Isoperimetric inequality from the poisson equation via curvature

Abstract

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L2‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L∞‐norm of the gradient Du can be bounded by the Lorentz norm LQ,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$. © 2011 Wiley Periodicals, Inc.

Gradient estimates for heat kernels and harmonic functions

“…In this section, following the central idea of [74,75] and using Theorem 3.2, we show that (RH p ) for p > 2 yields a Sobolev inequality or an isoperimetric inequality. Combining this and Theorem 1.9, we find a new necessary condition for quantitative regularity of harmonic functions and heat kernels, and for boundedness of Riesz transforms.…”

“…Applying the methods from [74,75], we show in Theorem 6.1 below that on a non-compact metric measure space (X, d, µ, E ) endowed with a "carré du champ" and satisfying (D Q ) and (UE), if additionally for some p 0 ∈ (2, ∞), (P p 0 , loc ) and one of the conditions (…”

“…Applying the methods from [74,75], we show in Theorem 6.1 below that on a non-compact metric measure space (X, d, µ, E ) endowed with a "carré du champ" and satisfying (D Q ) and (UE), if additionally for some p 0 ∈ (2, ∞), (P p 0 , loc ) and one of the conditions (RH p 0 ), (R p 0 ), (G p 0 ) holds, then the Sobolev inequality (S q,p ′ 0 ), where p ′ 0 < 2 is the Hölder conjugate of p 0 , q ≥ p ′ 0 satisfying 1/p ′ 0 − 1/q < 1/Q, is valid. An analogue for the isoperimetric inequality (p 0 = ∞) will also be established in Theorem 6.3.…”

An analytic proof of the planar quantitative isoperimetric inequality

“…In this paper, following the approach from [17], we consider the problem (2) in the plane, and give an estimation of the upper bound of the constant C via maximizing the L ∞ -norm of the gradient of solutions to the Poisson equation. Theorem 1.…”

A New Reversed Version of a Generalized Sharp Hölder's Inequality and Its Applications

“…For detailed expositions, the interested reader may consult [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”