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Vectorial Kato inequality for $p$-harmonic maps with optimal constant

Andreas Gastel, Katarzyna Mazowiecka, Michał Miśkiewicz

TL;DR

This work establishes the sharp vectorial Kato inequality for $p$-harmonic maps, identifying a piecewise sharp constant $\kappa(p,n)$ that governs the pointwise estimate $\kappa(p,n)\,|\nabla|\nabla u||^2 \le |\nabla^2 u|^2$ at points with $\nabla u\neq0$. The authors provide a coordinate-normal proof that reduces the problem to a two-by-two Hessian block and a quadratic minimization, yield the three-regime formula for $\kappa(p,n)$, and prove sharpness via explicit constructions (including scalar target examples and $n=2$). Building on this inequality, they derive regularity results for minimizing $p$-harmonic maps into spheres, giving explicit dimension/exponent criteria and concrete thresholds (e.g., $p_0\approx2.8685$ for $B^4\to S^4$ and $p\in[2,2.642]$ for $B^3\to S^3$). In a targeted later development, they introduce a gamma-regularization approach to stability and Bochner inequalities, enabling improved regularity for $p$-harmonic maps $B^3\to S^3$ up to $p_0\approx2.6427$ with an optimal gamma near $-0.6087$. The results advance the understanding of how optimal Kato-type constants influence regularity and provide sharper thresholds for singularity formation in vector-valued $p$-harmonic problems.

Abstract

We derive the sharp vectorial Kato inequality for $p$-harmonic mappings. Surprisingly, the optimal constant differs from the one obtained for scalar valued $p$-harmonic functions by Chang, Chen, and Wei. As an application we demonstrate how this inequality can be used in the study of regularity of $p$-harmonic maps. Furthermore, in the case of $p$-harmonic maps from $B^3$ to $\mathbb{S}^3$, we enhance the known range of $p$ values for which regularity is achieved. Specifically, we establish that for $p \in [2, 2.642]$, minimizing $p$-harmonic maps must be regular.

Vectorial Kato inequality for $p$-harmonic maps with optimal constant

TL;DR

This work establishes the sharp vectorial Kato inequality for -harmonic maps, identifying a piecewise sharp constant that governs the pointwise estimate at points with . The authors provide a coordinate-normal proof that reduces the problem to a two-by-two Hessian block and a quadratic minimization, yield the three-regime formula for , and prove sharpness via explicit constructions (including scalar target examples and ). Building on this inequality, they derive regularity results for minimizing -harmonic maps into spheres, giving explicit dimension/exponent criteria and concrete thresholds (e.g., for and for ). In a targeted later development, they introduce a gamma-regularization approach to stability and Bochner inequalities, enabling improved regularity for -harmonic maps up to with an optimal gamma near . The results advance the understanding of how optimal Kato-type constants influence regularity and provide sharper thresholds for singularity formation in vector-valued -harmonic problems.

Abstract

We derive the sharp vectorial Kato inequality for -harmonic mappings. Surprisingly, the optimal constant differs from the one obtained for scalar valued -harmonic functions by Chang, Chen, and Wei. As an application we demonstrate how this inequality can be used in the study of regularity of -harmonic maps. Furthermore, in the case of -harmonic maps from to , we enhance the known range of values for which regularity is achieved. Specifically, we establish that for , minimizing -harmonic maps must be regular.

Paper Structure

This paper contains 5 sections, 11 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Kato inequality
  3. Application of Kato's inequality to regularity of p-harmonic maps
  4. Regularity for minimizing p-harmonic maps uB3toS2
  5. Stability and Bochner inequalities

Key Result

Lemma 2.1

For any linear map $A \in L(\mathbb{R}^n, \mathbb{R}^d)$, there is an orthonormal basis $(b_1,\ldots,b_n)$ of $\mathbb{R}^n$ for which $(Ab_1,\ldots,Ab_n)$ are pairwise orthogonal in $\mathbb{R}^d$.

Figures (3)

  • Figure 1: A graph showing the dependence of $\kappa(p,n)$ on $p$ (assuming $d \ge 2$). The dashed curve illustrates the best constant $\kappa(p,n)$ in the scalar case $d = 1$ChaCheWei16. Note that the shape presented above corresponds to the asymptotic case $n \to \infty$, where the first two regimes have the same length. For small $n$, the middle regime can be very little (its length is $\approx 0.07$ for $n = 3$) or even absent (for $n = 2$, as found in MM). In general, $\kappa(p,n)$ is a piecewise quadratic function of class $C^1$.
  • Figure 2: The graph suggests that the function $\frac{2(p+2)}{3(4-p)} - (p-2)$ does not reach the value $\frac{3}{2}$ until $p = \frac{5}{2}$, while for $\kappa \ge \frac{5}{2}$, $\kappa(p,3)$ is simply $2$. The point of intersection is $p \approx 2.8685$.
  • Figure 3: The graph suggests that one can find some suitable $\gamma$ for each $p \le p_0$. However, no value of $\gamma$ is suitable for all $2 \le p \le p_0$.

Theorems & Definitions (28)

  • Lemma 2.1: an elementary fact from Linear Algebra
  • proof
  • Theorem 2.2: Kato's inequality for $p$-harmonic maps
  • Remark 2.3
  • proof : Proof of \ref{['thm:Kato']}
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Remark \ref{['rem:n0-properties']}
  • Definition 3.3
  • ...and 18 more