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On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

We prove a regularity result for weak minima of integral functionals of the form ∫ Ω F ( x , D u ) d x where F(x,ξ) is a Carathéodory function which grows as | ξ | p with some p > 1.

On weak type inequalities for rare maximal functions

K. Hare, A. Stokolos (2000)

Colloquium Mathematicae

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

A. M. Stokolos (2006)

Colloquium Mathematicae

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

Bianca Stroffolini (1995)

Studia Mathematica

We study very weak solutions of an A-harmonic equation to show that they are in fact the usual solutions.

On weighted inequalities for operators of potential type

Shiying Zhao (1996)

Colloquium Mathematicae

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...

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average A ₙ f ( x ) = 2 - n ∫ x x + 2 ⁿ f . The square function is defined as S f ( x ) = ( ∑ n = - ∞ ∞ | A ₙ f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 . The local version of this operator, namely the operator S ₁ f ( x ) = ( ∑ n = - ∞ 0 | A ₙ f ( x ) - A n - 1 f ( x ) | ² ) 1 / 2 , 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 L p into itself (p > 1) and L ∞ into BMO. We prove that the operator S not only maps L ∞ into BMO but it also maps BMO into BMO. We also prove that the L p boundedness still holds...

One-weight weak type estimates for fractional and singular integrals in grand Lebesgue spaces

Vakhtang Kokilashvili, Alexander Meskhi (2014)

Banach Center Publications

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.

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