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A Cantor set in the plane that is not σ-monotone

100%

Nekvinda A., Zindulka O.

Fundamenta Mathematicae

2011

tom 213

nr 3

221-232

A metric space (X,d) is monotone if there is a linear order < on X and a constant c such that d(x,y) ≤ cd(x,z) for all x < y < z in X, and σ-monotone if it is a countable union of monotone subspaces. A planar set homeomorphic to the Cantor set that is not σ-monotone is constructed and investigated. It follows that there is a metric on a Cantor set that is not σ-monotone. This answers a question raised by the second author.

Optimal estimates for the fractional Hardy operator

81%

Mizuta Y., Nekvinda A., Shimomura T.

Studia Mathematica

2015

tom 227

nr 1

1-19

Let $A_{α}f(x) = |B(0,|x|)|^{-α/n} ∫_{B(0,|x|)} f(t)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{α}$ is bounded from $L^{p}$ to $L^{p_{α}}$ with $p_{α} = np/(αp-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a 'source' space $S_{α,Y}$, which is strictly larger than X, and a 'target' space $T_{Y}$, which is strictly smaller than Y, under the assumption that $A_{α}$ is bounded from X into Y and the Hardy-Littlewood maximal operator M is bounded from Y into Y, and prove that $A_{α}$ is bounded from $S_{α,Y}$ into $T_{Y}$. We prove optimality results for the action of $A_{α}$ and the associate operator $A'_{α}$ on such spaces, as an extension of the results of Mizuta et al. (2013) and Nekvinda and Pick (2011). We also study the duals of optimal spaces for $A_{α}$.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Studia Mathematica

2 Nekvinda A.

1 Mizuta Y.

1 Shimomura T.

1 Zindulka O.

1 2015

1 2011

Wyniki wyszukiwania

A Cantor set in the plane that is not σ-monotone

Optimal estimates for the fractional Hardy operator