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Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p ( ℝ d ) (in the case p > 1 ), but (in the case when 1 / p ( · ) is log-Hölder continuous and p - = inf { p ( x ) : x ∈ ℝ d } > 1 ) on the variable Lebesgue spaces L p ( · ) ( ℝ d ) , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type ( 1 , 1 ) . In the present note we generalize Besicovitch’s covering theorem for the so-called γ -rectangles. We introduce a general maximal operator M s γ , δ and with the help of generalized Φ -functions, the strong- and weak-type...

Well-rounded sublattices of planar lattices

Michael Baake, Rudolf Scharlau, Peter Zeiner (2014)

Acta Arithmetica

A lattice in Euclidean d-space is called well-rounded if it contains d linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The task of enumerating well-rounded sublattices of a given lattice is of interest already in dimension 2, and has recently been treated by several authors. In this paper, we analyse the question more closely in the spirit of earlier work on similar sublattices and coincidence...

Windings of planar random walks and averaged Dehn function

Bruno Schapira, Robert Young (2011)

Annales de l'I.H.P. Probabilités et statistiques

We prove sharp estimates on the expected number of windings of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

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