On $L^p$--$L^q$ trace inequalities

Abstract

We give necessary and sufficient conditions in order that inequalities of the type $$ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma), $$ hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x, y) f(y) d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ 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 $K_{\mathcal D}(x, y) \sum_{Q\in{\mathcal D}} K(Q) \chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$. The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k\star f$ with positive radially decreasing kernel $k(|x-y|)$, the trace inequality $$ \| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \qquad f \in L^p(dx), $$ holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$, where $s = {\frac{q(p-1)}{p-q}}$. Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{\frac 1 {p-1}} \mu (B(x,r))^{\frac 1{p-1}} r^{n-1} dr,$ and $\bar{k}(r)=\frac1{r^n}\int_0^r k(t) t^{n-1} dt$. Analogous inequalities for $1\le q < p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$.