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

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \, \rho(y) \, d\sigma(y)\right|,$$ where $d\sigma$ denotes the normalised Lebesgue measure over the curves $\mathbb{C}$. Let $\triangle$ 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 $(\frac{1}{p}, \frac{1}{q}) \in (\triangle \setminus \{P, Q\}) \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\}$, there is a constant $B$ such that $\|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}$. Furthermore, if $m <5,$ then we have $\|\mathbb{M}f\|_{L^{5, \infty}(\mathbb{R}^2)} \leq B \|f\|_{L^{\frac{5}{2} ,1} (\mathbb{R}^2)}.$ We shall also consider a variable coefficient version of maximal theorem and we obtain the $L^p-L^q$ boundedness result for $(\frac{1}{p}, \frac{1}{q}) \in \triangle^{\circ} \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\},$ where $\triangle^{\circ}$ 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.Comment: 16 pages. revised version of the file. Several references have been modified. arXiv admin note: text overlap with arXiv:1510.08649, arXiv:1609.0814

Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation

In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_\gamma$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique continuation; (ii) the strong unique continuation property for some degenerate sublinear equations.Comment: revised version of the file, several references have been adde

A strong unique continuation property for the heat operator with Hardy type potential

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality $$|\Delta u - u_t| \leq \frac{M}{|x|^2} |u|,$$ with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.Comment: Revised paper: Lemma 2.2 has been adde