FALL 2020 Nonlinear Analysis Seminar Series

Associate Professor Dmitriy Stolyarov,
St. Petersburg State University and St. Petersburg Department of Steklov Mathematical Institute

Title: Hardy--Littlewood--Sobolev inequality for \(p=1\): Part 2, Proof.

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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