Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group

We prove the Hölder continuity for weak solutions to parabolic p-Laplace equations on the Heisenberg group. We deduce this result while considering an optimal tail condition.

💡 Research Summary

The paper establishes local Hölder continuity for weak solutions of a non‑local parabolic p‑Laplace equation posed on the Heisenberg group ℍⁿ. The equation under consideration is

∂ₜu(x,t) + p.v. ∫_{ℍⁿ} K(x,y,t) |u(x,t)−u(y,t)|^{p‑2}(u(x,t)−u(y,t)) dy = 0,

where the kernel K satisfies the two‑sided estimate

λ′ |y⁻¹∘x|^{-(Q+sp)} ≤ K(x,y,t) ≤ Λ′ |y⁻¹∘x|^{-(Q+sp)}

for almost every (x,y,t) and fixed constants λ′,Λ′>0. Here Q=2n+2 is the homogeneous dimension of ℍⁿ, s∈(0,1) and p>1. The authors assume that the solution u belongs to the natural energy space

u ∈ L^{p}{loc}(0,T; W^{s,p}{loc}(Ω)) ∩ L^{p‑1}{loc}(0,T; L^{p‑1}{sp}(ℍⁿ)) ∩ C_{loc}(0,T; L^{2}_{loc}(Ω)),

and that the “non‑local tail” of u satisfies an optimal integrability condition: for some ε>0

∫{0}^{T} \Big( ∫{ℍⁿ\B_R} |u(y,t)|^{p‑1} |y⁻¹∘x₀|^{-(Q+sp)} dy \Big)^{1+ε} dt < ∞

for every cylinder centred at (x₀,t₀). This L^{1+ε}{loc} tail condition is weaker than the L^{∞}{loc} tail used in earlier works and is shown to be sufficient for Hölder regularity.

The main result (Theorem 1.1) states that any locally bounded weak solution satisfying the above tail condition is locally Hölder continuous in space‑time. More precisely, there exist constants γ,β∈(0,1) depending only on the data (p,s,Q,λ′,Λ′) and ε such that for any parabolic cylinder Q_{r}=B_{r}(x₀)×(t₀−θ r^{sp},t₀] with θ>0,

ess osc_{Q_{r}} u ≤ γ ω (r/R)^{β},

where ω is a combined bound involving the supremum of u in a larger cylinder and the tail term. The exponent β is explicitly given as

β = min{ sp·ln(1−η)/ln((1−η)^{p‑2}λ sp), sp·ln(1−η)/ln( \barγ^{2‑p} λ sp), ε sp/