The aim of this paper is to obtain sharp estimates from below of the measure of the set of divergence of the m-fold Fourier series with respect to uniformly bounded orthonormal systems for the so-called G-convergence and λ-restricted convergence. We continue the study begun in a previous work.

Let ⁿ denote the usual n-torus and let $S{̃}_u^{\delta }\left(f\right)$, u > 0, denote the Bochner-Riesz means of order δ > 0 of the Fourier expansion of f ∈ L¹(ⁿ). The main result of this paper states that for f ∈ H¹(ⁿ) and the critical index α: = (n-1)/2, $li{m}_{R\to \infty }1/logR{\int }_0^R\left(\right||S{̃}_u^{\alpha }\left(f\right)-{f\left|\right|}_{H¹\left(ⁿ\right)})/(u+1)du=0$.

We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from $L_{\infty }$ to $L_{\infty }$ then it is also bounded from the classical Hardy space $H_p\left(T\right)$ to $L_p$ (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from $H_p\left(T\right)$ to $L_p$ (3/4 < p ≤ ∞) and is of weak type $(L_1,L_1)$. We define the two-dimensional dyadic hybrid Hardy space $H_1^{♯}\left(T^2\right)$ and verify that the maximal operator of the Cesàro means of a two-dimensional...

We investigate some convergence and divergence properties of the logarithmic means of quadratic partial sums of double Fourier series of functions, in measure and in the L Lebesgue norm.

We costruct functions in $H_w^1$ ($w\in A_1$) whose Fourier integral expansions are almost everywhere non-summable with respect to the Bochner-Riesz means of the critical order.

We consider the maximal function $\left|\right|\left(S^{a}f\right)\left[x\right]{\left|\right|}_{L^{\infty }[-1,1]}$ where $\left(S^{a}f\right){\left(t\right)}^{\wedge }\left(\xi \right)=e^{{it\left|\xi \right|}^a}f̂\left(\xi \right)$ and 0 < a < 1. We prove the global estimate $\left|\right|S^{a}f{\left|\right|}_{L²(ℝ,L^{\infty }[-1,1])}{\le C\left|\right|f\left|\right|}_{H^s\left(ℝ\right)}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

The two-dimensional classical Hardy spaces $H_p\left(ℝ\times ℝ\right)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p\left(ℝ\times ℝ\right)$ to $L_p\left({ℝ}^2\right)$ (1/2 < p ≤ ∞) and is of weak type $(H_1^{}\left(ℝ\times ℝ\right),L_1\left({ℝ}^2\right))$ where the Hardy space $H_1\left(ℝ\times ℝ\right)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^{(}ℝ\times ℝ)$ ⊃ $LlogL\left({ℝ}^2\right)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p\left(ℝ\times ℝ\right)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p\left(ℝ\times ℝ\right)$, the Fejér means...

The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w\left(t\right)={(1-t)}^{\alpha }{(1+t)}^{\beta }$. Almost exponentially localized polynomial elements (needlets) ${\phi }_{\xi }$, ${\psi }_{\xi }$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $⟨f,{\phi }_{\xi }⟩$ in respective sequence spaces.

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W(h_p,{\ell }_{\infty })$ to $W(L_p,{\ell }_{\infty })$. This implies the almost everywhere convergence of the Fejér means in a cone for all $f\in W(L₁,{\ell }_{\infty })$, which is larger than $L₁\left({ℝ}^d\right)$.