# FALL 2020 Nonlinear Analysis Seminar Series

### Date

Thursday, October 22, 2020 - 16:00 to 17:00

On Zoom

### 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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