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1
Content available remote

Some weighted inequalities for general one-sided maximal operators

100%
J. Martín-Reyes F., de la Torre A.
Studia Mathematica
|
1997
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tom 122
|
nr 1
1-14
EN
We characterize the pairs of weights on ℝ for which the operators $M^{+}_{h,k}f(x) = sup_{c>x}h(x,c) ʃ_{x}^{c} f(s)k(x,s,c)ds$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on ${(x,c): x < c}$, while k is defined on ${(x,s,c): x < s < c}$. If $h(x,c) = (c-x)^{-β}$, $k(x,s,c) = (c-s)^{α-1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M^{+}_{α,β}f = sup_{c>x} 1/(c-x)^{β} ʃ_{x}^{c} f(s)/(c-s)^{1-α} ds$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and 1 < p < ∞, weak and strong type are equivalent. If we take α = β we characterize the strong type weights for the operator $M^{+}_{α,α}$ introduced by W. Jurkat and J. Troutman in the study of $C_α$ differentiation of the integral.
2
Content available remote

One-sided discrete square function

100%
de la Torre A., Torrea J.
Studia Mathematica
|
2003
|
tom 156
|
nr 3
243-260
EN
Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf(x) = 2^{-n} ∫_{x}^{x+2ⁿ} f$. The square function is defined as $Sf(x) = (∑_{n=-∞}^{∞} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$. The local version of this operator, namely the operator $S₁f(x) = (∑_{n=-∞}^{0} |Aₙf(x) - A_{n-1}f(x)|²)^{1/2}$, 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 if one replaces Lebesgue measure by a measure of the form w(x)dx if, and only if, the weight w belongs to the $A⁺_{p}$ class introduced by E. Sawyer [8]. Finally we prove that the one-sided Hardy-Littlewood maximal function maps BMO into itself.
3
Content available remote

Weighted weak type inequalities for the ergodic maximal function and the pointwise ergodic theorem

80%
de la Torre A., Martín-Reyes F. J.
Studia Mathematica
|
1987
|
tom 87
|
nr 1
33-46
4
Content available remote

An atomic theory of ergodic $H^{p}$ spaces

50%
Caballero R., de la Torre A.
Studia Mathematica
|
1985
|
tom 82
|
nr 1
39-59
5
Content available remote

On the Pointwise Ergodic Theorem in $L_{p}$

41%
Martín-Reyes F. J., de la Torre A.
Studia Mathematica
|
1994
|
tom 108
|
nr 1
1-4
6
Content available remote

A dominated ergodic estimate for $L_{p}$ spaces with weights

41%
Atencia E., de la Torre A.
Studia Mathematica
|
1982
|
tom 74
|
nr 1
35-47
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