The order of growth of a harmonic function is determined by the growth of its gradient and conversely. We extend these results to solutions of certain subelliptic equations in John domains in Carnot groups. The modulus of the gradient is replaced by a local average of the horizontal gradient. In the harmonic case these quantities are equivalent. The proof uses recent integral inequalities assoc...

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show thatM is a hyperbolic manifold of constant sectional curvature −h 2 4 , provided M is asymptotically harmonic of constant h > 0.

We deal with the nonhomogeneous A-harmonic equation d∗A x, g du d∗h and the related conjugate A-harmonic equation A x, g du h d∗v. Some priori estimates about solutions to these equations are obtained, which generalize some existing results. Particularly, we obtain the same estimate given by Theorem 1 of Iwaniec 1992 for the weak solution to the first equation under weaker conditions by a simpl...

The quantum constraint equations for a relativistic three-dimensional harmonic oscillator are shown to find concise expression in terms of Lorentz covariant ladder operators. These ladder operators consist of two conjugate 4-vectors that are each constrained to generate three linearly independent combinations of ladder operator components for raising and lowering the eigenstates of the oscillat...

We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class...

It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a gr...