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Harnack inequality for Bessel operators

Giorgio Metafune, Luigi Negro, Chiara Spina

TL;DR

This work analyzes a singular parabolic operator $\mathcal{L}=\Delta_{x}+2a\cdot\nabla_xD_y+D_{yy}+\dfrac{c}{y}D_y$ in the half-space $\mathbb{R}^{N+1}_+$ with Neumann boundary at $y=0$, deriving uniqueness results under both pointwise and integral growth and establishing a Harnack inequality for nonnegative solutions. The authors leverage sharp heat-kernel estimates with respect to the weighted measure $d\mu(z)=y^c dx dy$, the doubling property of the corresponding volume, and Bessel-operator structure to obtain semigroup representations, mean-value inequalities, and Schauder estimates up to the boundary. A dual strategy is employed: first prove uniqueness under pointwise growth via barrier arguments, then translate integral-growth bounds into pointwise ones using parabolic regularity, culminating in a kernel-based Harnack inequality and Liouville-type results. The results extend known Harnack theory to nonzero drift $a$ and to the regime $-1<c<0$, offering a comprehensive framework for positive solutions through kernel bounds and semigroup methods.

Abstract

We prove uniqueness results and Harnack inequality for Bessel operators \begin{align*} %\label{def L transf alpha} D_t-Δ_{x} -2a\cdot\nabla_xD_y- D_{yy}- \frac cy D_y % \nonumber \\[1ex]&=y^α\sum_{i,j=1}^{N+1}a_{ij}D_{ij}+y^{α-1}\left(v,\nabla\right)-by^{α-2}. \end{align*} in the strip $[0,T]\times \mathbb{R}^{N+1}_+=\{0 \leq t \leq T, x \in \mathbb{R}^N, y>0\}$ under Neumann boundary conditions at $y=0$.

Harnack inequality for Bessel operators

TL;DR

This work analyzes a singular parabolic operator in the half-space with Neumann boundary at , deriving uniqueness results under both pointwise and integral growth and establishing a Harnack inequality for nonnegative solutions. The authors leverage sharp heat-kernel estimates with respect to the weighted measure , the doubling property of the corresponding volume, and Bessel-operator structure to obtain semigroup representations, mean-value inequalities, and Schauder estimates up to the boundary. A dual strategy is employed: first prove uniqueness under pointwise growth via barrier arguments, then translate integral-growth bounds into pointwise ones using parabolic regularity, culminating in a kernel-based Harnack inequality and Liouville-type results. The results extend known Harnack theory to nonzero drift and to the regime , offering a comprehensive framework for positive solutions through kernel bounds and semigroup methods.

Abstract

We prove uniqueness results and Harnack inequality for Bessel operators \begin{align*} %\label{def L transf alpha} D_t-Δ_{x} -2a\cdot\nabla_xD_y- D_{yy}- \frac cy D_y % \nonumber \\[1ex]&=y^α\sum_{i,j=1}^{N+1}a_{ij}D_{ij}+y^{α-1}\left(v,\nabla\right)-by^{α-2}. \end{align*} in the strip under Neumann boundary conditions at .
Paper Structure (8 sections, 23 theorems, 98 equations)

This paper contains 8 sections, 23 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The operator
  3. Remarks on the measure $d\mu(z)=y^c\,dx\,dy$
  4. Uniqueness under pointwise growth conditions
  5. Mean value inequalities and uniqueness under integral growth conditions
  6. Uniqueness of positive solutions
  7. Harnack inequality
  8. Consequences

Key Result

Theorem 2.1

(MNS-Singular-Half-Space) Let $0<\frac{m+1}{p}<c+1$. Then the operator endowed with domain $D(\mathcal{L})=W^{2,p}_{\mathcal{N}}(m)$ generates a bounded analytic semigroup $(e^{t\mathcal{L}})_{t \geq 0}$ in $L^p_m$.

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Proposition 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 16 more