[Paper Review] Sharp constants for composition with a bi-Lipschitz measure-preserving map

Frédéric Bernicot, Sahbi Keraani|arXiv (Cornell University)|Apr 26, 2012

Holomorphic and Operator Theory2 references3 citations

TL;DR

This paper establishes sharp, logarithmic bounds for the composition of functions with bi-Lipschitz, measure-preserving maps in BMO, Hardy, and Carleson spaces. It proves that the operator norm of composition grows at most logarithmically in the bi-Lipschitz constant $ K_ heta $, which is optimal for BMO, and applies this to improve a priori estimates for transport PDEs with rough vector fields.

ABSTRACT

In this note, we aim to describe sharp constants for the composition operator with a bi-Lipschitz measure-preserving map in several functional spaces (BMO, Hardy space, Carleson measures, ...). It is interesting to see how the measure preserving property allows us to improve these constants. Moreover, we will prove the optimality of our results for the BMO space and describe improved estimates for solutions of transport PDEs.

Motivation & Objective

• To determine sharp operator norm bounds for the composition map $ f o f imes heta $, where $ \theta $ is a bi-Lipschitz, measure-preserving diffeomorphism.
• To characterize the optimal growth rate of the norm of $ f \circ \theta $ in BMO, Hardy, and Carleson spaces under measure-preserving and bi-Lipschitz conditions.
• To prove the optimality of logarithmic growth in the bi-Lipschitz constant $ K_\theta $ for BMO norms.
• To apply these sharp estimates to improve a priori bounds for solutions of transport PDEs with divergence-free vector fields.
• To show that the logarithmic dependence on $ K_\theta $ is necessary and cannot be improved, even under measure preservation.

Proposed method

• Introduces the quantity $ K_\theta = \sup_{x \neq y} \left( \frac{|\theta(x) - \theta(y)|}{|x - y|} + \frac{|x - y|}{|\theta(x) - \theta(y)|} \right) $, which measures the bi-Lipschitz distortion of $ \theta $, with $ K_\theta \geq 2 $, equality iff $ \theta $ is an isometry.
• Uses the geometric lemma that measure-preserving bi-Lipschitz maps map balls to sets coverable by $ K_\theta^d $ balls of comparable radius, enabling volume and oscillation estimates.
• Applies John-Nirenberg inequality to show that $ \|f \circ \theta\|_{\text{BMO}} \lesssim \log(K_\theta) \|f\|_{\text{BMO}} $, with the logarithmic dependence being optimal.
• Establishes similar logarithmic bounds for $ \|f \circ \theta\|_{\text{Lip}_p(a)} \lesssim K_\theta^a \|f\|_{\text{Lip}_p(a)} $ and for Carleson measures $ \|\mu^{\sharp\theta}\|_{\mathcal{C}} \lesssim \log(K_\theta) \|\mu\|_{\mathcal{C}} $, with $ \mu^{\sharp\theta} $ the pullback measure.
• Applies duality between $ H^1 $ and BMO to derive $ \|f \circ \theta\|_{H^1} \lesssim \log(K_\theta) \|f\|_{H^1} $, showing that the image of an atom splits into $ \sim \log(K_\theta) $ atoms.
• Applies the results to transport PDEs: for $ \partial_t u + v \cdot \nabla u = 0 $, the solution satisfies $ \|u(t)\|_{\text{BMO}} \lesssim (1 + \|v\|_{L^1_t \text{Lip}}) \exp(ct) \|u_0\|_{\text{BMO}} $, improving upon naive exponential bounds in $ K_\theta $.

Experimental results

Research questions

• RQ1 What is the sharp dependence of the operator norm $ \|f \circ \theta\|_X $ on the bi-Lipschitz constant $ K_\theta $, when $ \theta $ is measure-preserving and $ X $ is BMO?
• RQ2 Can the logarithmic growth in $ K_\theta $ for the composition operator on BMO be improved, or is it optimal?
• RQ3 How do the composition norms behave in other function spaces such as Hardy spaces $ H^1 $, Lipschitz spaces $ \text{Lip}_p(a) $, and Carleson measures?
• RQ4 Can these sharp estimates be applied to obtain improved a priori bounds for solutions of transport PDEs with rough vector fields?
• RQ5 What is the role of measure preservation in enabling logarithmic growth instead of power-law growth in the composition operator norm?

Key findings

• For any $ f \in \text{BMO}({\mathbb{R}}^d) $, the composition $ f \circ \theta $ satisfies $ \|f \circ \theta\|_{\text{BMO}} \lesssim \log(K_\theta) \|f\|_{\text{BMO}} $, and this logarithmic dependence is optimal.
• For $ f \in \text{Lip}_p(a) $, the composition satisfies $ \|f \circ \theta\|_{\text{Lip}_p(a)} \lesssim K_\theta^a \|f\|_{\text{Lip}_p(a)} $, which is sharp and cannot be improved under the given assumptions.
• For a class $ \mathcal{S}C $ of Carleson measures, the pullback $ \mu^{\sharp\theta} $ satisfies $ \|\mu^{\sharp\theta}\|_{\mathcal{C}} \lesssim \log(K_\theta) \|\mu\|_{\mathcal{C}} $, showing logarithmic growth under measure preservation.
• By duality, the Hardy space norm satisfies $ \|f \circ \theta\|_{H^1} \lesssim \log(K_\theta) \|f\|_{H^1} $, implying that the image of an atom splits into $ \sim \log(K_\theta) $ atoms.
• For the transport equation $ \partial_t u + v \cdot \nabla u = 0 $ with divergence-free $ v $, the solution satisfies $ \|u(t)\|_{\text{BMO}} \lesssim (1 + \|v\|_{L^1_t \text{Lip}}) \exp(ct) \|u_0\|_{\text{BMO}} $, with the logarithmic dependence on $ K_\theta $ improving the growth estimate.
• The paper shows that the logarithmic bound is sharp and cannot be improved, even under measure preservation, by proving optimality via a construction based on a reference from [3].