2018/05/25 by Pérez, Carlos, Rela, Ezequiel · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1805.10388
We study weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit analysis on the dependence on the Ap constants of the involved weights. We obtain inequalities of the form ((1)/(w(Q))∫Q|f-fQ|qw )^(1)/(q)≤ Cwℓ(Q) ((1)/(w(Q))∫Q |∇ f|p w )^(1)/(p), with different quantitative estimates for both the exponent q and the constant Cw. We will derive those estimates together with a large variety of related results as a consequence of a general selfimproving property shared by functions satisfying the inequality (1)/(|Q|)∫Q |f-fQ| dμ≤ a(Q), for all cubes Q⊂ℝn and where a is some functional that obeys a specific discrete geometrical summability condition. We introduce a Sobolev-type exponent p^*w>p associated to the weight w and obtain further improvements involving Lp^*w norms on the left hand side of the inequality above. For the endpoint case of A1 weights we reach the classical critical Sobolev exponent p^*=(pn)/(n-p) which is the largest possible and provide different type of quantitative estimates for Cw. We also show that this best possible estimate cannot hold with an exponent on the A1 constant smaller than 1/p. We also provide an argument based on extrapolation ideas showing that there is no (p,p), p≥1, Poincaré inequality valid for the whole class of RH_∞ weights by showing their intimate connection with the failure of Poincaré inequalities, (p,p) in the range 0