Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 34, 2009, 279-289

HARDY TYPE INEQUALITY IN VARIABLE LEBESGUE SPACES

Humberto Rafeiro and Stefan Samko

Universidade do Algarve, Departamento de Matemática
Campus de Gambelas, 8005-139 Faro, Portugal; hrafeiro 'at' ualg.pt

Universidade do Algarve, Departamento de Matemática
Campus de Gambelas, 8005-139 Faro, Portugal; ssamko 'at' ualg.pt

Abstract. We prove that in variable exponent spaces L^p(.)(\Omega), where p(.) satisfies the log-condition and \Omega is a bounded domain in Rⁿ with the property that Rⁿ \ \overline\Omega has the cone property, the validity of the Hardy type inequality

||1/\delta(x)^\alpha \int_\Omega \varphi(y) / |x - y|^n-\alpha dy||_p(.) \le C||\varphi||_p(.), 0 < \alpha < min(1,n/p₊),

where \delta(x) = dist(x,\partial\Omega), is equivalent to a certain property of the domain \Omega expressed in terms of \alpha and \chi_\Omega.

2000 Mathematics Subject Classification: Primary 47B38, 42B35, 46E35.

Key words: Hardy inequality, weighted spaces, variable exponent.

Reference to this article: H. Rafeiro and S. Samko: Hardy type inequality in variable Lebesgue spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 279-289.

Full document as PDF file

Copyright © 2009 by Academia Scientiarum Fennica