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$L^{p}$ boundedness of maximal averages over hypersurfaces in $\mathbb {R}^{3}$

Michael Greenblatt

2012
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Transactions of the American Mathematical Society
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Extending the methods developed in the author's recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an L p boundedness theorem is proven for maximal operators over hypersurfaces in R 3 when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010). Newton
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... (2010). Newton polygons and adapted coordinates. We now give some relevant terminology which will be used throughout this paper. Below, R(x, y) denotes a smooth function defined on a neighborhood of the origin with nonvanishing Taylor expansion at the origin. Definition 1.2. Let R(x, y) = a,b R ab x a y b denote the Taylor expansion of R(x, y) at the origin. For any (a, b) for which R ab = 0, let Q ab be the quadrant {(x, y) ∈ R 2 : x ≥ a, y ≥ b}. Then the Newton polygon N (R) of R(x, y) is defined to be the convex hull of the union of all Q ab . In general, the boundary of a Newton polygon consists of finitely many (possibly zero) bounded edges of negative slope as well as an unbounded vertical ray and an unbounded horizontal ray. One often uses (t 1 , t 2 ) coordinates to write equations of lines relating to Newton polygons, so as to distinguish from the x, y variables of the domain of R(x, y). The line in the (t 1 , t 2 ) plane with equation t 1 = t 2 comes up so frequently it has its own name: Definition 1.4. The bisectrix is the line in the (t 1 , t 2 ) plane with equation t 1 = t 2 . A key role to follow in the above theorems as well as our theorems is played by the following polynomials. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MAXIMAL AVERAGES 1877 Definition 1.5. Suppose e is a compact edge of N (R). Define R e (x, y) by R e (x, y) = (a,b)∈e R ab x a y b . In other words R e (x, y) is the sum of the terms of the Taylor expansion of f corresponding to (a, b) ∈ e. Definition 1.6. Suppose R(x, y) has nonvanishing Taylor expansion at the origin such that R(0, 0) = 0 and ∇R(0, 0) = 0. Then R(x, y) is said to be in nonadapted coordinates if the bisectrix intersects N (R) in the interior of a compact edge e of N (R) such that R e (1, y) has a zero of order greater than d(R). If R(x, y) is not in nonadapted coordinates, then R(x, y) is said to be in adapted coordinates.

doi:10.1090/s0002-9947-2012-05697-2
fatcat:cxyu7pdghfc2lc6ggkwqrrxvbq