1,990 research outputs found
Singular measures and convolution operators
We show that in the study of certain convolution operators, functions can be
replaced by measures without changing the size of the constants appearing in
weak type (1,1) inequalities. As an application, we prove that the best
constants for the centered Hardy-Littlewood maximal operator associated to
parallelotopes do not decrease with the dimension.Comment: 8 page
Dimension dependency of the weak type bounds for maximal functions associated to finite radial measures
We show that the best constants appearing in the weak type (1,1) inequalities
satisfied by the centered Hardy-Littlewood maximal function associated to some
finite radial measures, such as the standard gaussian measure, grow
exponentially fast with the dimension.Comment: 7 pages, to appear in the Bull. London Math. So
The weak type bounds for the maximal function associated to cubes grow to infinity with the dimension
Let be the centered Hardy-Littlewood maximal function associated to
cubes in with Lebesgue measure, and let denote the lowest
constant appearing in the weak type (1,1) inequality satisfied by .
We show that as , thus answering, for the case
of cubes, a long standing open question of E. M. Stein and J. O. Str\"{o}mberg.Comment: A very similar version to this posting (but with fewer explicit
constants, in order to satisfy the requests of an anonymous referee) is to
appear in Ann. Math.; additionally, and with respect to the last version,
this one contains several simplifications and improvements in the exposition,
suggested by the refere
Besicovitch type properties in metric spaces
We explore the relationship in metric spaces between different properties
related to the Besicovitch covering theorem, with emphasis on geometrically
doubling spaces.Comment: 14 page
Strengthened Cauchy-Schwarz and H\"older inequalities
We present some identities related to the Cauchy-Schwarz inequality in
complex inner product spaces. A new proof of the basic result on the subject of
Strengthened Cauchy-Schwarz inequalities is derived using these identities.
Also, an analogous version of this result is given for Strengthened H\"older
inequalities
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