Sharp integral inequalities for harmonic functions

Abstract

Motivated by Carleman's proof of isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper half-space. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville-type theorems.

Q‐Curvature on a Class of Manifolds with Dimension at Least 5

“…It is worthwhile to note the similarity of Theorem 1.2 and 1.3 to classical Yamabe problem ( [LP, S]) and the integral equation considered in [HWY1,HWY2]. Indeed, the formulation of our approach follows that of [HWY2].…”

Subcritical approach to sharp Hardy–Littlewood–Sobolev type inequalities on the upper half space

“…In fact, for α = 2, (1.5) was first obtained by Hang, Wang and Yan [8]; and it can be seen as the higher dimensional version of Carleman's inequality and is closely related to the sharp isoperimetric inequality. Hang, Wang and Yan's inequality was also extended by Chen [2] with different kernel function (for viewing fractional Lapalacian operator on the boundary of the upper half space as a special map from Dirichlet data to Neumann data).…”

“…Hang, Wang and Yan's inequality was also extended by Chen [2] with different kernel function (for viewing fractional Lapalacian operator on the boundary of the upper half space as a special map from Dirichlet data to Neumann data). Chen's approach is along the same line as that in [8].…”

On Beckenbach–Radó type integral inequalities

“…In the classical case when n = 2, the claim holds for log f , see [5]. Our argument is similar to that of [5] except for the fact that instead of Carleman's inequality for analytic functions in [7] we apply sharp inequalities for harmonic functions by Hang et al [15,16]. …”

“…On page 20 of his book [29] Radó asked whether a similar characterization would be true in higher dimensions as well. In this work we give an answer to this question using recent sharp inequalities for harmonic functions by Hang et al [15,16]. We show that if f : R n → (0, ∞), n ≥ 3, is a continuous function, then f (n−2)/2 is subharmonic in R n if and only if B(x,r ) f (y) n dy…”