Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc

Bi-parameter potential theory and Carleson measures for the Dirichlet space on the bidisc, Discrete Analysis 2023:22, 58 pp.

Carleson measures arise naturally when considering harmonic or holomorphic extensions from the boundary of a domain to the interior of the domain. For instance, suppose one has an $L^p$ function $f$ on the real line for some $1 \leq p < \infty$, and let $u$ be its standard harmonic extension to the upper half-plane, given by convolution with the Poisson kernel. The function $u$ will not be expected to be an $L^p$ function on the upper half-plane with respect to the usual Lebesgue measure, but turns out to be an $L^p(\mu)$ function for other measures $\mu$ on the upper half-plane. Indeed, the celebrated Carleson embedding theorem asserts that a measure $\mu$ has this property if and only if it is what is now known as a “Carleson measure”, which means that for every interval $[x-r,x+r]$ on the real line, the measure $\mu( [x-r,x+r] \times [0,r])$ assigned to the rectangle $[x-r,x+r] \times [0,r]$ is bounded by a constant times the length of the interval. Analogous results hold when $f$ is an $H^p$ function on the circle (roughly speaking, an $L^p$ function with a holomorphic extension to the unit disc); this theorem has many applications in complex analysis and harmonic analysis, for instance to the corona problem of determining the spectrum of the Hardy space $H^\infty$ (viewed as a Banach algebra), and in the theory of functions of bounded mean oscillation (BMO). They are also instrumental in describing multipliers of Hardy spaces $H^p$: holomorphic functions $g$ with the property that multiplication by $g$ is a bounded operation on $H^p$, though the description is a bit more complicated, in which the intervals $[x-r,x+r]$ have to be replaced with finite unions of intervals, and the notion of length replaced with the more complicated notion of “Bessel capacity” from potential theory.

The original Carleson embedding theorem can be extended to higher dimensions without much difficulty (intervals get replaced by balls, and certain exponents get adjusted accordingly). However, when dealing with holomorphic functions of several complex variables, defined on the polydisc (the product of several copies of the unit disc), the situation becomes more delicate, even for functions of two complex variables on the bidisc, basically because intervals get replaced by axis-parallel rectangles of arbitrary eccentricity, which can no longer be interpreted as single-parameter balls in a metric space, but are instead genuinely bi-parameter objects. Many of the classical harmonic analysis techniques that can handle the geometry of single-parameter metric balls will fail in the bi-parameter setting if adapted naively, but over time many authors have come up with ingenious substitutes for the classical theory that can handle bi-parameter or multi-parameter settings. Often, even just stating the right generalizations correctly is a significant part of the problem.

In this paper, the authors state and prove the analogue of the Carleson embedding theorem and the multiplier characterization for the bidisc, where in both cases the characterization involves the Bessel capacity of finite unions of rectangles. This is achieved by first discretizing the problem to an analogous problem on the bitree (the product of two infinite dyadic trees), and then by carefully developing a bi-parameter capacity theory first on the bitree, and then on the bidisc. There are many technical subtleties, as some (but not all) of the classical one-parameter techniques are known to fail in the multi-parameter setting.