The modified \({\mathbf P}\)-integral and \({\mathbf P}\)-derivative and their applications
Let \({\mathbf P}:= (p_j)_{j\in\mathbb{Z}}\) be a symmetric sequence of integers with \(p_0:= 0\) and \(P\setminus\{p_0\}\subset [2,N]\). Let \(G_{\mathbf P}\) be a subring of \(\bigoplus_j \mathbb{Z}_{p_j}\) consisting of series \((x_j)\) with \(\operatorname{card}\{x_j; j< 0\}<\infty\). Set \(m_j:= p_1\dotsm p_j\) for \(j> 0\) and \(m_j:= {1\over m_{-j+1}}\) for \(j<0\) (with \(m_0:= 1\)) and define a map \(\varphi: G_{\mathbf P}\to\mathbb{R}_+\) given by NEWLINE\[NEWLINE\varphi(x):= \sum_j {x_j\over m_j}.NEWLINE\]NEWLINE This is a bijection up to a countable subset of \(G_p\) which allows to equip \(\mathbb{R}_+\) with a ring structure and define a family \(\chi_y: G_{\mathbf P}\to \mathbb{C}\), \(y\in\mathbb{R}_+\), of characters by NEWLINE\[NEWLINE\chi_y(x):= \exp\left(2\pi i\sum_{j> 0} {x_j\varphi_{-j}+ x_{-j} \varphi_j\over p_j}.\right)NEWLINE\]NEWLINE In turn, the family defines a Walsh-Fourier transform given for \(f\in L_1(\mathbb{R}_+)\) by NEWLINE\[NEWLINE\widehat f(x):= \int_{\mathbb{R}_+} f(y)\overline{\chi_y(x)}\,dyNEWLINE\]NEWLINE and then the authors define (\({\mathbf P}\)-) derivative and integral using the corresponding multipliers.NEWLINENEWLINE Using these one can define analogs of Lipschitz, Sobolev and BMO spaces on \(G_p\). They contain the following results. ``Criteria for a function to have a representation as the \({\mathbf P}\)-integral or \({\mathbf P}\)-derivate of an \(L^p\)-function are given, and direct and inverse approximation theorems for \({\mathbf P}\)-differentiable functions are established. A relation between the approximation properties of a function and the behaviour of \({\mathbf P}\)-derivatives of the appropriate approximate identity is obtained. Analogues of Lizorkin and Taibleson's results on embeddings between the domain of definitions of the \({\mathbf P}\)-derivative and Hölder-Besov classes are established. Some theorems on embedding into BMO, Lipschitz and Morrey spaces are proved.'' (From the author's abstract)
scientific article; zbMATH DE number 6084148