A new Jensen inequality for multivariate superquadratic functions is derived and proved. The derived Jensen inequality is then employed to obtain the general Hardy-type integral inequality for superquadratic and subquadratic functions of several variables.

Infinitely many, even scales of, equivalent conditions are derived to characterize the bilinear Hardy-type inequality under various ranges of parameters.

Some new multidimensional Hardy-type inequalites are proved and discussed. The cases with bilinear and iterated operators are considered and some equivalence theorems are proved.

A number of classical inequalities and convergence results related to Fourier coefficients with respect to unbounded orthogonal systems are generalized and complemented. All results are given in the case of Lorentz–Zygmund spaces.