The $(L^1,L^1)$ bilinear Hardy-Littlewood function and Furstenberg averages

On the Variation of the Hardy–littlewood Maximal Function

We show that a function f : R → R of bounded variation satisfies VarMf ≤ C Var f, where Mf is the centered Hardy–Littlewood maximal function of f . Consequently, the operator f 7→ (Mf) is bounded from W (R) to L(R). This answers a question of Hajłasz and Onninen in the one-dimensional case.

A Sharp Estimate for the Hardy-littlewood Maximal Function

The best constant in the usual L norm inequality for the centered Hardy-Littlewood maximal function on R is obtained for the class of all “peak-shaped” functions. A function on the line is called “peakshaped” if it is positive and convex except at one point. The techniques we use include variational methods. AMS Classification (1991): 42B25 0. Introduction. Let (0.1) (Mf)(x) = sup δ>0 1 2δ ∫ x+δ

On the Hardy - Littlewood majorant

The Hardy-Littlewood Function An Exercise in Slowly Convergent Series

The function in question is H(x) = ∑∞ k=1 sin(x/k)/k. In deference to the general theme of this conference, a summation procedure is first described using orthogonal polynomials and polynomial/rational Gauss quadrature. Its effectiveness is limited to relatively small (positive) values of x. Direct summation with acceleration is shown to be more powerful for very large values of x. Such values ...

Vector A2 Weights and a Hardy-littlewood Maximal Function

An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values in finite-dimensional Hilbert spaces. It is shown to be L bounded with respect to weights in the class A2 of Treil, thereby extending a theorem of Muckenhoupt from the scalar to the vector case. A basic chapter of the subject of singular integral operators is the weighted norm theory, which provides ...