Let
when
, and
when
.
We obtain the second-order horizontal Sobolev
-regularity of
-harmonic functions in the Heisenberg group
.
This improves the known range of
obtained by Domokos and Manfredi in ...2005.
For a strictly convex set K⊂R2 of class C2 we consider its associated sub-Finsler K-perimeter |∂E|K in H1 and the prescribed mean curvature functional |∂E|K−∫Ef associated to a continuous function f. ...Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, we prove that their characteristic curves are of class C2 and that this regularity is optimal. The result holds in particular when the boundary of E is of class C1.
This paper is devoted to a class of integral type Brezis-Nirenbreg problems on the Heisenberg group. They are a class of nonlinear integral equations on the bounded domains of Heisenberg group and ...related to the CR Yamabe problems on CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, the nonexistence and existence results are obtained.
We prove Lp estimates for a class of Littlewood-Paley functions on homogeneous groups including the Heisenberg groups under a sharp integrability condition on the kernels. The results will be shown ...by methods without use of Fourier transform estimates.
We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, via Krasnoselskii's genus. Finally, an ...interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.
Let Hn be the Heisenberg group, Q=2n+2 be the homogeneous dimension of Hn, we study the existence of solutions for the following Q-Laplacian elliptic ...system{−K(∫Ω|∇Hnu|Qdξ)ΔQu=λGu(ξ,u,v)ρ(ξ)℘inΩ;−K(∫Ω|∇Hnv|Qdξ)ΔQv=λGv(ξ,u,v)ρ(ξ)℘inΩ;u=0,v=0,on∂Ω, where Ω is an open, smooth and bounded subset of Heisenberg group Hn, K is a Kirchhoff type function, 0≤℘<Q and λ is a positive parameter, and nonlinear terms Gu,Gv have critical exponential growth behave like exp(β|s|QQ−1) as |s|→+∞ with some β>0.
We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2 bounded on vertical planes, with uniform ...constants, then it is also L2 bounded on all intrinsic graphs of compactly supported C1,α functions over vertical planes.
In particular, the result applies to the operator R induced by the kernelK(z)=∇H‖z‖−2,z∈H∖{0}, the horizontal gradient of the fundamental solution of the sub-Laplacian. The L2 boundedness of R is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that the intrinsic graphs mentioned above are non-removable. Apart from subsets of vertical planes, these are the first known examples of non-removable sets with positive and locally finite 3-dimensional measure.
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