Equivalent norms on analytic subspaces of \(L^ p\)
Let M be a positive integer, let \({\mathbb{R}}^ M\) denote the Euclidean space consisting of M-tuples of real numbers, and let \({\mathbb{C}}^ M\) denote the space consisting of M-tuples of complex numbers. If \(x=(x_ 1,x_ 2,...,x_ M)\in {\mathbb{R}}^ M\), \(h>0\), then the h-cube centred at x is \(Q(x,h)=\{t=(t_ 1,t_ 2,...,t_ M):x_ j-h\leq t_ j\leq x_ j+h\), \(j=1,2,...,M\}.\) In the main result of this paper the authors show that if \(0<p<\infty\), \(\mu\) is a Borel measure in \({\mathbb{R}}^ M\), there is a positive constant \(\nu =\nu (p,M)\) and finite constants \(k_ 1,k_ 2\), such that \[ (*1)\quad k_ 2^{-1}| Q(x,h)| \leq \mu (Q(x,h))\leq k_ 2| Q(x,h)| \quad for\quad | x| >k_ 1, \] then there is a finite constant c such that \[ (*2)\quad c^{-1}\int_{{\mathbb{R}}^ M}| f(x)|^ pdx\leq \int_{{\mathbb{R}}^ M}| f(x)|^ pd\mu (x)\leq c\int_{{\mathbb{R}}^ M}| f(x)|^ pdx \] for f in \(E^ p(T,M)=\{F\in L^ p({\mathbb{R}}^ M):F\) has extension from \({\mathbb{R}}^ M\) to \({\mathbb{C}}^ M\) as a function of exponential type \(T\}\). Results involving \(H^ p\)-spaces are also derived.
scientific article; zbMATH DE number 4017617