On some local properties of the sum of lacunary trigonometric series (Q1263777)

Let (1) \(a+\sum^{\infty}_{n=1}a_ n \cos (\lambda_ n+\psi_ n)\), where a, \(a_ n\), \(\lambda_ n\), \(\psi_ n\) are real numbers: \(\lambda_ n>0\), \(a_ n\geq 0\) and \(\lambda_{n+1}/\lambda_ n\geq \lambda >1\) for every \(n=1,2,... \). The sum of this series, whenever it exists, is denoted by \(\Phi\) (x). Let \({\hat \Phi}(x)=\sup_{k\geq 0}| \Phi_ k(x)|,\) where \(\Phi_ k\) is the k-th partial sum of (1). The series (1) is said to be summable if \(\sum^{\infty}_{n=1}| a_ n|^ 2<\infty\) and in this case we write \(\| \Phi \| =[a^ 2+\sum^{\infty}_{n=1}a^ 2_ n]^{1/2}.\) In {\S}{\S} 2-3, the author establishes certain local \(L_ p\)-norm inequalities connecting \(\Phi\) and \({\hat \Phi}\). In {\S}{\S} 4- 5, he gives similar estimates for \(\Phi (x;b,\xi)=b+\sum^{\nu}_{n=1}a_ n \cos (\lambda_ nx+\xi_ n)\) and shows that some estimates obtained are sharp in certain sense. In {\S} 6 some facts concerning module are treated. In {\S} 7 the problem of integrability of the sum of lacunary series with weights is investigated. Let \(p>0\), a non-zero modulus \(\omega\), and a summable lacunary series (1) be fixed. The following theorem is proved: In order that the integral \(\int_{+0}| \Phi (x)|^ p dx/\omega (x)\) be convergent, it is necessary and sufficient that all the following series converge: \[ \sum^{\infty}_{n=2}D_ n(\sum^{\infty}_{k=n}a^ 2_ n)^{p/2},\quad \sum^{\infty}_{n=2}D_ n| a+\sum^{n- 1}_{k=1}a_ k \cos \psi_ k|^ p, \] \[ \sum^{\infty}_{n=2}D_ n^{(p_ j)}| \sum^{\infty}_{k=1}a_ k\lambda^ j_ k \cos (\psi_ k+\pi j/2)|^ p,\quad j=1,2,..., \] where \(D_ n=\int_{I_ n}dx/\omega (x)\), \(D_ n(pf)=\int_{I_ n}(x^{p_ j}/\omega (x))dx\) and \(I_ n=[\pi \lambda^{-1},\pi \lambda]\).