172 research outputs found
On the regularity of maximal operators
We study the regularity of the bilinear maximal operator when applied to
Sobolev functions, proving that it maps with and , boundedly and continuously. The same result holds on when
. We also investigate the almost everywhere and weak convergence under the
action of the classical Hardy-Littlewood maximal operator, both in its global
and local versions.Comment: 10 page
Extremal functions in de Branges and Euclidean spaces
In this work we obtain optimal majorants and minorants of exponential type
for a wide class of radial functions on . These extremal
functions minimize the -distance to
the original function, where is a free parameter. To achieve this
result we develop new interpolation tools to solve an associated extremal
problem for the exponential function , where , in the general framework of de Branges
spaces of entire functions. We then specialize the construction to a particular
family of homogeneous de Branges spaces to approach the multidimensional
Euclidean case. Finally, we extend the result from the exponential function to
a class of subordinated radial functions via integration on the parameter
against suitable measures. Applications of the results presented
here include multidimensional versions of Hilbert-type inequalities, extremal
one-sided approximations by trigonometric polynomials for a class of even
periodic functions and extremal one-sided approximations by polynomials for a
class of functions on the sphere with an axis of symmetry
Some extremal functions in Fourier analysis, II
We obtain extremal majorants and minorants of exponential type for a class of
even functions on which includes and , where . We also give periodic versions of these results in which the
majorants and minorants are trigonometric polynomials of bounded degree. As
applications we obtain optimal estimates for certain Hermitian forms, which
include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev
inequalities. A further application provides an Erd\"{o}s-Tur\'{a}n-type
inequality that estimates the sup norm of algebraic polynomials on the unit
disc in terms of power sums in the roots of the polynomials.Comment: 40 pages. Accepted for publication in Trans. Amer. Math. So
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