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On a converse inequality for maximal functions in Orlicz spaces

100%

Kita H.

Studia Mathematica

1996

tom 118

nr 1

1-10

Let $Φ(t) = ʃ_{0}^{t} a(s)ds$ and $Ψ(t) = ʃ_{0}^{t} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_{1}^{∞} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_{s→∞}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_{0}$ such that $ʃ_{1}^{s} a(t)/t dt ≥ c_{1}b(c_{1}s)$ for all $s ≥ s_{0}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_{0}^{2π} Ψ((c_2)/(|⨍|_{𝕋}) |⨍(x)|) dx ≤ c_3 + c_{3} ʃ_{0}^{2π} Φ(1/(|⨍|_{𝕋})) Mf(x) dx$ for all $⨍ ∈ L^{1}(𝕋)$.

Resistance Conditions and Applications

61%

Kinnunen J., Silvestre P.

Analysis and Geometry in Metric Spaces

2013

tom 1

276-294

This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.

Ograniczanie wyników

1 Analysis and Geometry in Metric Spaces

1 Studia Mathematica

1 Kinnunen J.

1 Kita H.

1 Silvestre P.

1 2013

1 1996

Wyniki wyszukiwania

On a converse inequality for maximal functions in Orlicz spaces

Resistance Conditions and Applications