Weighted Norm Inequalities

Introduction In the rst part of the paper we study integral operators of the form (1) Kf(x) = v(x) x Z 0 k(x; y)u(y)f(y) dy; x > 0; where the real weight functions v(t) and u(t) are locally integrable and the kernel k(x; y) 0 satisses the following condition: there exists a constant D 1 such that Standard examples of a kernel k(x; y) 0 satisfying (2) are (i) k(x; y) = (x ? y) , 0 (ii) k(x; y) = log (1 + x ? y), k(x; y) = log x y ; 0 and their various combinations. Let 0 < p 1. We study (1) on R + = (0; 1), but any (a; b) R can be taken instead of (0; 1) without any loss of generality. Also, the dual 1 Z 0 jf(x)j p dx 1=p < 1 o : We consider K as a map from L p into L q and shall characterize the following problems: (B) L p ? L q boundedness, (C) L p ? L q compactness and measure of non-compactness, (S) L 2 ? L 2 Schatten-von Neumann ideal norms. Several factors aect the problem (B). First of these are restrictions imposed on the kernel k(x; y) 0. Another such factor is the range of parameters p and q, because of substantial diierence between the cases p q and q < p. Certain part is played also by the fact whether 1 p, q 1 or not. The cases when p = 1 or p = 1; q 1 and q = 1 or q = 1; p 1 follow trivially from known results ((KA], Chapter XI, Theorem 4). It also follows from the general theory of integral operators AS], Sc] that if 0 < p < 1 and K : L p ! L q is bounded, then k(x; y) = 0 almost everywhere. Among other factors, perhaps, the veriiability of a criterion is the most relevant. For instance, Muckenhoupt's criterion M] for the L p ? L p boundedness of (1) when k(x; y) = 1 penetrated many areas because of its explicit form, and being so easy to verify. On the other hand, the implicit \Schur's test" Kor], Sz], given for arbitrary kernel k(x; y) 0, 1 < q p < 1, had also had eeective applications Nik], Hern]. The problem (B) was intensively studied since 1988, …