On inequalities for derivatives of multivariate functions
W\({}_ 2^{{\bar \alpha}}(\gamma)\) denotes the class of functions x(t) defined on \(T^ n\), \(T=[-\pi,\pi]\), \(2\pi\)-periodic in each variable, for which \[ \| x(t)\|_ 2=^{def}(\int_{T^ n}| x(t)|^ 2dt)^{1/2}<\infty,\quad \int_{T}x(t)dt_ j=0, \] j\(=1,2,...,n\), \(t=(t_ 1,t_ 2,...,t_ n)\), and \[ \| x^{(\alpha^ k)}\|^ 2_ 2<\gamma_ k,\quad k=1,2,...,m;\quad \alpha^ k=(\alpha^ k_ 1,\alpha^ k_ 2,...,\alpha^ k_ n),\quad \gamma =(\gamma_ 1,\gamma_ 2,...,\gamma_ m), \] where \(x^{(\beta)}\), \(\beta =(\beta_ 1,...,\beta_ n)\), denotes the Weyl derivative. Th problem studied in the present paper is the following: Given \(\alpha^ 0=(\alpha^ 0_ 1,...,\alpha^ 0_ n)\), find the supremum \[ S(\gamma)=\sup \{\| x^{(\alpha^ 0)}\|^ 2_ 2:\;x\in W_ 2^{{\bar \alpha}}(\gamma)\},\quad {\bar \alpha}=(\alpha^ 1,\alpha^ 2,...,\alpha^ m). \]
inequality of Hardy-Littlewood-Pólya type for derivatives
scientific article; zbMATH DE number 4202627