On Boundedness of Weighted Hardy Operator in <inline-formula> <graphic file="1029-242X-2010-837951-i1.gif"/></inline-formula> and Regularity Condition

Abstract

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.