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

Abstract

We give necessary and sufficient conditions in order that inequalities of the type $$\Vert T_{K}f{\Vert }_{L^q(d\mu )}\le C\Vert f{\Vert }_{L^p(d\sigma )},f\in L^p(d\sigma ),$$ 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 ${\mathbb{R}}^n$, 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 ${\mathbb{R}}^n$, 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 $$\Vert T_{k}f{\Vert }_{L^q(d\mu )}\le C\Vert f{\Vert }_{L^p(dx)},f\in L^p(dx),$$ 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 $\bar{k}(r)=\frac{1}{r^n}{\int }_0^{r}k(t)t^{n-1}dt$. Analogous inequalities for $1\le q<p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$.

References in corpus (1)

• Quasilinear and Hessian equations of Lane-Emden type