Fractional type operators on the Heisenberg group

2024/04/08 by Rocha, Pablo
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.05195

Abstract

Let ρ(⋅) be the Koranyi norm on the Heisenberg group ℍⁿ ≡ (ℝ²ⁿ × ℝ, ⋅ ) defined by ρ(x,t) = ( |x|⁴ + 16 t² )^1/4, (x,t) ∈ ℍⁿ. For 0 ≤ α< Q:=2n+2, m ∈ ℕ ∩ (1 - \fracαQ, ∞ ), and m positive constants α₁, ..., α_m such that α₁ + ⋅ ⋅ ⋅ + α_m = Q - α, we consider the following generalization of the Riesz potential on ℍⁿ T_{α, m}f(x,t) = ∫_ℍⁿ f(y,s) ∏_j=1^m ρ((A_j y, r_j^-2 s)^-1 ⋅ ( x, t))^-α_j dy ds, where, in the case 0 < α< Q, the A_j's are matrices belonging to Sp (2n, ℝ) ∩ SO(2n) and r_j = 1 for every j=1, ..., m; for α= 0, we consider A_j = r_j^-1 I_{2n × 2n} for every j=1, ..., m, where the r_j's are positive constants such that r_i² - r_j² ≠ 0 if i ≠ j. In this note we study the behavior of these operators on variable Hardy spaces in ℍⁿ.

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