Singular Oscillatory Integrals on R^n

Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular oscillatory integral on R^n.Comment: final version, 10 pages, small typos corrected, one reference added. To appear in Math.

Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

Let T be a bounded linear operator acting on a complex Banach space X and (\lambda_n) a sequence of complex numbers. Our main result is that if |\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemer\'edi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (\lambda_n) is optimal among sequences such that |\lambda_n|/|\lambda_{n+1}| converges in [0,+\infty]. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.Comment: 18 pages; to appear in Math. Scand., this second version of the paper is significantly revised to deal with the more general case of a sequence of operators (\lambda_n T^n). The hypothesis of the theorem has been weakened. The numbering has changed, the main theorem now being Th. 3.8 (in place of Proposition 3.3). The changes incorporate the suggestions and corrections of the anonymous refere

A sharp estimate for the Hilbert transform along finite order lacunary sets of directions

Let $D$ be a nonnegative integer and ${\mathbf{\Theta}}\subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<\infty$, of the maximal directional Hilbert transform in the plane $$H_{{\mathbf{\Theta}}} f(x):= \sup_{v\in {\mathbf{\Theta}}} \Big|\mathrm{p.v.}\int_{\mathbb R }f(x+tv)\frac{\mathrm{d} t}{t}\Big|, \qquad x \in {\mathbb R}^2,$$ are comparable to $(\log\#{\mathbf{\Theta}})^\frac{1}{2}$. For vector fields $\mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $\mathsf{v}_D$, $$H_{\mathsf{v}_D,1} f(x):= \mathrm{p.v.} \int_{ |t| \leq 1 } f(x+t\mathsf{v}_D(x)) \,\frac{\mathrm{d} t}{t},$$ is $L^p$-bounded for all $1<p<\infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.Comment: 20 pages, 2 figures. Submitted. Changes: clarified the definition of D-lacunary set and streamlined the notatio

Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}} f(x) :=\sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|$$ and the halo function $\phi_{\mathcal{B}}(\alpha)$ on $(1,\infty)$ by $$\phi_{\mathcal B}(\alpha) :=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in \mathbb{R}^n : M_{\mathcal{B}} \chi_E (x) >1/\alpha\}|.$$ It is shown that if $\phi_{\mathcal{B}}(\alpha)$ satisfies the Solyanik estimate $\phi_{\mathcal B}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$ for $\alpha\in(1,\infty)$ sufficiently close to 1 then $\phi_{\mathcal{B}}$ lies in the H\"older class $C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on $\mathbb{R}^n$ lie in the H\"older class $C^{1/n}(1,\infty)$.Comment: 19 pages, 1 figure, minor typos corrected, incorporates referee's report, to appear in Adv. Mat