... Bx,kGtp−dt,4.206 Journal of Inequalities and ApplicationsApplying estimate 4.8 to both hypotheses a and b, by choosing of ε and δ, the right-handpart of 4.5 is less thanC7|x|−n/qx2−kεBx,k|t|βtftp−x,kdt1/p−x,k.4.9Simultaneously,Bx,k|t|βtftp−x,kdt≤Bx,k∩{t∈Rn:|t|βtft≥1}|t|βtftptdt ... 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 ... exponent,as well as the investigation of problems of regularity of nonlinear equations with nonstandardgrowth condition have become of late the arena of an intensive attack of many authors see11–18.2....