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Linear dimension‐free estimates in the embedding theorem for Schrödinger operators
Author(s) -
Dragičević Oliver,
Volberg Alexander
Publication year - 2012
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdr036
Subject(s) - mathematics , embedding , dimension (graph theory) , cartesian product , bilinear interpolation , pure mathematics , hilbert space , m. riesz extension theorem , riesz transform , differential operator , operator theory , bilinear transform , mathematical analysis , discrete mathematics , computer science , artificial intelligence , digital filter , statistics , filter (signal processing) , computer vision
We prove a bilinear embedding theorem for Schrödinger operators with nonnegative potentials. The embedding, acting on the cartesian product of L p (ℝ n ) and its dual, involves estimates that are independent of the dimension n and linear in terms of p . This feature is achieved by means of a particular Bellman function which satisfies three crucial properties. Connections with known results on the Heisenberg group as well as with results for the Hilbert transform along the parabola are also explored. We believe our approach is quite universal in the sense that one could apply it to a whole range of Riesz transforms arising from various differential operators.