[Paper Review] Ornstein's non-inequalities Riesz product approach

Krystian Kazaniecki, Michał Wojciechowski|arXiv (Cornell University)|Jun 27, 2014

Mathematics and Applications2 references3 citations

TL;DR

This paper presents a novel Riesz product-based approach to prove Ornstein's non-inequalities for differential operators with geometrically distinct multi-indices. By constructing a smooth trigonometric polynomial using iterated Riesz products and controlling directional growth via carefully chosen frequency vectors, the authors demonstrate that the L1 norm of one derivative can be made arbitrarily larger than the sum of L1 norms of other derivatives, even when they share the same homogeneity degree.

ABSTRACT

We provide a new technique to prove Ornstein's non-inequalities for derivatives with some geometrical dependence on their indexes.

Motivation & Objective

• To provide an alternative proof technique for Ornstein's non-inequalities in special cases involving differential operators with geometrically distinct multi-indices.
• To establish the failure of a priori L1 norm estimates between linearly independent homogeneous differential operators of the same degree.
• To construct smooth functions on the torus where one derivative dominates the sum of others in L1 norm, regardless of the prescribed bound K.
• To leverage Riesz product structures and frequency vector selection to control the relative growth of different derivative norms.

Proposed method

• Construct a modified Riesz product $ R_n(x) = -1 + \prod_{k=1}^n (1 + \cos(2\pi \langle x, a_k \rangle)) $ using integer frequency vectors $ a_k \in \mathbb{Z}^2 $.
• Inductively choose $ a_k $ such that the ratio $ \left| \frac{(q(2)^2 / q(1))^{l} - (\sigma_k / \sqrt{n})^l \right| \leq 1/3^n $ for all $ l \leq m $, ensuring controlled growth of derivative terms.
• Define a trigonometric polynomial $ Z(x) = \sum_{k=1}^n \sum_{q \in A_k \cup (-A_k)} \frac{1}{q(1)^4} \frac{1}{2^{r(q)}} e^{2\pi i \langle q, x \rangle} $, where $ A_k $ collects all frequency combinations with coefficients in \{-1,0,1\}.
• Show that $ D^{\alpha_0}Z = R_n $, so $ \|D^{\alpha_0}Z\|_{L_1} \leq 2 $, establishing boundedness of the base derivative.
• Decompose $ D^{\alpha_l}Z $ into two parts: $ I_l $, which captures deviation from idealized growth, and $ II_l $, which matches a generalized Riesz product structure.
• Use Lemma 1 to bound $ \|II_1\|_{L_1} \geq C\sqrt{n} $, and show $ \|II_m\|_{L_1} \leq 1 $ for $ m \geq 2 $, ensuring dominant contribution from the first derivative.

Experimental results

Research questions

• RQ1 Can a Riesz product-based construction demonstrate the failure of a priori L1 norm estimates between linearly independent homogeneous differential operators of equal degree?
• RQ2 Under what geometric conditions on multi-indices $ \alpha_j $ can one derivative’s L1 norm exceed the sum of others by an arbitrary factor K?
• RQ3 Is it possible to construct smooth functions on $ \mathbb{T}^2 $ such that a higher-order mixed derivative dominates all others in L1 norm?
• RQ4 How can frequency vectors $ a_k $ be selected inductively to control the relative growth of different derivative terms in a Riesz product framework?
• RQ5 What role does the parameter $ n $ play in amplifying the norm of a specific derivative relative to others in the constructed polynomial?

Key findings

• For every $ K > 0 $, there exists a smooth function $ f \in C^\infty(\mathbb{T}^2) $ such that $ \left\| \frac{\partial^5}{\partial x_1^3 \partial x_2^2} f \right\|_{L_1} \geq K \left( \sum_{j \neq 1} \| D^{\alpha_j} f \|_{L_1} \right) $, proving the desired non-inequality.
• The construction ensures $ \|D^{\alpha_0}Z\|_{L_1} \leq 2 $, while $ \|II_1\|_{L_1} \geq C\sqrt{n} $, and with $ n > 64K^2 C^{-2} $, this exceeds $ 8K+1 $, leading to the required norm dominance.
• The term $ I_l $, representing deviation from idealized growth, satisfies $ \|I_l\|_{L_1} \leq 1 $ due to the uniform error bound $ \leq 1/3^n $, ensuring negligible contribution.
• For $ m \geq 2 $, the term $ II_m $ satisfies $ \|II_m\|_{L_1} \leq n^{-m/2} \cdot n \leq 1 $, so its norm is bounded and does not interfere with the dominance of the first derivative.
• The key inequality $ \|II_1\|_{L_1} \geq C\sqrt{n} $ is derived from Lemma 1 and holds under the condition $ |a_{k+1}| > M_n |a_k| $, ensuring exponential growth of frequencies.
• The final result is achieved by choosing $ n > 64K^2 C^{-2} $, so that $ C\sqrt{n} > 8K+1 $, which guarantees the required norm ratio exceeds $ K $.