On the uniqueness of the uncentered ergodic maximal function
It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.
On weak minima of certain integral functionals
We prove a regularity result for weak minima of integral functionals of the form where F(x,ξ) is a Carathéodory function which grows as with some p > 1.
On weak type inequalities for rare maximal functions
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.
On weak type inequalities for rare maximal functions in ℝⁿ
The study of one-dimensional rare maximal functions was started in [4,5]. The main result in [5] was obtained with the help of some general procedure. The goal of the present article is to adapt the procedure (we call it "dyadic crystallization") to the multidimensional setting and to demonstrate that rare maximal functions have properties not better than the Strong Maximal Function.
On weakly A-harmonic tensors
We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.
On weighted dyadic Carleson's inequalities.
On weighted estimates for the Kakeya maximal operator
On weighted inequalities for certain fractional integral operators.
On weighted inequalities for ergodic operators
On weighted inequalities for operators of potential type
In this paper, we discuss a class of weighted inequalities for operators of potential type on homogeneous spaces. We give sufficient conditions for the weak and strong type weighted inequalities sup_{λ>0} λ|{x ∈ X : |T(fdσ)(x)|>λ }|_{ω}^{1/q} ≤ C (∫_{X} |f|^{p}dσ)^{1/p} and (∫_{X} |T(fdσ)|^{q}dω )^{1/q} ≤ C (∫_X |f|^{p}dσ )^{1/p} in the cases of 0 < q < p ≤ ∞ and 1 ≤ q < p < ∞, respectively, where T is an operator of potential type, and ω and σ are Borel measures on the homogeneous...
On weighted norm inequalities for the maximal function
On weighted norm inequalities for the Riesz transforms of functions with vanishing moments
On weighted weak type norm inequalities for one-sided oscillatory singular integrals
We consider one-sided weight classes of Muckenhoupt type and study the weighted weak type (1,1) norm inequalities for a class of one-sided oscillatory singular integrals with smooth kernel.
One-sided discrete square function
Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average . The square function is defined as . The local version of this operator, namely the operator , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps into itself (p > 1) and into BMO. We prove that the operator S not only maps into BMO but it also maps BMO into BMO. We also prove that the boundedness still holds...
One-Sided Littlewood-Paley Theory.
One-weight weak type estimates for fractional and singular integrals in grand Lebesgue spaces
We investigate weak type estimates for maximal functions, fractional and singular integrals in grand Lebesgue spaces. In particular, we show that for the one-weight weak type inequality it is necessary and sufficient that a weight function belongs to the appropriate Muckenhoupt class. The same problem is discussed for strong maximal functions, potentials and singular integrals with product kernels.
Operador de Hardy-Littlewood reiterado y clases de Orlicz de funciones maximales.
Operator valued BMO and commutators .
Operators Associated with Hermite Semigroup - A Survey.
Operators associated with representations of amenable groups singular integrals induced by ergodic flows, the rotation method and multipliers