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September 5, 2025

Stability and boundedness of weak solutions for fractional sub-Laplace equations

Key Points

Weak solutions to the Dirichlet problem show stability under fractional sub-Laplacian implications.
Boundedness of weak solutions is established for a non-local integro-differential operator.
Utilization of Vitali's convergence theorem and Prokhorov's theorem enhances understanding of solution behavior.
This research focuses on the Heisenberg group, enriching mathematical understanding in bounded domains.

Abstract

In this paper, we study the Dirichlet problem {Lsu(ξ)=f,ξ∈Ω,u(ξ)=g,ξ∈Hn∖Ω related to non-local integro-differential operator on the Heisenberg group Hn, where s∈(0,1), Ω⊂Hn is a bounded domain and Ls represents the fractional sub-Laplacian on Hn. By employing Vitali's convergence theorem and Prokhorov's theorem, we examine the stability of weak solutions to the integro-differential problem. Ultimately, the L∞ boundedness of weak solutions in relation to this Dirichlet problem is also established.

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