On classes of functions subharmonic in $\mathbb{R}^m$ which are bounded on certain sets

B. Ya. Levin ; V. N. Logvinenko

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 157-175

Résumé

Let $Z_j$ be Euclidean spaces of vectors $z_j=(z_{j,1},\dots,z_{j,n_j+1})$, $Z=\bigoplus\limits_{j=1}^pZ_j$, $X=\bigoplus\limits_{j=1}^p(z_{j,1},\dots,z_{j,n_j})$. A function $u:Z\to\mathbb{R}_+$, $u\not\equiv0$, is called logarithmically $p$-subharmonic, if $\log u(z)$ is upper semicontinuous and for any $j$ and for any $z_k$, $k\ne j$, either the function $z_j\to\log u(z_1,\dots,z_p)$ is subharmonic or $\log u(z_1,\dots,z_p)\equiv-\infty$. For such functions $u$ that satisfy the growth estimate $$ \log u(z)\leqslant\sigma\prod_{j=1}^p(1+|z_{j,n_j+1}|)+N\left(\sum_{\substack{1\leqslant j\leqslant p\\ 1\leqslant k\leqslant n_j}} z_{j,k}^2\right)^{1/2}+c,\quad \sigma, N\geqslant0,\quad c\in\mathbb{R}, $$ theorems are proved about the equivalence of $L^\infty(L^q)$-norm of restrictions $u\mid X$ and $u\mid E$ for some relatively dense subset $E$ of $X$. These theorems generalize well-known results of Cartwright and Plancherel–Polya.