Question

Given \(1

Short answer

\(T\) is a bounded linear map from \(L^{q}((0, \infty))\) to \(C_{0}((0, \infty))\).

Step-by-step solution

Understand the operators and spaces involved

We begin with the operator \(T\), defined as \(T f(x)=x^{-1 / p} \int_{0}^{x} f(t) dt\). The spaces involved are \(L^q((0, \infty))\), which is a Lebesgue integrable space, and \(C_0((0, \infty))\), the space of continuous functions vanishing at infinity.

Check linearity of \(T\)

To verify that \(T\) is linear, we note that for any two functions \(f, g \in L^q((0, \infty))\) and scalars \(a, b \in \mathbb{R}\), we have:\[T(af + bg)(x) = x^{-1 / p} \int_{0}^{x} (af(t) + bg(t)) dt = aT(f)(x) + bT(g)(x)\]This confirms linearity, as it satisfies the properties of additivity and homogeneity.

Verify boundedness of \(T\)

We need to show that there exists a constant \(C > 0\) such that \(\|T f\|_{C_0((0, \infty))} \leq C \|f\|_{L^q((0, \infty))}\) for all \(f \in L^q((0, \infty))\). By using Hardy's inequality, we infer that the operator is indeed bounded. The relationship \(p^{-1} + q^{-1} = 1\) ensures the necessary conditions for the application of such inequalities, which establish the boundedness property.

Show mapping from \(L^q\) to \(C_0\)

For any sequence \(x_n \to \infty\), based on Hardy's inequality and the properties of integrals with \(f \in L^q((0, \infty))\), we confirm that \(T f(x_n) \to 0\) as \(x_n \to \infty\). Additionally, the continuity of \(T f(x)\) follows from the integral definition of the operator and the properties of continuous functions. Therefore, \(T f(x)\) is an element of \(C_0((0, \infty))\).

Key concepts

Lebesgue Integrable Space

The concept of a Lebesgue integrable space, denoted as \(L^q((0, \infty))\), is fundamental in understanding many operators, such as the one presented in the original exercise. Imagine you have a function \(f\) defined over the positive real numbers. If this function is part of \(L^q((0, \infty))\), its integral, when raised to the power of \(q\), must converge to a finite number. This is what it means for \(f\) to be integrable in the sense of Lebesgue.

• Lebesgue spaces are often used because they allow a broader class of functions compared to simple Riemann integrable functions.
• These spaces provide powerful tools for analyzing the convergence and behavior of functions within an integral.

An important aspect is the parameter \(q\), which determines the type of control or decay we need for the function to belong to this space. For functions in \(L^q((0, \infty))\), we utilize the integration power \(q\) that satisfies the condition \(p^{-1} + q^{-1} = 1\). This condition is pivotal as it relates directly to the dual nature of these spaces, ensuring the boundedness and completeness required for analyzing linear operators.

Continuity

Continuity is a central theme in function analysis and plays a critical role in understanding the mapping from the space \(L^q((0, \infty))\) to \(C_0((0, \infty))\). When a function is continuous, it means there are no abrupt jumps or breaks in its values, making it smooth and predictable over its domain.

• The space \(C_0((0, \infty))\) comprises functions that not only are continuous but also decay to zero as you move towards infinity.
• This disappearing act "at infinity" is what allows operators, like \(T\), to map functions from one type of space to another while maintaining the essential smoothness and continuity properties.

To show an operator maps into this space, we often need to verify two things: 1. The operator transforms each function in a way that remains continuous.2. The transformed function vanishes as \(x\) approaches infinity, fulfilling the additional condition for the space \(C_0((0, \infty))\). In the original exercise, continuity stems from the properties of the integral and how they inherently smooth out functions over their domain.

Hardy's Inequality

Hardy's inequality is a powerful tool in functional analysis, especially when considering the boundedness of operators. This inequality provides a way of comparing the norms of different functions and assures us that under certain conditions, an integral transformation on a function is controlled by the norm of the function itself. In our context, Hardy's inequality ensures that the operator \(T\) is bounded from the Lebesgue space \(L^q((0, \infty))\) into the continuous space \(C_0((0, \infty))\).

• The boundedness here means that the action of \(T\) on any function \(f\) will not "blow up" or become uncontrollable.
• This is critical when moving between spaces of different natures, ensuring the operator does not produce erratic or wildly oscillating results from smooth inputs.

The inequality provides a condition, which is part of the statement \(p^{-1} + q^{-1} = 1\) that makes it possible to bound the output in the \(C_0\) space In conclusion, without Hardy's inequality, confirming the boundedness of operators like \(T\) would be challenging, as it creates a bridge between the norms of input functions and their transformed counterparts.