Weak-type estimate for a maximal function
Sanjay Patel
42B20Newton diagramCalderon-Zygmund decompositionvan der Corput's lemma
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 $P$.
Indiana University Mathematics Journal
2006
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10.1512/iumj.2006.55.2689
10.1512/iumj.2006.55.2689
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Indiana Univ. Math. J. 55 (2006) 341 - 368
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