= Carleson embedding theorem on the disk
{c}
{title2=$H^1(\mathbb D)\hookrightarrow L^1(\mu)\iff\mu\text{ is Carleson}$}
A positive <Borel measure> on the <unit disc> satisfies $\int|f|\,d\mu\le N\|f\|_1$ for every <H1 space on the unit disk> function exactly when it is a <Carleson measure>, with comparable optimal constants. The <normalized squared Cauchy kernel> tests necessity. For sufficiency, the boundary shadow of every point with $|f(z)|>t$ lies inside a component arc of $\{Nf>t\}$. The <geodesic-cap and Carleson-box comparison> bounds the measure of each shadow region by its component length, giving $\mu\{|f|>t\}\le C\|\mu\|_{\rm Carl}m\{Nf>t\}$. The <layer cake representation> and <H1 non-tangential maximal inequality on the disk> give the embedding. The broader kernel and covering mechanism is discussed in https://www.math.ucla.edu/~tao/247b.1.07w/notes6.pdf[Terence Tao's harmonic-analysis notes, Exercise 2 after Lemma 4.1]; the analytic endpoint here uses the fractional-power argument just given.
Back to article page
