Question

(A Generalized Hölder Inequality) Suppose that \(1 \leq p_{j} \leq \infty\) and \(\sum_{1}^{n} p_{j}^{-1}=\) \(r^{-1} \leq 1 .\) If \(f_{j} \in L^{p_{2}}\) for \(j=1, \ldots, n\), then \(\prod_{1}^{n} f_{j} \in L^{r}\) and \(\left\|\prod_{1}^{n} f_{j}\right\|_{r} \leq\) \(\prod_{1}^{n}\left\|f_{j}\right\|_{P_{j}}\). (First do the case \(n=2 .\) )

Short answer

For \(n=2\), the inequality \(\| f_1 f_2 \|_r \leq \| f_1 \|_{p_1} \| f_2 \|_{p_2}\) holds, as \(\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r} \).

Step-by-step solution

Understand the Problem

We are given a generalization of the Hölder Inequality. We need to prove that the product of functions \(\prod_{j=1}^{n} f_j\) belongs to the space \(L^r\) given certain conditions and to show that its norm satisfies a particular inequality. We start by considering the simpler case where \(n=2\).

Write Down the Conditions

We have two functions, \(f_1\) and \(f_2\), each belonging to a certain \(L^{p_j}\) space, which means \(f_j \in L^{p_j}\). The indices \(p_1\) and \(p_2\) are such that \(\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r}\) with \(1 \leq p_j \leq \infty\) and \(\frac{1}{r} \leq 1\).

Use Hölder's Inequality

For two functions \(f_1\) and \(f_2\), Hölder's inequality gives us:\[\| f_1 f_2 \|_r \leq \| f_1 \|_{p_1} \| f_2 \|_{p_2},\]when \(\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r}\). This comes from the properties of the \(L^p\) spaces and is a specific case of the inequality we are proving.

Generalize to n=2 Case

Since we are considering \(n=2\), we focus on \(\prod_{j=1}^{2} f_j = f_1 f_2\). Applying the case \(n=2\) directly utilizes Hölder's inequality, and thus the conditions are met for \(f_1 f_2 \in L^r\) and:\[\| f_1 f_2 \|_r \leq \| f_1 \|_{p_1} \| f_2 \|_{p_2}.\]

Verification

Verify that the condition \(\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r} \leq 1\) holds. This ensures that the generalized Hölder inequality can be applied directly, resulting in:\[\| f_1 f_2 \|_r \leq \| f_1 \|_{p_1} \| f_2 \|_{p_2}.\]With the case where \(n=2\), the solution satisfies the given inequality, hence confirming its validity in this specific case.

Key concepts

L^p spaces

When we talk about \( L^p \) spaces, we're referring to a set of functions that are defined on a specific domain and endowed with a particular measure. In more simple terms, \( L^p \) spaces help us understand the behavior of functions by considering them within a framework of integrability. This means we look at functions where the p-th power of their absolute value can be integrated.

Here's a breakdown:

• \( p = 1 \) would refer to the space of absolutely integrable functions, also known as \( L^1 \) space.
• \( p = 2 \) corresponds to what's also known as Hilbert space, or \( L^2 \), which is integral in understanding certain physics and engineering problems.
• As \( p \to \infty \), \( L^p \) considers the functions bounded almost everywhere.

A key property of \( L^p \) spaces is that they are complete, meaning they are mathematically well-behaved and include limits of convergent sequences of functions. This completeness plays a crucial role in ensuring stability and predictability when analyzing mathematical problems across various fields.

norms

In mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space. The most commonly used norms for functions include those measuring their 'size' in various ways related to their values over a domain. When applied to \( L^p \) spaces, the norm \( \| f \|_p \) is defined by:\[\| f \|_p = \left( \int |f|^p \right)^{1/p}.\]

Here's what you need to know about norms in \( L^p \) spaces:

• This expression calculates the 'average size' of the function when raised to the power \( p \) and then taking the \( p \)-th root.
• For \( p = \infty \), the norm corresponds to the supreme value, meaning the maximum absolute value of the function is typically used.
• In general, norms allow us to assess the "length" of functions in spaces similar to Euclidean spaces, which is crucial for comparing functions to each other.

These norms help us work with functions similarly to how we handle numbers in simpler algebra, letting us talk about functions in terms of distance and size. Used in equations, they give us powerful tools for generalizing concepts in calculus and analysis to more complex spaces.

generalized inequality

The term 'generalized inequality' refers to a broad class of inequalities that extend classical inequalities to more complex situations or higher dimensions. The Hölder Inequality is one such tool, often used in the context of \( L^p \) spaces. Simply put, it states that under certain conditions, the product of two functions will also be integrable, and we can establish an upper bound for this product.

The steps are simple:

• Identify the functions \( f_1 \) and \( f_2 \), making sure they belong to different \( L^{p_1} \) and \( L^{p_2} \) spaces respectively.
• Apply Hölder's inequality by ensuring \( \frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r} \). This guarantees that the joint operations of multiplication and integration are valid under the \( L^r \) space.
• Use the inequality: \( \| f_1 f_2 \|_r \leq \| f_1 \|_{p_1} \| f_2 \|_{p_2} \).

This generalization allows us to handle sums and products in function spaces very elegantly, which is extremely useful when dealing with more complicated integrals such as those found in physics and probability theory. The wisdom here is that it provides a structured method to ensure the properties of functions are preserved during such operations, and aids immensely in analytical calculations.