Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis
Tóm tắt
We study the boundedness problem for maximal operators
$ \mathcal{M} $
associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on
$ {L^p}\left( {{\mathbb{R}^3}} \right) $
if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the L
p
-boundedness of the associated maximal operator
$ \mathcal{M} $
, and a conjecture by Iosevich and Sawyer which relates the L
p
-boundedness of
$ \mathcal{M} $
to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an
$ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $
Fourier restriction theorem for S.
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