Absolutely summing operators on Bergman spaces over convex domains of finite type

Abstract

Let $Ω\subset\mathbb C^n$ be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For $1<p<\infty$ and $1\le q,r<\infty$, we characterize absolutely $r$-summing Carleson embeddings $A^p(Ω)\to L^q(μ)$ by diagonal operators formed from normalized masses on McNeal balls. For $1<p,q<\infty$ and $1\le r<\infty$, we obtain a corresponding big Hankel criterion using local $L^q$ distances to holomorphic functions, with the appropriate volume weights. Both criteria give upper and lower norm estimates. The proofs combine uniform local nuclear estimates, kernel synthesis, and an analytic decomposition at two fixed scales. Local Taylor polynomials give explicit rank and error bounds for the embeddings. These bounds transfer to general Hankel operators when $p=2$ or $q=2$. Suitable atomic embeddings have the same best approximation errors as their diagonal models, up to constants. A family of Hankel operators contains a complemented copy of the diagonal ideal and has matching approximation rates for power sequences in the Hilbert case. For positive Toeplitz operators on finite-type ellipsoids, we compute exact spectral asymptotics in three regimes, including the critical logarithmic factor and the leading constants for geometric symbols. These asymptotics give optimal approximation rates in the summing norm. Applications to composition and Volterra operators include the target endpoint $q=1$.