Question

If \(g \in L^{\infty}\), the operator \(T\) defined by \(T f=f g\) is bounded on \(L^{p}\) for \(1 \leq p \leq \infty\). Its operator norm is at most \(\|g\|_{\infty}\), with equality if \(\mu\) is semifinite.

Short answer

The operator norm of \( T \) is \( \|g\|_{\infty} \), with equality if \( \mu \) is semifinite.

Step-by-step solution

Understanding the Problem

We need to establish that the operator \( T \) defined by \( T f = fg \) is bounded on \( L^p \) spaces for \( 1 \leq p \leq \infty \), and determine its operator norm.

Defining Bounded Operators

An operator \( T: L^p \rightarrow L^p \) is bounded if there exists a constant \( C \) such that \( \, \|Tf\|_{L^p} \leq C \|f\|_{L^p} \, \) for all \( f \in L^p \). The smallest such constant \( C \) is the operator norm \( \|T\| \).

Express Operator Norm Calculation

We start by calculating \( \|Tf\|_{L^p} \). Since \( T f = fg \), we have \( \|Tf\|_{L^p} = \|fg\|_{L^p} = \|(fg)^p\|^{1/p} \).

Using Holder's Inequality

Applying Hӧlder's inequality to \( \|fg\|_{p} \):\[ \|fg\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^\infty}, \]multiplying the norm by \( \|g\|_{\infty} \) because \( g \) is essentially bounded.

Operator Norm Estimation

From the inequality \( \|fg\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^\infty} \), we find that the operator norm \( \|T\| \leq \|g\|_{L^\infty} \).

Equality Condition for Operator Norm

The operator norm reaches equality (i.e., \( \|T\| = \|g\|_{L^\infty} \)) if \( \mu \) is semifinite. This implies that there exists a subset \( E \) with finite positive measure where \( |f| \) approximates \( |g| \) almost everywhere, ensuring equality.

Key concepts

Operator Norm

An operator norm is a way to measure the 'size' or 'strength' of an operator, much like you might measure the length of a vector. In the context of bounded operators, which is what we are dealing with here, an operator is defined as bounded if there is a constant, say \( C \), such that the effect of applying the operator does not increase the size of the input by more than \( C \) times. Essentially, if you have an operator \( T \), and it acts on an element \( f \) in some space, like \( L^p \), the norm of the output is at most \( C \) times the norm of the input.
Let's put this into a formal definition: the operator \( T: L^p \rightarrow L^p \) is bounded if there exists a \( C \) such that \( \|Tf\|_{L^p} \leq C \|f\|_{L^p} \) for every \( f \) in \( L^p \). The operator norm \( \|T\| \) is then defined as the smallest value this \( C \) can take. In simple words, the operator norm expresses the worst-case scenario, the maximum stretching that this operator could achieve in terms of input-output ratios. In our situation, given a function \( g \) in \( L^{\infty} \), when it is used in the operator \( T \) defined by multiplication, the norm \( \|T\| \) is at most \( \|g\|_{\infty} \).
Achieving this bound exactly will depend on specific contexts, such as the measure properties in the space.

L^p Spaces

\(L^p\) spaces are essential concepts in functional analysis and describe a class of functions you can 'measure' or 'integrate' in some way. The \( p \) in \( L^p \) simply indicates the power to which you raise the absolute value of the function during integration.
For example:

• \( L^1 \) spaces are integrable functions where the absolute value is summed (integrated) over a defined domain.
• \( L^2 \) functions are square-integrable. These spaces are crucial in physics and engineering, notably in quantum mechanics.
• \( L^\infty \) is special as it describes essentially bounded functions, where instead of integrating the power of the absolute value, we measure its 'essential supremum'. This states the function never exceeds a particular bound except on a set of measure zero.

In the context of the operation \( T f = fg \), if \( g \) is in \( L^{\infty} \), it means \( g \) is bounded almost everywhere. This characteristic of \( g \) ensures that \( T \), the operator defined by multiplication, remains bounded across \( L^p \) spaces.

Holder's Inequality

Hölder's inequality is a fundamental mathematical tool that helps extend the notion of the dot product into more general settings like function spaces. It serves as a critical bridge to support how we handle situations involving integrals and product spaces.
Given functions \( f \) and \( g \) and the context of \( L^p \) and \( L^q \) spaces, where \( 1/p + 1/q = 1 \), Hölder's inequality asserts such an estimate:
\[ \int |fg| \leq \|f\|_{L^p}\|g\|_{L^q} \]
The inequality plays a significant role in proving the boundedness of operators like our example \( T f = fg \). Here, especially when \( g \) is in \( L^\infty \), we apply Hölder's inequality seamlessly because when integrating, one of the norms consolidates completely bounded behavior as seen:
\[ \|fg\|_{L^p} \leq \|f\|_{L^p}\|g\|_{L^\infty} \]
Such relationships allow us to affirm that the operator's norm observed from \( L^p \rightarrow L^p \) directions is at most the supremum norm of \( g \). This becomes an integral part, especially in mathematical analysis, stabilizing results under various functional operations.