We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...

Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have $$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic, as a Banach space, to $L^{p,\infty }({\Omega }^{\text{'}},{\sum }^{\text{'}},{\mu }^{\text{'}})$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $\Omega ={\Omega }_1\cup {\Omega }_2$ such that $({\Omega }_1,\Sigma \cap {\Omega }_1,\mu {|}_{\Sigma \cap {\Omega }_1})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $\sigma \subseteq {\Omega }_2$. In particular, $L^{p,\infty }(\Omega ,\sum ,\mu )$ is isomorphic to ${\ell }^{p,\infty }$.

For every element x** in the double dual of a separable Banach space X there exists the sequence $\left(x^{\left(2n\right)}\right)$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class $B_1\left(X\right)╲B_{1/2}\left(X\right)$ (resp. to the class $B_{1/4}\left(X\right)$) as the elements with the sequence $\left(x^{\left(2n\right)}\right)$ equivalent to the usual basis of ${\ell }^1$ (resp. as the elements with the sequence $(x^{(4n-2)}-x^{\left(4n\right)})$ equivalent to the...

Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^{*}$ of a Banach space X, then ${\ell }^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\ell }^1$ contains a copy of ${\ell }^1$ that is complemented in ${\ell }^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1[0,1]$ contains a copy of ${\ell }^1$ that is complemented in $L^1[0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of...

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 $\left(e_n^{\zeta }\right)$,...

It is shown that the weak $L^p$ spaces ${\ell }^{p,\infty },L^{p,\infty }[0,1]$, and $L^{p,\infty }[0,\infty )$ are isomorphic as Banach spaces.

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.

$(s_{jk}:j,k=0,1,...)$ 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...

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact...