The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_p\left({ℝ}^d\right)$ (in the case $p>1$), but (in the case when $1/p(·)$ is log-Hölder continuous and $p_{-}=inf\{p\left(x\right):x\in {ℝ}^d\}>1$) on the variable Lebesgue spaces $L_{p(·)}\left({ℝ}^d\right)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch’s covering theorem for the so-called $\gamma$-rectangles. We introduce a general maximal operator $M_s^{\gamma ,\delta }$ and with the help of generalized $\Phi$-functions, the strong-...

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\le p<\infty$. More precisely, given any measurable set $E_0$, the estimate $$w\left(\{x\in {ℝ}^n:M^{+,d}\left({𝒳}_{E_0}\right)\left(x\right)>t\}\right)\le \frac{C{\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}^p}{t^p}v\left(E_0\right)$$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}\left(ℛ\right)$, that is, $$\frac{\left|E\right|}{\left|Q\right|}\le {\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}{\left(\frac{v\left(E\right)}{w\left(Q\right)}\right)}^{1/p}$$ for every dyadic cube $Q$ and every measurable set $E\subset Q^{+}$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...

Let $M_S$ denote the strong maximal operator. Let $M_x$ and $M_y$ denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that ${\int }_Q{M}_y{M}_{x}h<\infty$ but ${\int }_Q{M}_x{M}_{y}h=\infty$. It is shown that if f is a function supported on Q such that ${\int }_Q{M}_y{M}_{x}f<\infty$ but ${\int }_Q{M}_x{M}_{y}f=\infty$, then there exists a set A of finite measure in ℝ² such that ${\int }_A{M}_{S}f=\infty$.

The main aim of this paper is to prove that the maximal operator ${\sigma }^{}$ is bounded from the Hardy space $H_{1/2}$ to weak-$L_{1/2}$ and is not bounded from $H_{1/2}$ to $L_{1/2}$.

Let $R$ be a commutative ring with unity. The notion of maximal non $\lambda$-subrings is introduced and studied. A ring $R$ is called a maximal non $\lambda$-subring of a ring $T$ if $R\subset T$ is not a $\lambda$-extension, and for any ring $S$ such that $R\subset S\subseteq T$, $S\subseteq T$ is a $\lambda$-extension. We show that a maximal non $\lambda$-subring $R$ of a field has at most two maximal ideals, and exactly two if $R$ is integrally closed in the given field. A determination of when the classical $D+M$ construction is a maximal non $\lambda$-domain is given. A necessary condition...

The notion of maximal non-pseudovaluation subring of an integral domain is introduced and studied. Let $R\subset S$ be an extension of domains. Then $R$ is called a maximal non-pseudovaluation subring of $S$ if $R$ is not a pseudovaluation subring of $S$, and for any ring $T$ such that $R\subset T\subset S$, $T$ is a pseudovaluation subring of $S$. We show that if $S$ is not local, then there no such $T$ exists between $R$ and $S$. We also characterize maximal non-pseudovaluation subrings of a local integral domain.

Let $U₁,...,U_d$ be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of $\mathbb{Z}{₊}^d$ and the associated collection of rectangular parallelepipeds in ${ℝ}^d$ with sides parallel to the axes and dimensions of the form $n₁\times \cdots \times n_d$ with $(n₁,...,n_d)\in \Gamma .$ The associated multiparameter geometric and ergodic maximal operators $M_{}$ and $M_{\Gamma }$ are defined respectively on $L¹\left({ℝ}^d\right)$ and L¹(Ω) by $M_{}g\left(x\right)=su{p}_{x\in R\in }1/\left|R\right|{\int }_R\left|g\left(y\right)\right|dy$ and $M_{\Gamma }f\left(\omega \right)=su{p}_{(n₁,...,n_d)\in \Gamma }1/n₁\cdots n_d{\sum }_{j₁=0}^{n₁-1}\cdots {\sum }_{j_d=0}^{n_d-1}\left|f\left(U{₁}^{j₁}\cdots U_d^{j_d}\omega \right)\right|$. Given a Young function Φ, it is shown that $M_{}$ satisfies the weak type estimate ...

Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in D^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$. ...

Let Y be a Banach space and let $S\subset L_p$ be a subspace of an $L_p$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S\left(Y\right)\subset L_p\left(Y\right)$. We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{-tB}$ is a positive contraction...

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.

Let $R$ be a ring, $n$ a fixed non-negative integer, $𝒲ℐ$ the class of all left $R$-modules with weak injective dimension at most $n$, and $𝒲ℱ$ the class of all right $R$-modules with weak flat dimension at most $n$. Using left (right) $𝒲ℐ$-resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that $-\otimes -$ is right balanced on ${ℳ}_R\times {}_Rℳ$ by $𝒲ℱ\times 𝒲ℐ$, and investigate the global right $𝒲ℐ$-dimension of ${}_Rℳ$ by right derived functors of $\otimes$.

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d}f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We give some rather weak sufficient condition for $L^p$ boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ on the product spaces $ℝⁿ\times {ℝ}^m$ (1 < p < ∞), which improves and extends some known results.