Carleson inequalities on parabolic Hardy spaces (Q515223)

For \(\alpha\in(0,1]\), consider the parabolic differential operator \(L^{(\alpha)}:=\partial_t+(-\frac{\partial^2}{\partial x_1^2}-\dots- \frac{\partial^2}{\partial x_n^2})^\alpha\). One says that a continuous function on \({\mathbb R}_+^{n+1}=\{(x,t):x\in{\mathbb R}^n ,t>0\}\) is \(L^{(\alpha)}\)-harmonic if \(L^{(\alpha)}u=0\) in the sense of distributions. For \(p\in(1,\infty)\), let \(h_\alpha^p\) be the space of all \(L^{(\alpha)}\)-harmonic functions \(u\) satisfying \[ \|u\|_{h_\alpha^p}=\sup\limits_{t>0}\left(\int_{{\mathbb R}^n}|u(x,t)|^pdx\right)^{1/p}<\infty. \] One says that a positive Borel measure \(\mu\) on \({\mathbb R}_+^{n+1}\) satisfies a \((p,q)\)-Carleson inequality with \(p,q\in(1,\infty)\) if the inclusion mapping from \(h_{\alpha}^p\) to \(L^q({\mathbb R}_+^{n+1},d\mu)\) is bounded. Let \[ T^{(\alpha)}(x,t)=\big\{(y,s)\in{\mathbb R}_+^{n+1}:|y-x|^{2\alpha}+s\leq t\big\}. \] For \(\tau>0\), a positive Borel measure \(\mu\) on \({\mathbb R}_+^{n+1}\) is said to be a \(T_\tau\)-Carleson measure if there exists a constant \(C>0\) such that \(\mu(T^{(\alpha)}(x,t))\leq Ct^{(n/2\alpha)\tau}\) for all \((x,t)\in{\mathbb R}_+^{n+1}\). The main result of the paper says that if \(1<p\leq q<\infty\) and \(\mu\) is a positive Borel measure on \({\mathbb R}_+^{n+1}\), then \(\mu\) satisfies a \((p,q)\)-Carleson inequality if and only if the measure \(\mu\) is a \(T_{q/p}\)-Carleson measure.