Fractional Sobolev Space: $H^{1/2}$ Interpolation Proof

Hey guys! Today, we're diving deep into a fascinating topic in functional analysis and partial differential equations: proving that the fractional Sobolev space $H^{\frac{1}{2}}$ is equivalent to the interpolation space $(L^2, H^1)_{\frac{1}{2},2;K}$ using the K-method. This is super useful in a bunch of areas, especially when you're dealing with regularity results for PDEs. So, let's break it down step by step!

Understanding Fractional Sobolev Spaces

First off, what exactly is a fractional Sobolev space? Well, Sobolev spaces, in general, are function spaces that incorporate information about the derivatives of functions, making them essential in the study of differential equations. When we talk about a fractional Sobolev space like $H^{\frac{1}{2}}$, we're essentially extending the idea of Sobolev spaces to allow for non-integer orders of differentiation. This might sound a bit weird at first, but it’s incredibly powerful. Think of it as being able to measure smoothness in a more refined way than just looking at integer derivatives.

To formally define $H^{\frac{1}{2}}(\mathbb{R}^n)$, we often use the Fourier transform. A function $f$ belongs to $H^{\frac{1}{2}}(\mathbb{R}^n)$ if its Fourier transform $\hat{f}(\xi)$ satisfies:

$$\int_{\mathbb{R}^n} (1 + |\xi|^2)^{\frac{1}{2}} |\hat{f}(\xi)|^2 d\xi < \infty$$

This definition tells us that $H^{\frac{1}{2}}$ consists of functions whose Fourier transforms decay sufficiently rapidly to ensure that the integral converges. The term $(1 + |\xi|^2)^{\frac{1}{2}}$ acts as a weight, penalizing high frequencies. Essentially, functions in $H^{\frac{1}{2}}$ have a certain amount of “smoothness” characterized by this fractional order.

Alternatively, we can define $H^{\frac{1}{2}}$ using a more direct approach involving differences. For instance, we can say that $f \in L^2(\mathbb{R}^n)$ belongs to $H^{\frac{1}{2}}(\mathbb{R}^n)$ if

$$\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|f(x) - f(y)|^2}{|x - y|^{n+1}} dx dy < \infty$$

This definition captures the idea that functions in $H^{\frac{1}{2}}$ don't change too rapidly. The integrand measures the average difference between the function's values at two points, scaled by the distance between those points. If the integral is finite, it means that the function is, in a sense, “half-differentiable”. Understanding these different characterizations of $H^{\frac{1}{2}}$ is crucial for grasping its properties and how it relates to other function spaces.

Introduction to Interpolation Spaces and the K-Method

Now, let's switch gears and talk about interpolation spaces. In general, interpolation theory provides a way to construct intermediate spaces between two given spaces. These intermediate spaces inherit properties from the original spaces in a controlled manner. In our case, we're interested in the interpolation space $(L^2, H^1)_{\frac{1}{2},2;K}$, which is constructed from $L^2$ (the space of square-integrable functions) and $H^1$ (the Sobolev space of functions with square-integrable first derivatives).

The K-method is a specific technique for defining interpolation spaces. It relies on the K-functional, which measures how well a function can be approximated by elements in the two original spaces. For $f \in L^2 + H^1$, the K-functional is defined as:

$$K(t, f) = \inf_{f = f_0 + f_1} (||f_0||_{L^2} + t ||f_1||_{H^1})$$

where $f_0 \in L^2$ and $f_1 \in H^1$. In simpler terms, we're trying to decompose $f$ into two parts: one that's well-behaved in $L^2$ and another that's well-behaved in $H^1$. The parameter $t$ controls the trade-off between the $L^2$ norm of $f_0$ and the $H^1$ norm of $f_1$. A small $t$ emphasizes the $L^2$ norm, while a large $t$ emphasizes the $H^1$ norm. The infimum is taken over all possible decompositions of f.

Using this K-functional, we define the interpolation space $(L^2, H^1)_{\theta, q;K}$ as the set of all functions $f \in L^2 + H^1$ such that:

$$||f||_{(L^2, H^1)_{\theta, q;K}} = \left( \int_0^{\infty} (t^{-\theta} K(t, f))^q \frac{dt}{t} \right)^{\frac{1}{q}} < \infty$$

In our particular case, we have $\theta = \frac{1}{2}$ and $q = 2$, so we're looking at functions $f$ for which

$$||f||_{(L^2, H^1)_{\frac{1}{2}, 2;K}} = \left( \int_0^{\infty} (t^{-\frac{1}{2}} K(t, f))^2 \frac{dt}{t} \right)^{\frac{1}{2}} < \infty$$

This integral essentially averages the K-functional over all scales $t$, giving us a measure of how well $f$ can be approximated by functions in $L^2$ and $H^1$ simultaneously. The interpolation space $(L^2, H^1)_{\frac{1}{2},2;K}$ consists of functions that strike a balance between belonging to $L^2$ and belonging to $H^1$.

Proving the Equivalence: $H^{\frac{1}{2}} = (L^2,H^1)_{\frac{1}{2},2;K}$

Alright, now for the main event: proving that $H^{\frac{1}{2}} = (L^2,H^1)_{\frac{1}{2},2;K}$. This involves showing that the norms are equivalent, meaning there exist constants $C_1$ and $C_2$ such that for all $f$:

$$C_1 ||f||_{H^{\frac{1}{2}}} \leq ||f||_{(L^2,H^1)_{\frac{1}{2},2;K}} \leq C_2 ||f||_{H^{\frac{1}{2}}}$$

This equivalence tells us that $H^{\frac{1}{2}}$ and $(L^2,H^1)_{\frac{1}{2},2;K}$ are essentially the same space, just with possibly different ways of measuring the