In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The considered cases encompass a ...very wide class of equations with isotropic growth conditions that are generalizations of the p-Laplacian and include equations with polynomial or exponential type growth. Some more general conditions have also been explored.
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the ...published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
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.
On uniform measures in the Heisenberg group Chousionis, Vasilis; Magnani, Valentino; Tyson, Jeremy T.
Advances in mathematics (New York. 1965),
03/2020, Volume:
363
Journal Article
Peer reviewed
Open access
We study uniform measures in the first Heisenberg group H equipped with the Korányi metric dH. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an ...affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. We also show that each 3-uniform measure which is supported on a vertically ruled surface is proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane, and that no quadratic x3-graph can be the support of a 3-uniform measure. According to a result of Merlo, every 3-uniform measure is supported on a quadratic variety; in conjunction with our results, this shows that all 3-uniform measures are proportional to spherical 3-Hausdorff measure restricted to an affine vertical plane. We establish our conclusions by deriving asymptotic formulas for the measures of small extrinsic balls in (H,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in H.
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.
The goal of this article is to establish local Lipschitz continuity of solutions for a class of sub-elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the ...growth and ellipticity condition is a natural generalization of the sub-elliptic p-Laplace equation and more general quasilinear equations with polynomial or exponential type growth.
Let L=−ΔHn+μ be a generalized Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sub-Laplacian, and μ is a nonnegative Radon measure satisfying certain conditions. In this paper, we ...first establish some estimates of the fundamental solution and the heat kernel of L. Applying these estimates, we then study the Hardy spaces HL1(Hn) introduced in terms of the maximal function associated with the heat semigroup e−tL; in particular, we obtain an atomic decomposition of HL1(Hn), and prove the Riesz transform characterization of HL1(Hn). The dual space of HL1(Hn) is also studied.
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