| dc.description.abstract | Our goal in this paper is to find a characterization of n-dimensional bilinear Hardy inequalities parallel to integral(B(0,.)) f.integral(B(0,.)) g parallel to(q,u(0,infinity) )<= C parallel to f parallel to(p1,v1,Rn)parallel to g parallel to(p2,v2,Rn), f, g is an element of M+(R-n), parallel to integral c(B(0,.)) f.integral c(B(0,.)) g parallel to(q,u(0,infinity) )<= C parallel to f parallel to(p1,v1,Rn)parallel to g parallel to(p2,v2,Rn), f, g is an element of M+(R-n), when 0 < q <= infinity, 1 <= p1, p2 <= infinity and u and v1, v 2 are weight functions on (0,infinity ) and , R-n, respectively. Obtained results are new when p(i) = 1 or p(i) =infinity, i = 1, 2, or 0 < q <= 1 even in 1-dimensional case. Since the solution of the first inequality can be obtained from the characterization of the second one by usual change of variables we concentrate our attention on characterization of the latter. The characterization of this inequality is easily obtained for p(1) <= q using the characterizations of multidimensional weighted Hardy-type inequalities while in the case q < p(1) the problem is reduced to the solution of multidimensional weighted iterated Hardy-type inequality. To achieve our goal, we characterize the validity of multidimensional weighted iterated Hardy-type inequality parallel to parallel to integral cB((0,s)) h(z)dz parallel to(p,u,(0,t))parallel to(q,mu,(0,infinity) <= c parallel to h parallel to(theta,v,(0,infinity),) h is an element of M+(R-n) where 0 < p, q < infinity, 1 <= theta <= infinity, u is an element of W (0, infinity ), v is an element of W(R-n) and mu is a non-negative Borel measure on (0, infinity). We are able to obtain the characterization under the additional condition that the measure mu is non-degenerate with respect to U-q/p. | en_US |
