arXiv subjects
Publications and source records attributed to Paul Hagelstein.
We provide a condition on a set of directions $Ω\subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_Ω$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. The techniques of proof extend ideas of Bateman and Katz involving probabilistic construction of Kakeya-type sets involving sticky maps and Bernoulli percolation.
$K_σ$ sets involving sticky maps $σ$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on $L^p(\mathbb{R}^2)$ for all $1 \leq p < \infty$. We indicate limits to this approach by showing that, given $ε> 0$ and a natural number $N$, there exists a tree $\mathcal{T}_{N, ε}$ of finite height that is lacunary of order $N$ but such that, for \emph{every} sticky map $σ: \mathcal{B}^{h(\mathcal{T}_{N, ε})} \rightarrow \mathcal{T}_{N, ε}$, one has $|K_σ \cap ((1,2) \times \mathbb{R})| \geq 1 - ε$.
Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewise constant functions $f$ with nonzero and zero values alternating. The above inequality strengthens a recent result of Bilz and Weigt \cite{BW} proved for indicator functions of bounded variation vanishing at $\pm\infty$. We conjecture that the inequality holds for all functions of bounded variation, representing a stronger version of the existing conjecture ${\rm Var} (Mf)\le {\rm Var} (f)$. We also obtain the discrete counterpart of our theorem, moreover proving a transference result on equivalency between both settings that is of independent interest.
We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in $L^2(\mathbb{R})$ $\big($and hence that of the Hermite operator in $L^2\big(\mathbb{R}; e^{-x^2}dx)\big)\big)$ in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in $L^2(\mathbb{R})$.
Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that $M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^2$ must differentiate $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$.
Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$ |\{x \in \mathbb{R}^n : M_{\mathcal{B}}f(x) > α\}|\leq C_n \int_{\mathbb{R}^{n}} \frac{|f|}α\left(1+\log^+\frac{|f|}α\right)^{n-1} $$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several classes of rare bases including Córdoba, Soria and Zygmund bases.
Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. In this article we develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multi-parameter time-frequency analysis techniques. As applications we prove sharp or quantified bounds for Rubio de Francia type square functions of conical multipliers and of multipliers adapted to rectangles pointing along $N$ directions. A suitable combination of these estimates yields a new and currently best-known logarithmic bound for the Fourier restriction to an $N$-gon, improving on previous results of A. Cordoba. Our directional Carleson embedding extends to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.
Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ consists of parallelepipeds with side lengths of the form $s, 2^j s, t $, where $s, t > 0$ and $j$ lies in a nonempty subset $S$ of the integers. In this paper, we prove the following: If $S$ is a finite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α\left(1 + \log^+ \frac{|f|}α\right)\;$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))} = 0\;.$$ On the other hand, if $S$ is an infinite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α \left(1 + \log^+ \frac{|f|}α\right)^{2}$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))^2} = 0\;.$$
Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ contains parallelepipeds with side lengths of the form $s, \frac{2^N}{s} , t $, where $s, t > 0$ and $N$ lies in a nonempty subset $S$ of the natural numbers. We show that if $S$ is an infinite set, then the associated geometric maximal operator $M_\mathcal{B}$ satisfies the weak type estimate $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α \left(1 + \log^+ \frac{|f|}α\right)^{2}$$ but does not satisfy an estimate of the form $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))^2} = 0\;.$$
A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result in the context of ergodic theory that provides a generalization of Birkhoff's Ergodic Theorem. In particular, we show that if $f$ is a measurable function on a standard probability space and $T$ is an invertible measure-preserving transformation on that space, then the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^*f$ is finite a.e.
We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing $T$-cycles of a differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + \cdots + a_N = 1$. Following an approach of Morgül, we associate to each periodic orbit of $f$, $N \in \mathbb{N}$, and $a_1,\ldots,a_N$ an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of $f$, there exist $N$ and $a_1,\ldots,a_N$ whose associated polynomial is Schur stable, and we find the minimal $N$ that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejér kernels found in classical harmonic analysis.