Question

For which real values of \(a\) and \(b\) is \(|x|^{a}|\log | x||^{b}\) integrable over \(\left\\{x \in \mathbb{R}^{n}\right.\) : \(\left.|x|<\frac{1}{2}\right\\}\) ? Over \(\left\\{x \in \mathbb{R}^{n}:|x|>2\right\\}\) ?

Short answer

\(a > 1-n\), \(b \le 0\) for \(|x| < \frac{1}{2}\); \(a < -n\) for \(|x| > 2\).

Step-by-step solution

Understanding the Problem

The problem requires determining conditions on the parameters \(a\) and \(b\) such that the function \(|x|^{a}|\log |x||^{b}\)\ is integrable over two sets: one where \(|x| < \frac{1}{2}\) and another where \(|x| > 2\). We will need to analyze both regions separately to find suitable values of \(a\) and \(b\).

Region 1: Analyzing \(|x| < \frac{1}{2}\)

In this region, the function \(|x|^{a}|\log |x||^{b}\) is integrable if the integral \(\int_{|x|<\frac{1}{2}} |x|^{a} |\log |x||^{b} \, dx\) converges. The key is the behavior near \(x = 0\), where \(|x| < \frac{1}{2}\) implies a small \(|x|\). As \(|x|\) approaches zero, \(\log |x|\) becomes large and negative. We focus on whether \(|x|^{a}\) can counteract \(|\log |x||^{b}\) at zero.

Solving for Integrability in \(|x| < \frac{1}{2}\)

We approximate the behavior near zero. For convergent integrals, the critical condition is \(a + n - 1 > 0\), where \(n\) is the dimension since the function behaves like a product of a power of \(|x|\) and a potentially singular function. Additionally, for the logarithmic term to be integrable, \(b\) must be less than zero (i.e., \(\int |\log |x||^{b}\) should converge as \(|x| \to 0\)). Therefore, \(b \le 0\) is required to prevent the logarithm from diverging too aggressively.

Region 2: Analyzing \(|x| > 2\)

The function \(|x|^{a}|\log |x||^{b}\) is integrable in this region if \(\int_{|x|>2} |x|^{a} |\log |x||^{b} \, dx\) is finite. Here, as \(|x| \to \infty\), the power term \(|x|^{a}\) can either decay or grow depending on \(a\). Similar to the previous analysis, we need \(a + n - 1 < 0\) for the integral to converge at infinity, and the logarithmic term should not diverge, so \(b\) can be any finite real value.

Solving for Integrability in \(|x| > 2\)

In the context of \(|x| > 2\), the convergence condition simplifies to ensure the decay of the integrand. Hence, \(a < -n\) is necessary for the convergence of \(|x|^{a}\) to counteract \(|x|\)'s growth beyond 2. The logarithmic term does not add stronger restrictions here as it diminishes beyond a certain points due to \(|\log |x||\).

Conclusion

For \(|x| < \frac{1}{2}\), the conditions are \(a > 1 - n\) and \(b \le 0\). For \(|x| > 2\), the condition is \(a < -n\). These constraints ensure the function is integrable within the given regions by accommodating both the power and logarithmic behaviors effectively.

Key concepts

Integrability Conditions

When we talk about integrability, we are focusing on whether an integral converges or diverges. This means checking if the value of the integral is finite or infinite. In real analysis, finding conditions for integrability involves understanding how the function behaves over specific regions, such as near zero or towards infinity.
To determine integrability, we often analyze the behavior of the function as it approaches critical points such as zero or infinity. If a function grows too fast or too slow at these points, it might not be integrable.

For the function \(|x|^a|\log |x||^b\) over regions like \(|x| < \frac{1}{2}\), we check how the power \(|x|^a\) handles a small \(|x|\) and whether the logarithmic term \(|\log |x||^b\) becomes too large. Basically, \(a + n - 1 > 0\) ensures the function remains small enough to keep the integral finite, while \(b \leq 0\) keeps the logarithmic growth in check.
On the other hand, for \(|x| > 2\), \(a < -n\) makes sure that the function decays quickly enough as \(|x|\) increases to infinity, ensuring a convergent integral.

Power and Logarithmic Functions

Both power and logarithmic functions play a key role in understanding the integrability of functions. These functions have distinct growth behaviors that we need to grasp fully to manage convergence.
Power functions, denoted as \(|x|^a\), indicate a growth or decay based on the power \(a\). When \(a\) is positive, \(|x|^a\) grows as \(|x|\) increases, while if \(a\) is negative, it decays. This behavior is crucial when determining integrability, especially as \(|x|\) tends towards zero or infinity.

• At \(|x| < \frac{1}{2}\), the concern is whether \(|x|^a\) can counteract the rise of \(|\log |x||^b\) near zero.
• For \(|x| > 2\), \(|x|^a\) should decay swiftly enough to manage the potential unboundness of \(|x|\).

Meanwhile, logarithmic functions like \(|\log |x||^b\) can become very large as \(|x|\) approaches zero or gradually small as \(|x|\) moves away from zero. The exponent \(b\) determines this growth rate:

• If \(b \leq 0\), the logarithm behaves more tamely around zero, which helps in maintaining a convergent integral.
• Moving towards \(|x| > 2\), the effect of \(|\log |x||\) diminishes, especially for larger \(|x|\).

Convergent Integrals

Understanding convergent integrals is essential when working with functions over unbounded domains or regions where behavior changes drastically. A convergent integral is one where the accumulated area under the curve is finite. This is significant as it determines whether the function can be sufficiently 'controlled' over the given region.
In the context of \(|x|^a|\log |x||^b\), convergence depends on both \(a\) and \(b\) as well as the dimension \(n\). For the region \(|x| < \frac{1}{2}\), it is important that \(a + n - 1 > 0\) to tame the rise of the function as \(|x|\) approaches zero.

This inequality ensures the power term doesn't allow the function to grow unbounded near zero. Additionally, \(b \leq 0\) means the logarithmic function doesn't escalate too rapidly, making the integral more likely to converge.
In \(|x| > 2\), convergence requires \(a < -n\), which allows \(|x|^a\) to decay as \(|x|\) approaches infinity. This decay ensures the function's value shrinks fast enough to keep the integral finite over its domain.
Mastering the understanding of these conditions allows students to handle more complex integrals by anticipating and compensating for potentially divergent behaviors.