Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators

Let ν = (ν ₁, … , ν ₙ) ∈ (-1/2, ∞ )ⁿ ν = ( ν 1 , … , ν n ) ∈ ( - 1 / 2 , ∞ ) n , with n ≥ 1 n ≥ 1 , and let Δ _ν Δ ν be the multivariate Bessel operator defined by Δ ν = -∑ ⱼ₌₁ⁿ( ∂ ²/∂ xⱼ² - ν ⱼ² - 1/4/xⱼ² ) . Δ ν = - ∑ j = 1 n ∂ 2 ∂ x j 2 - ν j 2 - 1 / 4 x j 2 . In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator Δ _ν Δ ν . We then study the higher-order Riesz transforms associated with Δ _ν Δ ν . First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with Δ _ν Δ ν .