© The London Mathematical Society 2006
ON Lp–Lq TRACE INEQUALITIES
Departament de Matemàtica Aplicada i Anàlisi, Facultat de Matemàtiques, Universitat de Barcelona Gran Via 585, 08071 Barcelona, Spain cascante{at}ub.edu ortega{at}ub.edu
Department of Mathematics, University of Missouri-Columbia Columbia, MO 65211, USA igor{at}math.missouri.edu
Received 25 May 2005.
We give necessary and sufficient conditions in order that inequalities of the type
 |
Rn K(x, y) f(y) d 
(y) with nonnegative kernels, and measures d µ and d on 
n, in the case where p > q > 0 and p > 1.
An important model is provided by the dyadic integral operator with kernel K
(x, y) = 
Q
K(Q)
Q(x) Q(y), where = {Q} is the family of all dyadic cubes in n, and K(Q) are arbitrary nonnegative constants associated with Q .
The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator Tk f = k 
f with positive radially decreasing kernel k(|x – y|), the trace inequality
 |
holds if and only if 
k[µ] Ls (dµ), where s = q(p – 1)/(p – q). Here k[µ] is a nonlinear Wolff potential defined by 
, and 
. Analogous inequalities for 1 
q < p were characterized earlier by the authors using a different method which is not applicable when q < 1.