Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space $X^{\xi }$ generated by subsequences $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$, respectively, of the natural Schauder basis $\left(e_n^{\xi }\right)$ of $X^{\xi }$ are isomorphic if and only if $\left(e_{l_n}^{\xi }\right)$ and $\left(e_{m_n}^{\xi }\right)$ are equivalent. Further, $X^{\xi }$ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of $\left(e_n^{\xi }\right)$. It is also shown that there exists a complemented subspace spanned by a block basis of $\left(e_n^{\xi }\right)$, which is not isomorphic to a subspace generated by a subsequence of $\left(e_n^{\zeta }\right)$,...

We prove that for every closed locally convex subspace E of $L_0$ and for any continuous linear operator T from $L_0$ to $L_0/E$ there is a continuous linear operator S from $L_0$ to $L_0$ such that T = QS where Q is the quotient map from $L_0$ to $L_0/E$.

We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in L^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_1^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)=e^{{i\left|y\right|}^a}$, a > 0, a ≠ 1, map B into itself.

Let $q,q_1,\cdots ,q_s\ge 2$ be integers, and let ${\alpha }_1,{\alpha }_2,\cdots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence $${(\left\{{\alpha }_m{q}^n\right\},\cdots ,\left\{{\alpha }_{m+s-1}{q}^n\right\})}_{m,n=1}^{MN}$$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${\left(xn\right)}_{n=1}^{MN}$ in $s$-dimensional unit cube $(s,M,N=1,2,\cdots )$. We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\left\{{\alpha }_1{q}_1^n\right\},\cdots ,\left\{{\alpha }_s{q}_s^n\right\})}_{n=1}^N$ (Korobov’s problem).

$(s_{jk}:j,k=0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $\left(s_{jk}\right)$ converges in Pringsheim’s sense. These conditions are satisfied if $\left(s_{jk}\right)$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $\left(s_{jk}\right)$ is summable (C,1,1) to a finite limit and there exist constants $n_1>0$ and H such that $jk(s_{jk}-s_{j-1,k}-s_{j-1,k}+s_{j-1,k-1})\ge -H$, $j(s_{jk}-s_{j-1,k})\ge -H$ and $k(s_{jk}-s_{j,k-1})\ge -H$ whenever $j,k>n_1$, then...

C. Watari [12] obtained a simple characterization of Lipschitz classes $Li{p}^{\left(p\right)}\alpha \left(W\right)(1\ge p\ge \infty ,\alpha >0)$ on the dyadic group using the $L^p$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B_{p,q}^{\alpha }$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B_{p,q}^{\alpha }$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality...

Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: ${\left|\right|s*\left|\right|}_{p,\mu }\le C_{p,\alpha ,\beta }{{\Sigma }_{j=0}^{\infty }{\Sigma }_{k=0}^{\infty }{\left(j̅\right)}^{p-\alpha -2}{\left(k̅\right)}^{p-\beta -2}{\left|c_{jk}\right|}^p}^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...

Let G be the Walsh group. For $f\in L^1\left(G\right)$ we prove the a. e. convergence σf → f(n → ∞), where ${\sigma }_n$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $\sigma *f≔su{p}_n\left|{\sigma }_{n}f\right|.$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $\parallel \sigma *f{\parallel }_1\le c\parallel \left|f\right|{\parallel }_H$, where H is the Hardy space on the Walsh group.

We give a counterexample showing that $\overline{(I-T*)L_{\infty }}\cap L_{\infty }^{+}=0$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.