On weighted Hardy inequalities in mixed norms

On weighted iterated Hardy-type inequalities

“…Note that inequality (1.6) has been completely characterized in [18] and [19] in the case 0 < p < ∞, 0 < q ≤ ∞, 1 ≤ s ≤ ∞ by using difficult discretization and anti-discretization methods. Inequalities (1.5) -(1.6) and (1.7) -(1.8) were considered also in [38] and [39], but characterization obtained there is not complete and seems to us unsatisfactory from a practical point of view.…”

Weight boundedness of a class of quasilinear operators on the cone of monotone functions

“…The best constant C T in the inequality satisfies the estimate here, A 1 and A 2 are the best constants in the inequalities For 0 < c < d ≤ ∞, 0 < t, p < ∞, and h ∈ ᑧ + , we set [14,15].…”

“…In the case r = q, these inequalities were studied by many authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references in survey [11]), but the case r ≠ q has not been investi gated; handling it has required a fresh idea borrowed from [14,15].…”

Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces

“…In particular, for some parameters p, q, r this case of (1.2) was solved in [16], [17] and (1.4) in [7]. Complete solution of this case is given in [30], [31]. By a new method we characterize the inequalities (1.1)-(1.8) with a kernel k(x, y) satisfying (1.9) for all parameters 1 ≤ p ≤ ∞, 0 < r ≤ ∞, 0 < q ≤ ∞.…”