We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

Given a positive measure μ in ${ℝ}^n$, there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{\mu }f\left(x\right)=su{p}_{x\in B}1/\mu \left(B\right){ʃ}_B\left|f\right|d\mu$, where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in ${ℝ}^n$. We give some necessary and sufficient conditions for $M_{\mu }$ to be bounded from $L^1\left(d\mu \right)$ to $L^{1,\infty }\left(d\mu \right)$.

For d > 1, let $\left(S_{d}f\right)(x,t)={ʃ}_{{ℝ}^n}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, $x\in {ℝ}^n$, where f̂ is the Fourier transform of $f\in S\left({ℝ}^n\right)$, and $(S_d*f)\left(x\right)=su{p}_{0<t<1}\left|\left(S_{d}f\right)(x,t)\right|$ its maximal operator. P. Sjölin ([11]) has shown that for radial f, the estimate (*) $({ʃ}_{\left|x\right|<R}|(S_d*f)\left(x\right){{|}^{p}dx)}^{1/p}\le C_R\parallel f{\parallel }_{H_{1/4}}$ holds for p = 4n/(2n-1) and fails for p > 4n/(2n-1). In this paper we show that for non-radial f, (*) fails for p > 2. A similar result is proved for a more general maximal operator.

In this paper we prove that the maximal operator $${\tilde{\sigma }}^{\kappa ,*}f:=\underset{n\in \mathbb{P}}{sup}\frac{|{\sigma }_n^{\kappa }f|}{{log}^2(n+1)},$$ where ${\sigma }_n^{\kappa }f$ is the $n$-th Fejér mean of the Walsh-Kaczmarz-Fourier series, is bounded from the Hardy space $H_{1/2}\left(G\right)$ to the space $L_{1/2}\left(G\right).$

MSC 2010: 42A32; 42A20In this paper we have defined a new class of double numerical sequences. If the coefficients of a double cosine or sine trigonometric series belong to the such classes, then it is verified that they are Fourier series or equivalently their sums are integrable functions. In addition, we obtain an estimate for the mixed modulus of smoothness of a double sine Fourier series whose coefficients belong to the new class of sequences mention above.

We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_p$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

We prove and discuss some new $(H_p,L_p)$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\{q_k:k\ge 0\}$. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund...

For a Lebesgue integrable complex-valued function $f$ defined on ${ℝ}^{+}:=[0,\infty )$ let $\widehat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\widehat{f}\left(y\right)\to 0$ as $y\to \infty$. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1\left({ℝ}^{+}\right)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We...

We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.