A Necessary and Sufficient Condition for Hardy's Operator in the Variable Lebesgue Space


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MAMEDOV F., ZEREN Y.

ABSTRACT AND APPLIED ANALYSIS, 2014 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2014
  • Doi Number: 10.1155/2014/342910
  • Title of Journal : ABSTRACT AND APPLIED ANALYSIS

Abstract

The variable exponent Hardy inequality ??????????
??
??(??)−1

??
0
??(??)????
???????? ??????(.)(0,??)
≤ ??
??????????
??
??(??)
??
???????? ??????(.)(0,??), ?? ≥
0 is proved assuming that the exponents
?? : (0, ??) → (1,∞), ?? : (0, ??) → R not rapidly oscilate near origin and 1/??
??
(0)−?? > 0.Themain result is a necessary and sufficient
condition on ??, ?? generalizing known results on this inequality.

The variable exponent Hardy inequality parallel to x(beta(x)-1) integral(x)(0) f(t)dt parallel to(LP(.)(0,l)) <= C parallel to x(beta(x)) f parallel to(LP(.)(0,l)), f >= 0 is proved assuming that the exponents p : (0,l) -> (1, infinity), beta : (0, l) -> R not rapidly oscilate near origin and 1/p'(0) - beta > 0. The main result is a necessary and sufficient condition on p, beta generalizing known results on this inequality.