SELF-IMPROVEMENT OF (θ, p) POINCAR´E INEQUALITY FOR p > 0

Abstract

Classical Poincar´e (θ, p)-inequality on Rn   1 μ(B) ˆ B  f(y) − 1 μ(B) ˆ B f dμ  θ dμ(y)   1/θ . rB   1 μ(B) ˆ B |∇f|p dμ 
   1/p , (rB is the radius of ball B ⊂ Rn) has a self-improvement property, that is (1, p)-inequality, 1 < p < n, implies the «stronger»  q, p)-inequality (Sobolev-Poincar´e), where 1/q = 1/p − 1/n (inequality A . B means that A 6 cB with some inessential constant c). Such effect was investigated in a series of papers for the inequalities of more general type   1 μ(B) ˆ B |f(y) − SBf|θ dμ(y)   1/θ  . η(rB)   1 μ(B) ˆ σB gp dμ   1/p for functions on metric measure spaces. Here f ∈ Lθ loc, g ∈ Lp loc, and SBf is some number depending on the ball B and on the function f, η is some positive increasing function, σ > 1. Usually mean value of the function f on a ball B is chosen as SBf, and the case p > 1 is considered. We investigate self-improvement property for such inequalities on quasimetric measure spaces with doubling condition with parameter γ > 0. Unlike previous papers on this topic we consider the case θ, p > 0. In this case functions are not required to be summable, and we take SBf = I(θ) B f. Here I(θ) B f is the best approximation of the function f in Lθ(B) by constants. We prove that if η(t)t−α increases with some α > 0, then for 0 < p < γ/α and θ > 0 (θ, p)-inequality Poincar´e implies (q, p)-inequality with 1/q > 1/p − γ/α. If p > γ(γ + α)−1 (then the function f is locally integrable) then it implies also (q, p)-inequality with mean value instead of the best approximations I(θ) B f. Also we consider the cases αp = γ and αp > γ. If αp = γ, then (q, p)-inequality with any q > 0 follows from Poincar´e (θ, p)-inequality and moreover some exponential Trudinger type inequality is true. If αp > γ then Poincar´e (θ, p)-inequality implies the inequality |f(x) − f(y)| . η(d(x, y))[d(x, y)]−γ/p . [d(x, y)]α−γ/p for almost all x and y from any fixed ball B (. does depend on B).