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‖Lq(Rd)≲‖∇f‖Lp(Rd),f∈C∞0(Rd) holds true provided 1p−1q=1d and 1≤p<d. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality ‖Iα[g]‖Lq(Rd)≲‖g‖Lp(Rd),1p−1q=αd, 1<p<q<∞, here Iα is the Riesz potential of order α, i.e. a Fourier multiplier with the symbol |⋅|−α. It is easy to see by plugging g=δ0 (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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