We study mapping properties of the centered Hardy–Littlewood maximal operator $$\mathcal {M}$$ acting on Lorentz spaces $$L^{p,q}({\mathfrak {X}})$$ in context certain non-doubling metric measure $${\mathfrak {X}}$$ . The special class for which these are very peculiar is introduced and many examples given. In particular, each $$p_0, q_0, r_0 \in (1, \infty )$$ with $$r_0 \ge q_0$$ we construct...

Birkhoff used what has been called ([7]) 'Hilbert's projective metric' or ([9]) the 'Cayley-Hilbert metric'. In each case, it proved possible to obtain sharp estimates for the contraction constant of a positive linear operator with respect to the ' almost' metric. Subsequently, several authors generalized and sharpened the original results and established a close connection between the Birkhoff...

The analysis of holomorphic sections of high powers L of holomorphic ample line bundles L → M over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. Any polarized Kähler metric g with respect to the ample line bundle L corresponds to the Ricci curvature of a hermitian metric h on L. Any orthonormal basis {SN 0 , ..., S dN} of H(M,L ) induces a holomo...

Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F ) is a Borel subset of M if F is closed in L. We show that then f−1(y) is a Kσ set for all except countably many y ∈M , that M is also Luzin, and that the Borel classes of the sets f(F ), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the cla...

We show that if m,n ∈ N and k ∈ {1, . . . , n} satisfy m > n 3/2 √ k then for every p ∈ [2,∞) and f : Z4m → R we have 1 ( n k ) ∑ S⊆{1,...,n} |S|=k E [∣∣f(x + 2m∑j∈S εjej)− f(x)∣∣p] mp

Abstract. Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still questions which remain unknown. The Duffin-Schaeffer Conjecture is an attempt to answer all of these questions in full, and it has withstood more than...

This is essentially a note on Section 7 of Perelman’s first paper on Ricci flow. We list some basic properties of the index form for Perelman’s L-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse’s index theorem for Perelman’s L-length holds. As a corollary we get the finiteness of the number of the L-conjugate points along a finite L-geodesi...