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On the lower bound and upper bound of the Sum of Eigenvalues of the Fractional-Logarithmic Laplacian

H. Hajaiej

Abstract

We establish a lower bound and an upper bound to the sum of the Fractional-Logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior.

On the lower bound and upper bound of the Sum of Eigenvalues of the Fractional-Logarithmic Laplacian

Abstract

We establish a lower bound and an upper bound to the sum of the Fractional-Logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior.
Paper Structure (3 sections, 7 theorems, 174 equations)

This paper contains 3 sections, 7 theorems, 174 equations.

Table of Contents

  1. Introduction and Main Results
  2. Lower Bound Estimate
  3. Upper Bound Estimate

Key Result

Proposition 2.2

Let $\rho>0$, let $L$ be slowly varying at infinity, and let $\left\{a_{k}\right\}_{k \geq 0}$ be a nonnegative sequence which is ultimately monotone. Assume that for some constant $c>0$. Then

Theorems & Definitions (14)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.1: Remark 2.6
  • Corollary 2.6
  • ...and 4 more