In this paper, we investigate the relationship between the shortest bisector and the area on 2-sphere. Let L denote the length of the shortest bisector and A the area of the 2-sphere. We show that if the inequality L^2 ≤π A holds for 2-spheres for which no shortest bisector is a simple closed geodesic, then it also holds for all 2-spheres.
In this paper, we have established some comprehensive results regarding the bounds of the zeros of quaternionic polynomials and obtained significant extensions of certain classical results. Our results also yield proper refinements of some known zero bounds.
Recent work has established necessary and sufficient conditions for the boundedness of inhomogeneous Calderón–Zygmund operators on the local Hardy spaces h^p(ℝ^n) in the range max{n/(n+ε ),n/(n+δ )}
We study stability and Hölder regularity up to the boundary for the quaternionic m-Hessian equation H_m(u):=(Δ u)^m∧β ^n-m=f dV on bounded strongly m-hyperconvex domains in ℍ^n . We prove the capacity estimate Vol (K)≤ C_τCap_m(K,Ω )^τ , 1<τ <2n/2n-m, for every compact set K⋐Ω , using Sroka’s quaternionic Monge–Ampère L^q , q>2 , estimate and the Maclaurin comparison between the n- and m-Hessian operators. It implies that every density f∈ L^p , p>2n/m , is dominated by m-Hessian capacity with a positive gain. A sublevel capacity inequality and a De Giorgi iteration then yield one-sided L^1 -to- L^∞ stability and density stability. If solvable data have uniformly bounded L^p densities converging in L^1 and boundary values converging uniformly, the corresponding solutions converge uniformly to a unique continuous weak solution with the limiting Hessian measure and boundary value. Assuming smooth solvability for strictly positive smooth data, we obtain continuous solvability for arbitrary nonnegative L^p densities and continuous boundary data. If the boundary datum and a subsolution are C^0,a , then the solution is Hölder continuous on Ω with every exponent 0<γ
