails whenever there exists a set A of infinite measure in the σ -field Σ with that has no subset B ∈ Σ that satisfies: 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 ∥ 1 A ∥ ∞ = 1 , but every g ∈ L 1 ( 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 1 A is zero and the extremal equality fails.

For p = ∞ , the supremum is in general not attained. As an example, let S = N , Σ = P ( N ) and μ the counting measure. Define:

Then ∥ f ∥ ∞ = 1. For g ∈ L 1 ( μ , N ) 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 || f₁ + f₂ ||_p ≤ || f₁ ||_p + || f₂ ||_p for all f₁ and f₂ in L^p(μ), see Minkowski inequality.

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

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

Generalization of Hölder's inequality

Assume that r ∈ (0, ∞) and p₁, …, p_n ∈ (0, ∞] such that

∑ k = 1 n 1 p k = 1 r .

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

∥ ∏ k = 1 n f k ∥ r ≤ ∏ k = 1 n ∥ f k ∥ p k .

In particular,

f k ∈ L p k ( μ ) ∀ k ∈ { 1 , … , n } ⟹ ∏ k = 1 n f k ∈ L r ( μ ) .

Note: For r ∈ (0, 1), contrary to the notation, || . ||_r is in general not a norm, because it doesn't satisfy the triangle inequality.

Interpolation

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

1 p = ∑ k = 1 n θ k p k .

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

f k = | f | θ k , 1 ⩽ k ⩽ n .

Then by the above generalization of Hölder's inequality,

∥ f ∥ p = ∥ ∏ k = 1 n f k ∥ p ⩽ ∏ k = 1 n ∥ f k ∥ p k θ k = ∏ k = 1 n ∥ f ∥ p k θ k .

In particular, taking θ₁ = θ and θ₂ = 1 − θ, in the case n = 2, we obtain the interpolation result (Littlewood's inequality)

∥ f ∥ p θ ⩽ ∥ f ∥ p 1 θ ⋅ ∥ f ∥ p 0 1 − θ ,

for θ ∈ ( 0 , 1 ) and

1 p θ = θ p 1 + 1 − θ p 0 .

A similar application of Hölder gives Lyapunov's inequality: If

p = θ p 0 + ( 1 − θ ) p 1 , θ ∈ ( 0 , 1 ) ,

then

∥ f ∥ p p ⩽ ∥ f ∥ p 0 p 0 θ ⋅ ∥ f ∥ p 1 p 1 ( 1 − θ ) .

Both Littlewood and Lyapunov imply that if f ∈ L p 0 ∩ L p 1 , then f ∈ L p for all p 0 < p < p 1 .

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,

∥ f g ∥ 1 ⩾ ∥ f ∥ 1 p ∥ g ∥ − 1 p − 1 .

If

∥ f g ∥ 1 < ∞ and ∥ g ∥ − 1 p − 1 > 0 ,

then the reverse Hölder inequality is an equality if and only if

∃ α ⩾ 0 | f | = α | g | − p p − 1 μ -almost everywhere .

Note: The expressions:

∥ f ∥ 1 p and ∥ g ∥ − 1 p − 1 ,

are not norms, they are just compact notations for

( ∫ S | f | 1 p d μ ) p and ( ∫ S | g | − 1 p − 1 d μ ) − ( p − 1 ) .

Conditional Hölder inequality

Let (Ω, F, ℙ) be a probability space, G ⊂ F 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,

E [ | X Y | | G ] ≤ ( E [ | X | p | G ] ) 1 p ( E [ | Y | q | G ] ) 1 q P -almost surely.

Remarks:

If a non-negative random variable Z has infinite expected value, then its conditional expectation 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 ∞.

Hölder's inequality for increasing seminorms

Let S be a set and let F ( S , C ) be the space of all complex-valued functions on S. Let N be an increasing seminorm on F ( S , C ) , meaning that, for all real-valued functions f , g ∈ F ( S , C ) we have the following implication (the seminorm is also allowed to attain the value ∞):

∀ s ∈ S f ( s ) ⩾ g ( s ) ⩾ 0 ⇒ N ( f ) ⩾ N ( g ) .

Then:

∀ f , g ∈ F ( S , C ) N ( | f g | ) ⩽ ( N ( | f | p ) ) 1 p ( N ( | g | q ) ) 1 q ,

where the numbers p and q are Hölder conjugates.

Remark: If (S, Σ, μ) is a measure space and N ( f ) is the upper Lebesgue integral of | f | then the restriction of N to all Σ-measurable functions gives the usual version of Hölder's inequality.

References

Hölder's inequality Wikipedia

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