A reference on Besov-Sobolev spaces

From Lecture one, the Besov-Sobolev spaces defined there are also called weighted Dirichlet spaces, which have been studied by many authors. The following paper

Theory of Bergman Spaces in the Unit Ball of $C^n$,

which is published here:

Mémoires de la SMF 115 (2008), vi+103 pages

(Webpage: http://smf4.emath.fr/Publications/Memoires/2008/115/html/smf_mem-ns_115..... )

treated these spaces as weighted Bergman spaces. The above paper can be downloaded from the following link in Arxiv.org


Ruhan Zhao

Besov-Sobolev, Weighted Dirichlet and Weighted Bergman

Ruhan, thanks for pointing out the reference to Besov-Sobolev spaces, weighted Dirichlet space and weighted Bergman spaces. I will be sure to add this to the references in the lectures. There definitely are many connections between these concepts that could, and in some instances will, be highlighted in the course of the seminar

For the participants following along, I will briefly explain the connections between weighted Bergman space and the Besov-Sobolev spaces that I had defined in the lecture notes. I personally prefer the notation for the Besov-Soboev space $B_\sigma^2$ since it encapsulates all these different spaces at once.

Recall that we had defined the Besov-Sobolev space of order $\sigma\geq 0$ via the norm:
\left\Vert f\right\Vert_{B_\sigma^2}^2:=\left\vert f(0)\right\vert^2+\int_{D}\left\vert f'(z)\right\vert^2 (1-\left\vert z\right\vert^2)^{2\sigma} dA(z).

Currently in the Seminar, we are focusing on the choice of $\sigma=\frac{1}{2}$ since this is the Hardy space of analytic functions. In the later lectures we will turn to $0\leq\sigma<\frac{1}{2}$, and the value $\sigma=0$ (the Dirichlet space) in particular, since the story for these spaces will mimic that for the Hardy space. Calling these spaces "weighted Dirichlet spaces'' makes perfect sense since we want the derivative of $f$ to be integrable with respect to a certain radial weight. Note that it this situation we are measuring the size of $f$ by integrating its derivative values.

However, when $\frac{1}{2}<\sigma$ then the story is slightly different. It turns out we don't need the derivative to study the space of functions, and in fact only need the function to be integrable with respect to a weight. This change makes somethings much easier, but conversely makes other things much more difficult.

In this case the norm for $B_\sigma^2$ is simply an equivalent norm for the weighted Bergman space. Here, we define the weighted Bergman space $A_\beta^2$ to be those analytic functions normed by the following expression:
\left\Vert f\right\Vert_{A_\beta^2}^2:=\int_{D}\left\vert f(z)\right\vert^2 (1-\left\vert z\right\vert^2)^{\beta}dA(z).

Note that in the notation we have set up, when $\sigma=1$ then we have that $B_{1}^2=A^2$, the standard Bergman space, via an equivalent norm. Indeed, we have for $f(z)=\sum_{n=0}^\infty a_n z^n$ that
\left\Vert f\right\Vert_{A^2}^2=\sum_{n=0}^\infty\frac{1}{n+1}\left\vert a_n\right\vert^2,
\left\Vert f\right\Vert_{B_1^2}^2=\left\vert a_0\right\vert^2+ \sum_{n=1}^\infty\frac{4n^2}{n(n+1)(n+2)}\left\vert a_n\right\vert^2\approx \sum_{n=0}^\infty\frac{1}{n+1}\left\vert a_n\right\vert^2.
Both of these computations follow from evaluating $z^n$ in the appropriate space and an obvious orthogonality argument. The computation of the norm of $f\in B_1^2$ reduces to a computation of a beta function. But, these two expressions clearly show that we have $B_{1}^2=A^2$ with equivalent norms. More generally, when $\frac{1}{2}<\sigma$ we will have that $B_\sigma^2=A^2_{\beta(\sigma)}$ with equivalent norms, and so we can call the spaces "weighted Bergman spaces''.

The Bergman spaces are very interesting objects to study and one can ask similar questions as what appears in the lectures. For example, characterizing the Carleson measures for $A_\beta^2$ turns out to be relatively easy and is similar to the story for the Hardy space. Carleson measures can be characterized by simply testing on the reproducing kernel for the space (something that fails in the weighted Dirichlet spaces). But, characterizing the interpolating sequences (which will come up a little later) turns out to be much more subtle and requires more work (this characterization is a beautiful result of Seip). And, the interpolating sequences for the weighted Dirichlet spaces turn out to be more closely related to that for the Hardy space. While it is definitely good to be aware of the results for Bergman spaces, in general the story turns out to be different as compared the spaces when $0\leq\sigma\leq\frac{1}{2}$.

While we can study all these spaces of functions via the family of norms given by the Besov-Sobolev spaces, certain questions become easier/harder depending on the choice of $\sigma$. In my opinion, the key take away point is that, in the range $0\leq\sigma<\frac{1}{2}$ the spaces $B_\sigma^2$ all behave roughly the same, the space $\sigma=\frac{1}{2}$ is very special, and the spaces when $\frac{1}{2}<\sigma$ all behave roughly the same. But, the ranges less than $\frac{1}{2}$ and greater than $\frac{1}{2}$ are very different from each other. So, while we can refer to them as "weighted Dirichlet'' or "weighted Bergman'' one should be careful to make sure that it is known which parameter $\sigma$ is playing a role. It is for this reason that I prefer the notation for $B_\sigma^2$ to describe them all at the same time.