FALL 2020 Nonlinear Analysis Seminar Series

Date
Location
Description
Associate Professor Dmitriy Stolyarov, St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute
Title: Hardy--Littlewood--Sobolev inequality for p=1
Abstract:
The classical Sobolev embedding theorem says that the inequality \|f\|_{L_q(\mathbb{R}^d)}\lesssim \|\nabla f\|_{L_p(\mathbb{R}^d)},\quad f \in C_0^\infty(\mathbb{R}^d) holds true provided \frac{1}{p} - \frac{1}{q} = \frac{1}{d} and 1 \leq p < d. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality \|\mathrm{I}_\alpha [g]\|_{L_q(\mathbb{R}^d)} \lesssim \|g\|_{L_p(\mathbb{R}^d)},\quad \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d},\ 1 < p < q < \infty, here \mathrm{I}_{\alpha} is the Riesz potential of order \alpha, i.e. a Fourier multiplier with the symbol |\cdot|^{-\alpha}. It is easy to see by plugging g = \delta_0 (the Dirac's delta) in the role of g that the HLS inequality is false at the endpoint p=1. However, the Sobolev embedding is true in this case, as it was proved by Gagliardo and Nirenberg. The folklore principle, supported by the results of Bourgain--Brezis, Van Schaftingen, and many others, says that the HLS inequality becomes valid when we somehow separate the function g from the set of delta-measures. We will discuss this effect in more details and state new results in this direction.
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