Analytic semigroups in weighted $L_1$-spaces on the half-line generated by singular or degenerate operators

TL;DR

This paper identifies parameter ranges for which a class of singular or degenerate differential operators generate analytic semigroups on weighted $L_1$ spaces on the half-line, advancing understanding of their functional analytic properties.

Contribution

It determines the specific parameter conditions under which these operators generate analytic semigroups in weighted $L_1$ spaces, extending previous results to a broader class of singular or degenerate operators.

Findings

Parameter ranges for semigroup generation are explicitly characterized.

Operators with singular or degenerate coefficients are shown to generate analytic semigroups.

Results apply to weighted $L_1$ spaces with specific weight functions.

Abstract

Ranges of the real-valued parameters $\alpha$, $a$, $b$, and $m$ are identified for which the operator $$\mathcal{A}_{\alpha}(a,b)f(x):=x^\alpha\left(f''(x)+\frac{a}{x}f'(x)+\frac{b}{x^2}f(x)\right), \quad x>0,$$ generates an analytic semigroup in $L_1((0,\infty),x^m\mathrm{d}x)$.