Abstract
Abstract We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.
Highlights
We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups
Many well-known multilinear inequalities commonly used in analysis, such as multilinear Hölder’s inequality, Loomis–Whitney inequality and the sharp Young convolution inequality, can be seen as instances of a broader family of estimates: the so called Brascamp–Lieb inequalities
By means of the heat semigroup {etL}t>, we introduce the nonlinear heat ow v(t, x) = etLf p
Summary
Many well-known multilinear inequalities commonly used in analysis, such as multilinear Hölder’s inequality, Loomis–Whitney inequality and the sharp Young convolution inequality, can be seen as instances of a broader family of estimates: the so called Brascamp–Lieb inequalities. This constant depends on the maps Bj and the exponents pj and is called the Brascamp–Lieb constant These inequalities were extensively studied in the last years, starting from the works of Rogers [20] Brascamp, Lieb and Luttinger [7] and Brascamp and Lieb [6], where the authors studied the rank-one case, that is the case where nj = for all j, using rearrangement techniques. In particular they proved that the Brascamp– Lieb constant is the same if one restricts the inputs to Gaussians, a result known as Lieb’s Theorem.
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