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Magnetic Hardy inequalities with singular integral weights

Hynek Kovarik, Pier Cristoforo Rossaro

Abstract

In this paper we present Hardy type inequalities for magnetic Dirichlet forms with singular integral weights. We analyze the local and global optimality of the integral weight and discuss several examples in details. An application of our results to spectral estimates for magnetic Schrödinger operators is provided as well.

Magnetic Hardy inequalities with singular integral weights

Abstract

In this paper we present Hardy type inequalities for magnetic Dirichlet forms with singular integral weights. We analyze the local and global optimality of the integral weight and discuss several examples in details. An application of our results to spectral estimates for magnetic Schrödinger operators is provided as well.
Paper Structure (7 sections, 10 theorems, 112 equations)

This paper contains 7 sections, 10 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. The background
  3. Main results
  4. Regular magnetic fields
  5. Singular magnetic fields
  6. Examples
  7. Eigenvalues of magnetic Schrödinger operators

Key Result

Theorem 1.1

Let $B\neq 0$ satisfy ass-regular, and let $A:\mathbb{R}^2\to\mathbb{R}^2$ be a vector field such that $\nabla \times A =B$. Then there exists a constant $C_B$ such that for all $u\in d_0(Q_A)$ we have where

Theorems & Definitions (28)

  • Theorem 1.1: Hardy inequality for regular magnetic fields
  • Remark 1.2: Optimality at zero
  • Remark 1.3: Optimality at infinity
  • Remark 1.4
  • Theorem 1.6: Hardy inequality for singular magnetic fields
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 18 more