Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^n$

Gianfranco Marletta; Fulvio Ricci

ArticleOriginal scientific text

Title

Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^{n}$

Authors ¹, ¹

Affiliations

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Abstract

Given a hypersurface

x_{n} = Ꮁ (x_{1} ..., x_{n - 1})

in

ℝ^{n}

, where Ꮁ is homogeneous of degree d>0, we define the two-parameter maximal operator

M f (x) = \supset_{a, b > 0} \int_{s \in ℝ^{n - 1}, | s | < 1}

|f(x - (as, bᎱ(s)))|ds

. W e p r o v e t \hat{if} d \neq 1 and t h e h y p e r s u r f a c e h a s n o n - v a n i s h \in g G a u s s i a n c u r v a t u r e a w a y o m t h e or i g \in, t h e n M i s b o u n d e d o n

L^p

if and o n l y if p > \frac{n}{n - 1} . I f d = 1, i . e . if t h e s u r f a c e i s a c o \neq, t h e s a m e c o n c l u s i o n h o l d s \in dim e n s i o n n \geq 3 if t h e s u r f a c e h a s n - 1 n o n - v a n i s h \in g p r \in c i p a l c u r v a t u r e s a w a y o m t h e or i g \in and i t \int e r \sec t s t h e h y p e r p l a \neq

x_n = 0!$! only at the origin.

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Title

Two-parameter maximal functions associated with homogeneous surfaces in ℝn

Affiliations

Abstract

Bibliography

Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^{n}$