$L^p-L^q$ estimates for maximal operators associated to families of finite type curves

Abstract

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x):=\underset{1\le t\le 2}{\sup }\left|{\int }_{\mathbb{C}}f(x-ty)\rho (y)d\sigma (y)\right|,$$ where $d\sigma$ denotes the normalised Lebesgue measure over the curves $\mathbb{C}$. Let $△$ be the closed triangle with vertices $P=(\frac{2}{5},\frac{1}{5}),Q=(\frac{1}{2},\frac{1}{2}),R=(0,0).$ In this paper, we prove that for $(\frac{1}{p},\frac{1}{q})\in (△\setminus \{P,Q\})\cap \left\{(\frac{1}{p},\frac{1}{q}):q>m\right\}$, there is a constant $B$ such that $\Vert \mathbb{M}f{\Vert }_{L^q({\mathbb{R}}^2)}\le B\Vert f{\Vert }_{L^p({\mathbb{R}}^2)}$. Furthermore, if $m<5,$ then we have $\Vert \mathbb{M}f{\Vert }_{L^{5,\infty }({\mathbb{R}}^2)}\le B\Vert f{\Vert }_{L^{\frac{5}{2},1}({\mathbb{R}}^2)}.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the $L^p-L^q$ boundedness result for $(\frac{1}{p},\frac{1}{q})\in {△}^{\circ }\cap \left\{(\frac{1}{p},\frac{1}{q}):q>m\right\},$ where ${△}^{\circ }$ is the interior of the triangle with vertices $(0,0),(\frac{1}{2},\frac{1}{2}),(\frac{2}{5},\frac{1}{5}).$ An application is given to obtain $L^p-L^q$ estimates for solution to higher order, strictly hyperbolic pseudo-differential operators.

arXiv commentThere was an error on the range of $(p,q).$

References in corpus (1)

• Maximal averages associated to families of finite type surfaces