IUMJ

Title: Weak-type estimate for a maximal function

Authors: Sanjay Patel

Issue: Volume 55 (2006), Issue 1, 341-368

Abstract:

Let $ P(s,t) $ denote a real-valued polynomial of real variables $s$ and $t$. In this paper we show that the maximal function $\mathcal{M}$ defined by \[\mathcal{M}f(x) = \sup_{0 < h,k < 1} \frac{1}{hk} \left|\int_0^h \int_0^k f(x-P(s,t))\,{\mathrm d}s \,{\mathrm d}t \right|\] is weak-type 1-1 with a bound dependent on the coefficients of $P$.