We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.

By a Fourier multiplier technique on Cantor-like Abelian groups with characters of finite order, the norms from L² into $L^{2l}$ of certain embeddings of character sums are computed. It turns out that the orders of the characters are immaterial as soon as they are at least four.

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp Lp estimates for p > 4, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.

Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?

We study two rather different problems, one arising from Diophantine geometry and one arising from Fourier analysis, which lead to very similar questions, namely to the study of the ranks of matrices with entries either zero or $\left(\right(xy/q\left)\right),(0\le x,y\<q)$, where $\left(\right(u\left)\right)=u-\left[u\right]-1/2$ denotes the “centered” fractional part of $x$. These ranks, in turn, are closely connected with the non-vanishing of the Dirichlet $L$-functions at $s=1$.

We study the relationship between the growth rate of an integer sequence and harmonic and functional properties of the corresponding sequence of characters. In particular we show that every polynomial sequence contains a set that is $\Lambda \left(p\right)$ for all $p$ but is not a Rosenthal set. This holds also for the sequence of primes.

In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the $\mathbb{Z}{₂}^d$ case by establishing a result of exponential integrability corresponding to the case p = +∞.

In this paper, we give a generalization of Fefferman-Stein inequality for the fractional one-sided maximal operator: $${\int }_{-\infty }^{+\infty }{M}_{\alpha }^{+}\left(f\right){\left(x\right)}^{p}w\left(x\right)dx\le A_p{\int }_{-\infty }^{+\infty }{\left|f\left(x\right)\right|}^p{M}_{\alpha p}^{-}\left(w\right)\left(x\right)dx,$$ where $0<\alpha <1$ and $1<p<1/\alpha$. We also obtain a substitute of dual theorem and weighted norm inequalities for the one-sided fractional integral $I_{\alpha }^{+}$.

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s &lt; n, [resp. 0 &lt; s &lt; n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 &lt; p &lt; ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

Let φ: R → [0,∞) an integrable function such that φχ(-∞,0) = 0 and φ is decreasing in (0,∞). Let τhf(x) = f(x-h), with h ∈ R {0} and fR(x) = 1/R f(x/R), with R &gt; 0. In this paper we characterize the pair of weights (u, v) such that the operators Mτhφf(x) = supR&gt;0|f| * [τhφ]R(x) are of weak type (p, p) with respect to (u, v), 1 &lt; p &lt; ∞.

The inequality (*) $({\sum }_{\left|n\right|=1}^{\infty }{\sum }_{\left|m\right|=1}^{\infty }{\left|nm\right|}^{p-2}{\left|f̂(n,m)\right|}^p{)}^{1/p}\le C_p\parallel ƒ{\parallel }_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series...

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 of $s_{mn}(x,y)$...