An Interpolation Result for A1 Weights With Applications

Content Overview

Abstract & Introduction

Notation and Preliminary Results

Proof of our main theorem

Applications to fractional Poincar´e inequalities

References

Abstract.

We characterize the real interpolation space between weighted L1 and W1,1 spaces on arbitrary bounded domains, when the weights are positive powers of the distance to the boundary multiplied by an A1 weight. As an application of this result we obtain weighted fractional Poincar´e inequalities with sharp dependence on the fractional parameter s (for s close to 1) and show that they are equivalent to a weighted Poincar´e inequality for the gradient.

1. Introduction

2. Notation and Preliminary Results

3. Proof of our main theorem

4. Applications to fractional Poincar´e inequalities

In the forthcoming results we will make use of two well-known properties of weighted norms contained in the following lemma. We include a proof for the sake of completeness.

References

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