2006/11/13 by Carme Cascante, Cascante, Carme, Joaquı́n M. Ortega +3
Mathematics · #31C45 #46E35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.math/0611378
openalex publication_date 2006/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give necessary and sufficient conditions in order that inequalities of the type ‖ TK f‖Lq(dμ)≤ C ‖f‖Lp(dσ), f ∈ Lp(dσ), hold for a class of integral operators TK f(x) = ∫Rn K(x, y) f(y) d σ(y) with nonnegative kernels, and measures d μ and dσ on \Rn, in the case where p>q>0 and p>1. An important model is provided by the dyadic integral operator with kernel K\mathcal D(x, y) ∑_Q∈\mathcal D K(Q) χQ(x) χQ(y), where \mathcal D=\Q\ is the family of all dyadic cubes in \Rn, and K(Q) are arbitrary nonnegative constants associated with Q ∈\mathcal D. 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 ‖ Tk f‖Lq(dμ)≤ C ‖f‖Lp(d x), f ∈ Lp(dx), holds if and only if \mathcal Wk[μ] ∈ Ls (dμ), where s = (q(p-1))/(p-q). Here \mathcal Wk[μ] is a nonlinear Wolff potential defined by \mathcal Wk[μ](x)=∫0+∞ k(r) k(r)^\frac 1 p-1 μ(B(x,r))^\frac 1p-1 rn-1 dr, and k(r)=\frac1rn∫0r k(t) tn-1 dt. Analogous inequalities for 1≤ q < p were characterized earlier by the authors using a different method which is not applicable when q<1.