arXiv subjects

Aleksandar Bulj

Publications and source records attributed to Aleksandar Bulj.

7 recordsLinked to original sources

arXiv2026

On hyperbolic corners and unit-area triangles in planar sets of large measure

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $\Omega(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erd\H{o}s, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

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arXiv2026

Reverse square function estimates for degenerate curves and its applications

Building on the classical work of C\'{o}rdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $\delta$-neighborhood of the curve $\{(\xi,\xi^a): |\xi|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty)\backslash\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schr\"{o}dinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.

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arXiv2025

Fourier extension estimates on a strip in $\mathbb{R}^2$

Given a smooth curve with nonzero curvature $Σ\subset \mathbb{R}^2$, let $E_Σ$ denote the associated Fourier extension operator. For both general compact curves and the parabola, we characterize the pairs $(p,q)\in [1,\infty]^2$ for which the estimates $\|E_Σf\|_{L^q(Ω)}\leq C\|f\|_{L^p(Σ)}$ and $(\mathcal{R}(|E_Σf|^{q}))^{\frac{1}{q}}\leq C\|f\|_{L^p(Σ)}$ hold, where $Ω$ is a strip in $\mathbb{R}^2$ and $\mathcal{R}$ denotes the Radon transform. This work continues the study of mass concentration of $x\mapsto E_Σf(x)$ near lines in $\mathbb{R}^2$, initiated by Bennett and Nakamura and later extended by Bennett, Nakamura, and the second author, where expressions of the form $(\mathcal{R}(|E_Σf|^{2}))^{\frac{1}{2}}$ were studied.

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arXiv2024

Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure

Let $μ$ be a translation invariant measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ and let $λ$ denote the Lebesgue measure on $\mathbb{R}^d$. If there exists an open set $U$ such that $0<μ(U)=λ(U)<\infty$, it is a simple exercise to show that $μ=λ|_{\mathcal{B}(\mathbb{R}^d)}$. Is the same conclusion true if $U$ is merely a Borel set? The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by Hirst.

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arXiv2023

Generic norm growth of powers of homogeneous unimodular Fourier multipliers

For an integer $d\ge 2$, $t\in \mathbb{R}$ and a $0$-homogeneous function $Φ\in C^{\infty}(\mathbb{R}^{d}\setminus\{0\},\mathbb{R})$, we consider the family of Fourier multiplier operators $T_Φ^t$ associated with symbols $ξ\mapsto \exp(itΦ(ξ))$ and prove that for a generic phase function $Φ$, one has the estimate $\lVert T_Φ^t\rVert_{L^p\to L^p} \gtrsim_{d,p, Φ}\langle t\rangle ^{d\lvert\frac{1}{p}-\frac{1}{2}\rvert}$. That is the maximal possible order of growth in $t\to \pm \infty$, according to the previous work by V. Kovač and the author and the result shows that the two special examples of functions $Φ$ that induce the maximal growth, given by V. Kovač and the author and independently by D. Stolyarov, to disprove a conjecture of Maz'ya actually exhibit the same general phenomenon.

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arXiv2022

Asymptotic behavior of $L^p$ estimates for a class of multipliers with homogeneous unimodular symbols

We study Fourier multiplier operators associated with symbols $ξ\mapsto \exp(iλϕ(ξ/|ξ|))$, where $λ$ is a real number and $ϕ$ is a real-valued $C^\infty$ function on the standard unit sphere $\mathbb{S}^{n-1}\subset\mathbb{R}^n$. For $1<p<\infty$ we investigate asymptotic behavior of norms of these operators on $L^p(\mathbb{R}^n)$ as $|λ|\to\infty$. We show that these norms are always $O((p^\ast-1) |λ|^{n|1/p-1/2|})$, where $p^\ast$ is the larger number between $p$ and its conjugate exponent. More substantially, we show that this bound is sharp in all even-dimensional Euclidean spaces $\mathbb{R}^n$. In particular, this gives a negative answer to a question posed by Maz'ya. Concrete operators that fall into the studied class are the multipliers forming the two-dimensional Riesz group, given by the symbols $r\exp(iφ) \mapsto \exp(iλ\cosφ)$. We show that their $L^p$ norms are comparable to $(p^\ast-1) |λ|^{2|1/p-1/2|}$ for large $|λ|$, solving affirmatively a problem suggested in the work of Dragičević, Petermichl, and Volberg.

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arXiv2022

Multi-parameter maximal Fourier restriction

The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by ellipsoids. Along the lines of the same proof, we also establish a $d$-parameter Menshov--Paley--Zygmund-type theorem for the Fourier transform on $\mathbb{R}^d$. Such a result is interesting for $d\geq2$ because, in a sharp contrast with the one-dimensional case, the corresponding endpoint $L^2$ estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.

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