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 $L_{1/2}\left(G\right).$

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...

The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^1$ is not locally integrable in general. However, in view of the duality between $H^1$ and $BMO$, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic...

We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood- Paley theory on M. This in...

This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^p$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^p$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques...

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators ${ℳ}^{+}$ and ${ℳ}^{-}$. More precisely, we prove that ${ℳ}^{+}$ and ${ℳ}^{-}$ map $W^{1,p}\left(ℝ\right)\to W^{1,p}\left(ℝ\right)$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $\mathrm{BV}\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ boundedly and map $l^1\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$, that is, $$\mathrm{Var}\left(M^{+}\left(f\right)\right)\le \mathrm{Var}\left(f\right)\text{and}\mathrm{Var}\left(M^{-}\left(f\right)\right)\le \mathrm{Var}\left(f\right)$$ if $f\in \mathrm{BV}\left(\mathbb{Z}\right)$, where $\mathrm{Var}\left(f\right)$ is the total variation of $f$ on $\mathbb{Z}$ and $\mathrm{BV}\left(\mathbb{Z}\right)$ is the set of all functions $f:\mathbb{Z}\to ℝ$ satisfying $\mathrm{Var}\left(f\right)<\infty$.

We consider the boundedness of certain singular integral operators that arose in the study of Sobolev spaces on Lipschitz curves, [P1]. The standard theory available (David and Journé's T1 Theorem, for instance; see [D]) does not apply to this case becuase the operators are not necessarily Calderón-Zygmund operators, [Ch]. One of these operators gives an explicit formula for the resolvent at λ = 1 of the dyadic paraproduct, [Ch].

We give $A_p$ type conditions which are sufficient for two-weight, strong $(p,p)$ inequalities for Calderón-Zygmund operators, commutators, and the Littlewood-Paley square function $g_{\lambda }^{*}$. Our results extend earlier work on weak $(p,p)$ inequalities in [13].

The uniqueness theorem for the ergodic maximal operator is proved in the continous case.

It is proved that the ergodic maximal operator is one-to-one.

It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.

We prove a regularity result for weak minima of integral functionals of the form ${\int }_{\Omega }F(x,Du)dx$ where F(x,ξ) is a Carathéodory function which grows as ${\left|\xi \right|}^p$ with some p > 1.

The properties of rare maximal functions (i.e. Hardy-Littlewood maximal functions associated with sparse families of intervals) are investigated. A simple criterion allows one to decide if a given rare maximal function satisfies a converse weak type inequality. The summability properties of rare maximal functions are also considered.