In the "ax+b" group of affine mappings of the line, we examine right maximal functions with respect to the left Haar measure. Each such maximal function is defined with respect to the right-translates of a one-parameter family of sets that shrinks to e. Family consisting of inverted rectangles turn out to be interesting here. In this cases the maximal function is always bounded on L^p. We prove that the weak type (1,1) property holds for certain rectangle shapes but not for others. The proof involves a maximal function on R^2 that it is not translation-invariant. The latter maximal function is taken with respect to a family of rectangles in the plane whose eccentricities depend on their location.
A note on maximal functions on a solvable Lie group
GIULINI, SAVERIO;
1990-01-01
Abstract
In the "ax+b" group of affine mappings of the line, we examine right maximal functions with respect to the left Haar measure. Each such maximal function is defined with respect to the right-translates of a one-parameter family of sets that shrinks to e. Family consisting of inverted rectangles turn out to be interesting here. In this cases the maximal function is always bounded on L^p. We prove that the weak type (1,1) property holds for certain rectangle shapes but not for others. The proof involves a maximal function on R^2 that it is not translation-invariant. The latter maximal function is taken with respect to a family of rectangles in the plane whose eccentricities depend on their location.
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