On Boundedness of Weighted Hardy Operator in and Regularity Condition

Aziz Harman,FarmanImran Mamedov

doi:10.1155/2010/837951

Abstract

We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces . Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponents , necessary and sufficient conditions are proved for the boundedness of the Hardy operator from into . Also a separate statement on the exactness of logarithmic conditions at zero and at infinity is given. This shows that logarithmic regularity conditions for the functions at the origin and infinity are essentially one.

Highlights

The object of this investigation is the Hardy-type weighted inequality|x|β · −n/p · −n/q · Hf ≤ C |x|β · f, Hf x Lq · RnLp · Rn f y dy |y|≤|x|1.1 in the norms of generalized Lebesgue spaces Lp · Rn
For the one-dimensional Hardy operator in 1, the necessary and sufficient condition was obtained for the exponents β, p, q
We prove that the logarithmic regularity conditions are essential one for such kind of inequalities to hold

Summary

Introduction

1.1 in the norms of generalized Lebesgue spaces Lp · Rn. This subject was investigated in the papers 1–7. For the one-dimensional Hardy operator in 1 , the necessary and sufficient condition was obtained for the exponents β, p, q. The logarithmic condition was assumed in an arbitrarily small neighborhood of zero, where an additional restriction p x ≥ p 0 was imposed on the exponent. The exact condition was found in 1 They proved this result by using of interpolation approaches. We consider the multidimensional case, and the condition β x const is not obligatory, while the necessary and sufficient condition is obtained by a set of exponents p, q, β without imposing any preliminary restrictions on their values Theorems 3.1 and 3.2. In Theorem 3.3, it has been proved that logarithmic conditions at zero and at infinity are exact for the Hardy inequality to be valid in the case q p. Problems of the boundedness of classical integral operators such as maximal and singular operators, the Riesz potential, and others in Lebesgue spaces with variable exponent, as well as the investigation of problems of regularity of nonlinear equations with nonstandard growth condition have become of late the arena of an intensive attack of many authors see 11–18

Lebesgue Spaces with a Variable Exponent

The Main Results

Proofs of the Main Results

10 We come to a contradiction

Exactness of the Logarithmic Conditions