In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Be...In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.展开更多
A complementarity hypothesis concerning outsider and insider perspectives of a gargantuan black hole is proposed. The two thought experiments presented herein are followed by a brief discussion of a new interpretation...A complementarity hypothesis concerning outsider and insider perspectives of a gargantuan black hole is proposed. The two thought experiments presented herein are followed by a brief discussion of a new interpretation of black hole interior “space-and-time-reversal”. Specifically, it is proposed that the “singularity” space of the black hole interior is time-like and the expansion time of the black hole interior is space-like. The resemblance of this new insider interpretation to our own expanding and redshifting big bang universe is compelling.展开更多
文摘In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.
文摘A complementarity hypothesis concerning outsider and insider perspectives of a gargantuan black hole is proposed. The two thought experiments presented herein are followed by a brief discussion of a new interpretation of black hole interior “space-and-time-reversal”. Specifically, it is proposed that the “singularity” space of the black hole interior is time-like and the expansion time of the black hole interior is space-like. The resemblance of this new insider interpretation to our own expanding and redshifting big bang universe is compelling.
