Glasgow Mathematical Journal

Table of Contents - September 2012 - Volume 54, Issue 03  

Research Article

BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

ADAM OSȨKOWSKIa1

a1 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland e-mail: ados@mimuw.edu.pl

Abstract

Let μ be a Borel measure on ℝ. The paper contains the proofs of the estimates

\begin{linenomath} $$ ||\mathcal{M}_\mu f||_{L^{q,\infty}(A,\mu)}\leq c_{p,q}||f||_{L^p(\R,\mu)}\,\mu(A)^{1/q-1/p},\qquad 1\leq p<\infty,\,q\in (0,p], $$ \end{linenomath}

and
\begin{linenomath} $$ ||\mathcal{M}_\mu f||_{L^{q,\infty}(A,\mu)}\leq C_{p,q}||f||_{L^{p,\infty}(\R,\mu)}\,\mu(A)^{1/q-1/p},\quad 1<p<\infty, q\in (0,p]. $$ \end{linenomath}

Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and c p,q , and C p,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for c p,q and C p,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝ n with respect to a general product measure.

(Received November 20 2011)

(Accepted January 25 2012)

(Online publication March 30 2012)

2000 Mathematics Subject Classification

  • Primary: 42B25;
  • Secondary: 42B35;
  • 46E30