Two-parameter maximal functions associated with degenerate homogeneous surfaces in ℝ³

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Title

Authors ¹, ¹, ²

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Department of Mathematics, Wrocław University, Plac Grunwaldzki 2, 50-384 Wrocław, Poland

We consider the two-parameter maximal operator

M f (x) = \supset_{a, b > 0}

ʃ_{|s| < 1} |f(x-(as,bΓ(s)))|ds

o n a h o m o \geq \neq o u s s u r f a c e

x_3 = Γ(x_1,x_2)

\in

ℝ^3

. W e a s \sum e t \hat{t} h e c u r v a t u r e o f t h e \leq v e l s e t

Γ(x_1,x_2) = 1

h a s a d e \geq \neq r a c y o f f \in i t e or d e r k a t a g i v e n p \oint . W e p r o v e t \hat{t} h e o p e r a \to r M i s b o u n d e d o n

L^p

if and o n l y if

p > max{3/2, 2k/(k+1)}!$!.

[MR] G. Marletta and F. Ricci, Two-parameter maximal functions associated with homogeneous surfaces in $ℝ^{n}$ , this issue, 53-65.