Fourier-analytic derivation of Hardy-type oscillatory identity for p-circle domains by generalized Bessel functions

Using Fourier-analytic methods, we derive a Hardy-type oscillatory identity for the family of planar domains ∈ R² |x₁|ᵖ + |x₂|ᵖ ≤ rᵖ\ { x ∈ R 2 ∣ | x 1 | p + | x 2 | p ≤ r p } with positive p satisfying (2/p) ∈ N ( 2 / p ) ∈ N . This identity generalizes the classical Hardy’s identity for the Euclidean disk, arising from the Poisson summation formula. For these domains of non-smooth curves (including the astroid), the rotational symmetry of the Fourier transform is lost, so it cannot be expressed using classical Bessel functions. To address this, we introduce a family of generalized Bessel functions naturally arising from the Fourier transform of p-radial functions. These functions capture the oscillatory structure induced by the loss of rotational symmetry and provide the analytic framework for Hardy-type identities between these domains. While classical motivations relate to estimating lattice point errors in number theory, we focus here on deriving the identities themselves and understanding their structure.