Question

Let $f$ be a continuous function on $$\begin{gathered} {\mathbb{R}}^n\mathrm{\setminus } \\ 0 \end{gathered}$$ that is homogeneous of degree $-n$ (i.e., $f(rx)=r^{-n}f(x))$ and has mean zero on the unit sphere (i.e., $\int fd\sigma =0$ where $\sigma$ is surface measure on the sphere). Then $f$ is not locally integrable near the origin (unless $f=0)$, but the formula $$⟨PV(f),\varphi ⟩=\underset{ϵ\to 0}{lim}{\int }_{|x|>ϵ}f(x)\varphi (x)dx(\varphi \in C_c^{\mathrm{\infty }})$$ defines a distribution $PV(f)-"PV"$ stands for "principal value" - that agrees with $f$ on $$\begin{gathered} {\mathbb{R}}^n\mathrm{\setminus } \\ 0 \end{gathered}$$ and is homogeneous of degree $-n$ in the sense of Exercise 9 .

Short answer

The principal value defines a distribution, capturing $f$'s behavior away from the origin.

Step-by-step solution

Understanding the problem

We need to determine whether the given function $f$ can define a distribution through a principal value integral. We know $f$ is homogeneous of degree $-n$ and has mean zero on a sphere. We aim to show that $f$ defines a distribution $⟨PV(f),\varphi ⟩$.

Understanding homogeneity condition

Since $f$ is homogeneous of degree $-n$, for any $r>0$ and vector $x\in {\mathbb{R}}^n$, we have $f(rx)=r^{-n}f(x)$. This property will help us in considering integrals over domains scaled by $r$.

Applying mean zero condition

The condition $\int fd\sigma =0$ implies that $f$ averages out to zero over the unit sphere. This property will be crucial in analyzing how $f$ behaves under integrals with small neighborhoods removed, characteristic of principal value integrals.

Recognize non-local integrability near origin

Despite not being locally integrable near the origin, given that $f$ can blow up as $|x|o0$, we can't perform direct integration. Instead, we use the principal value to handle this singularity by computing limits as $ϵ\to 0$.

Setting up the principal value integral

We use the formula $⟨PV(f),\varphi ⟩=\underset{ϵ\to 0}{lim}{\int }_{|x|>ϵ}f(x)\varphi (x)dx$, which effectively ignores a small ball around the origin where $f$ might not be integrable.

Show homogeneity in distribution sense

To show the distribution is homogeneous of degree $-n$, we verify that scaling $\varphi (rx)$ in the integral gives $r^{-n}⟨PV(f),\varphi ⟩$, consistent with the original homogeneity of $f$.

Show $PV(f)$ satisfies distribution properties

Verify $⟨PV(f),\varphi ⟩$ defines a linear functional on the space of test functions $C_c^{\mathrm{\infty }}$. By linearity and boundedness of $⟨PV(f),\varphi ⟩$ with respect to the test function space topology, it is indeed a distribution.

Conclude behavior at origin

Since the distribution is well-defined but $f$ itself is not integrable near the origin, $PV(f)$ effectively captures the behavior of $f$ away from the origin while extending to a meaningful entity at points near and including the origin.

Key concepts

Homogeneous Functions

In mathematics, a homogeneous function is a special type of function that exhibits consistent scaling behavior. Specifically, a function $f$ is said to be homogeneous of degree $n$ if, for any vector $x$ in ${\mathbb{R}}^n$ and any positive real number $r$, the relation $f(rx)=r^{n}f(x)$ holds true. This property implies that when the input variables are scaled by a factor, the function itself scales by a specific power of that factor.

Now, let's consider our given exercise where the function $f$ is described as homogeneous of degree $-n$. This means that under scaling, the function contracts or shrinks as $r$ increases because of the negative degree. Thus, $f(rx)=r^{-n}f(x)$, essentially describing how the function behaves in terms of size and growth when moving farther from or closer to the origin.

Understanding these properties is crucial as it allows us to deal with certain mathematical scenarios where direct evaluation might be challenging, such as integration over infinite domains or regions near singularities.

Principal Value

In the context of integrals, the 'principal value' is a technique used to deal with situations where a function is not well-behaved or exhibits singularities, like nearby the origin. This approach helps in defining integrals for functions that otherwise might be non-integrable due to infinite or undefined behavior near certain points.

In our problem, the principal value in connection with function $f$ is introduced as a way to overcome the non-local integrability near the origin. The principal value of an integral is symbolically represented as $⟨PV(f),\varphi ⟩=\underset{ϵ\to 0}{lim}{\int }_{|x|>ϵ}f(x)\varphi (x)dx$.

This effectively circumvents the singularity at the origin by removing a small ball of radius $ϵ$, allowing us to compute the behavior of $f$ away from its problematic zone. Consequently, the principal value integral captures the essence and behavior of $f$ over its functional domain, despite the integrability issues near the origin.

Locally Integrable

The term 'locally integrable' in mathematical analysis refers to functions that can be integrated over every compact subset of a domain. For a function to be locally integrable, it must produce finite values when integrated over any finite-sized region within its domain.

However, as seen in the exercise, even if a function is well-behaved over several parts of its domain, it may still have issues close to certain points—in our case, the origin. Here, the function $f$ is not locally integrable at the origin due to its singularity; i.e., $f$ becomes problematic where $|x|\to 0$.

This challenge underscores the need for approaches like the principal value, which allows us to interpret and handle the behavior of such functions, enabling them to be used within distributions. Thus, even functions that misbehave at points (like the origin) can be analyzed and manipulated effectively using strategies like principal values to extend their applicability.