On L p – L q Trace Inequalities

We give necessary and sufficient conditions in order that inequalities of the type‖ T K f ‖ L q ( d μ ) ⩽ C ‖ f ‖ L p ( d σ ) ,   f ∈ L p ( d σ ) ,hold for a class of integral operators T K f ( x ) = ∫ Rn K ( x, y ) f ( y ) d σ( y ) with nonnegative kernels, and measures d μ and d σ on ℝ 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 )χ Q ( x ) χ Q ( y ), where = { Q } is the family of all dyadic cubes in ℝ 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 T k f = k ⋆ f with positive radially decreasing kernel k (| x − y |), the trace inequality‖ T K f ‖ L q ( d μ ) ⩽ C ‖ f ‖ L p ( d x ) ,   f ∈ L p ( d x ) ,holds if and only if k [μ] ∈ L s ( d μ), where s = q ( p − 1)/( p − q ). Here k [μ] is a nonlinear Wolff potential defined byW k [ μ ] ( x ) = ∫ 0 + ∞ k ( r ) k *¯( r )/ ( p − 1 )μ( B ( x ,   r ) )1 / ( p − 1 )r n − 1 d r , andk *¯ ( r ) = ( 1 / r n ) ∫ 0 r k ( t )t n − 1 d t . Analogous inequalities for 1 ⩽ q < p were characterized earlier by the authors using a different method which is not applicable when q < 1.