Maximal operator for pseudo-differential operators with homogeneous symbols

TL;DR

This paper establishes a Sjölin-type maximal estimate for pseudo-differential operators with homogeneous symbols, introducing a novel phase decomposition formula that avoids time translation, advancing the understanding of maximal estimates in harmonic analysis.

Contribution

It introduces a new phase decomposition formula for pseudo-differential operators that does not involve time translation, providing a different approach from previous methods.

Findings

Established a Sjölin-type maximal estimate for homogeneous pseudo-differential operators.

Developed a phase decomposition formula independent of time translation.

Provided new Cotlar type estimates relevant to the maximal operator.

Abstract

The aim of the present paper is to obtain a Sj\"{o}lin-type maximal estimate for pseudo-differential operators with homogeneous symbols. The crux of the proof is to obtain a phase decomposition formula which does not involve the time traslation. The proof is somehow parallel to the paper by Pramanik and Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a maximal dyadic sum operator on $R^n$, Illinois J. Math, {\bf 47} (2003), no. 3, 775--813). In the present paper, we mainly concentrate on our new phase decomposition formula and the results in the Cotlar type estimate, which are different from the ones by Pramanik and Terwilleger.