A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities

Abstract

The main result of this paper supports a conjecture by Pérez and Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincaré-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by Pérez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range p < 1.As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects. Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting. Q∈Q 1 w(Q)ˆQ M(wχ Q ) dx,

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

“…The seminorm |𝑓| W𝑠,𝑝 (Ω) was introduced in the context of Poincaré and Sobolev-Poincaré inequalities in John domains in [9] and further results on its relevance for these inequalities were obtained in [2,6,7,11]. It is equivalent to the usual Gagliardo seminorm given by…”

“…The seminorm $$|f|_{\widetilde{W}^{s,p}(\Omega )}$$ was introduced in the context of Poincaré and Sobolev–Poincaré inequalities in John domains in [9] and further results on its relevance for these inequalities were obtained in [2, 6, 7, 11]. It is equivalent to the usual Gagliardo seminorm given by $$$$\begin{equation*} |f|_{ W^{s,p}(\Omega )}^p = \int _\Omega \int _\Omega \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, dy \,dx \end{equation*}$$$$when Ω is a Lipschitz domain (see [5, Equation (13)]) or, more generally, a uniform domain (see [12, Corollary 4.5]).…”

“…The weighted space $$ W^{s,2}(\Omega , 1, \delta ^{\beta })$$ was used in Lipschitz domains in [1] to establish regularity results and to develop efficient numerical methods for nonlocal equations involving the fractional Laplace operator, while the spaces $$\widetilde{W}^{s,p}(\Omega , d^\alpha , d^\beta )$$ have been proved useful for Poincaré and Sobolev–Poincaré inequalities in John domains [2, 11]. But, to the best of our knowledge, this is the first result where they are studied in the context of interpolation spaces.…”

Quantitative John-Nirenberg inequalities at different scales

“…Note that Y (Q) := w r (Q), r > 1 is a possible choice of Y in the above theorem and thus the particular case of a constant functional in the main theorem [MP20] proves that w is an A ∞,wr (dµ) weight for any r > 1. Evidently, the Fujii-Wilson A ∞ (dµ) weights studied for instance in [HPR12] are A ∞,Y (dµ) weights for the functional Y defined by Y (Q) := w(Q) for every cube Q in R n .…”

“…The need of this condition for a self-improving result like Theorem B has been investigated in [MP20], where the first author studies alternative arguments avoiding the A ∞ (dµ) condition on the weight to get a self-improving like that. Although the results there are not fully satisfactory in the sense that they do not recover the improvement (2.4) without the A ∞ (dµ) condition, they are good enough to get a new unified approach for getting classical and fractional weighted Poincaré-Sobolev inequalities.…”