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1 answer
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How to reduce the compact support to the case of small diameters in Tao's "A sharp bilinear restriction estimate for paraboloids"

I am reading Terence Tao's paper "A sharp bilinear restriction estimate for paraboloids" to prove the bilinear restriction estimate on paraboloids. In Section 3, he assumes that $\text{diam}(...
7 votes
2 answers
1k views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
6 votes
1 answer
591 views

For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...