On the nonlinear Brascamp–Lieb inequality

Abstract

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Decoupling for certain quadratic surfaces of low co‐dimensions

“…Moreover, from [2] we know that the function is continuous (with values in ). Indeed, it is even Hölder continuous [1].…”

An Algebraic Brascamp–Lieb Inequality

“…At this point we then apply Lemma 3 to rescale the inner integral so that we may then apply Theorem 7. Finally, using the bound on the degree of Z (S j ) ⊃ H j 15, we obtain (14), completing the proof.…”

“…It is common in applications to encounter nonlinear variants where the linear maps L j are replaced with nonlinear maps, as observed in [6,8,13,20] for example. Significant progress on inequalities of this type was made in [14], where the authors establish the following highly general local nonlinear Brascamp-Lieb inequality.…”

“…We shall assume for the remainder of the paper without loss of generality that the maps B j have finite degree, since the case of infinite degree holds vacuously, and that BL T x M (dB(x), p) < ∞ for all x ∈ M, in particular that B j is a submersion on M. We may do this firstly because we may remove the set of non-smooth points harmlessly since it is closed and null, so M is still an open subset of an algebraic variety, and secondly we may remove the set of smooth points at which the weight arising in (2) vanishes, i.e. those x ∈ M such that BL(dB(x), p) = ∞, since this set is closed by continuity of the reciprocal of the Brascamp-Lieb constant (Theorem 5.2 of [14]). We shall begin by reducing to the case where d = n, i.e.…”

“…. , H m of algebraic varieties such that, if substituted into (13), then this inequality would yield (14). These manifolds may be thought of as the unions of 'discrete foliations' of the preimages B −1 j (V j ) via the fibres of B j .…”