2023/03/14 by David Kalaj, João P. G. Ramos, Kalaj, David +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2303.08069
openalex publication_date 2023/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Assume that Δh is the hyperbolic Laplacian in the unit ball \mathbbB and assume that Φn is the unique radial solution of Poisson equation Δh log Φn =-4 (n-1)2 satisfying the condition Φn(0)=1 and Φn(ζ)=0 for ζ∈ ∂\mathbbB. We explicitly solve the question of maximizing Rn(f,Ω)= \frac∫Ω|f(x)|2 Φnα(|x|) dτ(x)‖f‖2B2α, over all f \inB2α and Ω⊂ \mathbbB with τ(Ω) = s, where dτ denotes the invariant measure on \mathbbB, and ‖f‖_B2α2 = ∫_\mathbbB |f(x)|2 Φnα(|x|) dτ(x) < ∞. This result extends the main result of Tilli and the second author \citeramostilli to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with several additional technical difficulties arising from the definition of the weights Φn through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-sunharmonic functions and one for the Wavelet transform is only available in dimension one.