关于验证分段径向函数$f_\epsilon$满足V₀性质但不满足V∞性质的技术求助

Hey everyone, I'm currently stuck on working through a problem involving verifying two specific properties for a piecewise radial function, and I'd really appreciate some guidance on where to start. Let me break down the problem details first:

Background Definitions #

For any integrable function $f$, we define two key properties:

• $V_0$ property:
  $$ \lim_{r \to 0} \sup_{x \in \Bbb R^n} r^{-\lambda} \int_{B(x,r)} |f(y)|^p , dy = 0;$$
• $V_\infty$ property:
  $$ \lim_{r \to \infty} \sup_{x \in \Bbb R^n} r^{-\lambda} \int_{B(x,r)} |f(y)|^p , dy = 0,$$
  where $0 < \lambda < n$ and $1 \leqslant p < \infty$ are fixed constants.

My Goal #

I need to show that the piecewise function
$$ f_\epsilon(x) = \begin{cases} |x|^{\frac{\lambda - n}{p} + \epsilon} \quad \text{ if } |x| \leqslant 1 \ |x|^{\frac{\lambda-n}{p}} \quad , , , \text{ if } |x| \geqslant 1 \end{cases},$$
with $\epsilon > 0$, satisfies the $V_0$ property but does not satisfy the $V_\infty$ property.

My Current Approach & Stumbling Block #

Normally, when dealing with radial functions, my go-to intuition is to check if the function is radially decreasing. If it is, I can ignore the $\sup_{x \in \Bbb R^n}$ term entirely and just work with integrals over the ball $B(0,r)$, which simplifies things a lot. But in this case, I'm pretty sure this function isn't radially decreasing, so that strategy won't work here.

The only other approach I could think of was to directly compute the integral over $B(x,r)$ by splitting it into two parts based on the piecewise definition of $f_\epsilon$. I ended up with:
$$ \int_{B(x,r)} |f_\epsilon(y)|^p , dy = \int_{B(x,r) \cap B(0,1)} |y|^{\lambda - n + p\epsilon} , dy + \int_{B(x,r) \setminus B(0,1)} |y|^{\lambda - n} , dy$$
(Note: I simplified the exponents by raising $f_\epsilon$ to the $p$-th power, which makes the integrals a bit cleaner.)

But after splitting the integral like this, I'm stuck. I don't see how to proceed from here to analyze the limits as $r \to 0$ and $r \to \infty$, especially with that $\sup_{x \in \Bbb R^n}$ term still hanging around.

I'm mainly looking for a clear starting point or a different strategy to tackle this problem. Thanks in advance for any help or insights you can share!

备注：内容来源于stack exchange，提问作者xyz