On inequalities between norms of partial derivatives on convex domains

2025/05/02 by Alexander Plakhov, Vladimir Yu. Protasov, Plakhov, Alexander +1
Mathematics · #26D10 #49K21 #52A10 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2505.01611

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider inequalities between L_p-norms of partial derivatives, p∈ [1,+∞], for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? If not, what is the upper bound? We show that for p=1, the ratio is always bounded and find sharp estimates, while for p>1, the answer depends on the geometry of the domain.

Citations

• Symmetry-breaking in a generalized Wirtinger inequality
• Inequalities for the derivatives

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