Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 785-805

SHARP INTEGRABILITY EXPONENTS AND CONSTANTS FOR MUCKENHOUPT AND GEHRING WEIGHTS AS SOLUTION TO A UNIQUE EQUATION

Arturo Popoli

Università degli Studi di Napoli "Federico II"
Dipartimento di Matematica e Applicazioni "R. Caccioppoli"
Via Cintia, 80126 Napoli, Italy; arturo.popoli 'at' unina.it

Abstract. The sharp results for the self-improving and the transition properties of Gehring RH_q and Muckenhoupt A_p weights are unified and improved into corresponding sharp results for weights satisfying a general reverse Hölder inequality. We show that the optimal exponents of integrability as well as the best constants in the integral inequalities can be obtained by mean of the unique algebraic equation

(x/(x – q))^1/q =B(x/(x – p))^1/p

holding for the so called B_p^q class (see 1.10) which contains the Gehring and Muckenhoupt classes as particular cases.

2010 Mathematics Subject Classification: Primary 46E30; Secondary 26D15.

Key words: Reverse Hölder's inequalities, Muckenhoupt weights, Gehring lemma.

Reference to this article: A. Popoli: Sharp integrability exponents and constants for Muckenhoupt and Gehring weights as solution to a unique equation. Ann. Acad. Sci. Fenn. Math. 43 (2018), 785-805.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4351