On the interrelation of the generalized Besov-Nikols'kiǐ and Weyl-Nikols'kiǐ classes of functions
The authors generalize former results of B. V. Simonov on embedding theorems for the Besov-Nikol'skij class and the Weyl-Nikol'skij class by considering spaces with more general weights in \(t\). The functions considered are \(2\pi\) periodic, measurable functions. The Besov-Nikol'skij space is defined by finiteness of the quantity \[ \int_0^1 \left(\omega_k(f,t)_p/t^r \right)^{\theta} \frac{dt}t, \] where \[ \omega_{\beta}(f,t)_p = \sup_{| h| \leq t} \Bigl\| \sum_{\nu = 0}^{\infty} (-1)^{\nu} \binom {\beta}{\nu} f(x + (\beta - \nu)h)\Bigr\| _p, \] with \[ \binom{\beta}{\nu} = \frac {\beta(\beta -1)\cdots (\beta - \nu +1)}{\nu !} \] and \(k>r>0\). The authors consider spaces in which the function \(\alpha(t) = t^{-r\theta -1}\) is replaced by a function for which there is a real number \(\sigma\) such that \[ \int _0^1 \alpha(t) t^{\sigma - \varepsilon} dt = +\infty,\quad \forall \varepsilon>0, \] while \[ \int _0^1 \alpha(t) t^{\sigma} dt < +\infty. \] The generalized Besov-Nikol'skiĭ class \(B(p, \theta, k, \alpha)\) is the set of \(f \in L_p\) for which \[ \int_0^1 \alpha(t) \omega_k^{\theta}(f,t)_p dt \] is finite. A function \(f\) belongs to the Weyl-Nikol'skij class if \(f \in L_p\) and its Fourier series \[ \frac{a_0(f)}2 + \sum_{n=1}^{\infty} (a_n(f) \cos nx + b_n(f) \sin nx), \] after multiplication by \(\lambda_n\), \[ \sum_{n=1}^{\infty} \lambda_n(a_n(f) \cos nx + b_n(f) \sin nx), \] where \[ \lambda_n = \left( \int_{1/n}^1 \alpha(t) dt + n^{k\theta} \int_0^{1/n} \alpha(t) t^{k\theta} dt \right)^{1/\theta}, \] is the Fourier series of some \(\phi \in L_p\). In the case of \(\alpha(t) = t^{-r\theta -1}\), \(\lambda_n = n^r\), and \(\phi\) is a fractional derivative of \(f\). The embedding theorems involve the spaces \(WH(\alpha, \beta, \psi, p, \theta, k)\) and \(B_0H(\alpha, \beta, \psi, p, \theta, k)\), where \(f \in WH\) if \(f \in W(p, \theta, k, \alpha)\) and \[ \omega_{\beta}(f, \delta)_p \leq C \psi(\delta), \quad 0< \delta \leq 1, \] and \(f \in B_0H\) if \(f \in L_p\) and \[ \int_0^{\delta} t^{-\theta k -1} \int_0^t \alpha(u)u^{\theta k}du \omega_{k + \beta}^{\theta}(f,t)_p dt + \delta^{\theta \beta} \int_{\delta}^1 t^{\theta \beta - 1} \int_t^1\alpha(u)du \omega_{k + \beta}^{\theta}(f,t)_p dt\leq C \psi^{\theta}(\delta), \] where \(0 < \theta < +\infty\), \(\beta > 0\), \(k \geq \sigma/\theta\). The results are that if \(0 < \theta \leq \min(2,p)\), \[ B_0H(\alpha, \beta, \psi, p, \theta, k) \subseteq WH(\alpha, \beta, \psi, p, \theta, k) \] and the subset relationship is reversed if \(\max(2,p) \leq \theta <+ \infty\). In particular there is equality when \(\theta = p = 2\). Embeddings are also given for the intersection of the \(B_0H\) and \(WH\) spaces with spaces of functions whose Fourier series are lacunary cosine series, and whose Fourier series have coefficients \(a_n\) such that \(n^{\tau} a_n\) is decreasing.
scientific article; zbMATH DE number 1134604