Estimates for maximal functions associated to hypersurfaces in ℝ³ with height 𝕙<2: Part I
Annotatsiya
In this article, we continue the study of the problem of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -boundedness of the maximal operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated to averages along isotropic dilates of a given, smooth hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite type in 3-dimensional Euclidean space. An essentially complete answer to this problem was given about eight years ago by the third and fourth authors in joint work with M. Kempe [Acta Math 204 (2010), pp. 151–271] for the case where the height <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the given surface is at least two. In the present article, we turn to the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h greater-than 2 period"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>2.</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">h>2.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> More precisely, in this Part I, we study the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h greater-than 2 comma"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h>2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> assuming that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is contained in a sufficiently small neighborhood of a given point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript 0 Baseline element-of upper S"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo> ∈ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x^0\in S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at which both principal curvatures of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanish. Under these assumptions and a natural transversality assumption, we show that, as in the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h greater-than-or-equal-to 2 comma"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h\ge 2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the critical Lebesgue exponent for the boundedness of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> remains to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript c Baseline equals h comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p_c=h,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> even though the proof of this result turns out to require new methods, some of which are inspired by the more recent work by the third and fourth authors on Fourier restriction to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S period"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Results on the case where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">h>2</mml:annotation> </mml:semantics> </mml:math>