Besicovitch showed that a compact set in $\mathbb{R}^n$ which contains a unit line segment in every direction can have measure $0$. These constructions also work over other metric spaces like the $p$-adics and profinite integers. It is conjectured that it is impossible to construct sets with measure $0$ which contain a unit $2$-disk in every direction in $\mathbb{R}^n$. We prove that over the $p$-adics and profinite integers any set which contains a 2-flat in every direction must have positive measure. The main ingredients are maximal Kakeya estimates for $(\mathbb{Z}/N\mathbb{Z})^n$ proven in [Dha22] and adapting Fourier analytic arguments of Oberlin [Obe05]. In general, we prove $L^{n-1}$ to $L^{n-1}$ estimates for the maximal operator corresponding to $2$-flats.
We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ there are uniform bounds (independent of the measure) for the weak type $(p,p)$ of the centered maximal operator, then $X$ satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type $(1,1)$ of the centered operator. Thus, the following characterization is obtained: the centered maximal operator satisfies uniform weak type $(1,1)$ bounds if and only if the space $X$ has the Besicovitch intersection property. In $\mathbb{R}^d$ with any norm, the constants c…
We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{iλφ}\|_{A_p(\mathbb T^m)}$ as $|λ|\rightarrow \infty, ~λ\in\mathbb R,$ for $C^1$ -smooth real functions $φ$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estim…
We establish a number of uncertainty inequalities for the additive group of a finite affine plane, showing that for $p$ prime, a nonzero function $f\colon\mathbb F_p^2\to\mathbb C$ and its Fourier transform $\hat f\colon\widehat{\mathbb F_p^2}\to\mathbb C$ cannot have small supports simultaneously. The "baseline" of our investigation is the well-known Meshulam's bound, which we sharpen, for the particular groups under consideration, taking into account not only the sizes of the support sets $\mathrm{supp}\,f$ and $\mathrm{supp}\,\hat f$, but also their structure. Our results imply in particula…
We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $\|f\|_{A(\mathbb T)}=\|\hat{f}\|_{l^1(\mathbb Z)}$. According to the known Beurling--Helson theorem, if $ϕ: \mathbb T\rightarrow\mathbb T$ is a continuous mapping such that $\|e^{inϕ}\|_{A(\mathbb T)}=O(1), ~n\in\mathbb Z,$ then $ϕ$ is linear. It was conjectured by Kahane that the same conclusion about $ϕ$ is true under the assumption that $\…
We study completeness of the spaces $\mathcal{P}_s^=$ of probability measures in $\mathbb{R}^N$ which have equal (prescribed) moments up to order $s \in \mathbb{N}$, endowed with the metric $d_s(μ,ν)=\sup_{x \in \mathbb{R}^N\setminus 0}\frac{|\hat μ(x)-\hat ν(x)|}{|x|^s}$, where $\hat μ$ is the characteristic function of $μ$. We prove that the spaces $(\mathcal{P}_s^=,d_s)$ are complete if $s$ is even and construct suitable counterexamples to completeness for all odd $s$. This solves an open problem formulated by J. Carrillo and G. Toscani in 2007.
Let $SP$ be the set of upper strongly porous at $0$ subsets of $\mathbb R^{+}$ and let $\hat I(SP)$ be the intersection of maximal ideals $I \subseteq SP$. Some characteristic properties of sets $E\in\hat I(SP)$ are obtained. It is shown that the ideal generated by the so-called completely strongly porous at $0$ subsets of $\mathbb R^{+}$ is a proper subideal of $\hat I(SP).$
We consider several problems related to the restriction of $(\nabla^k) \hat{f}$ to a surface $Σ\subset \mathbb R^d$ with nonvanishing Gauss curvature. While such restrictions clearly exist if $f$ is a Schwartz function, there are few bounds available that enable one to take limits with respect to the $L_p(\mathbb R^d)$ norm of $f$. We establish three scenarios where it is possible to do so: $\bullet$ When the restriction is measured according to a Sobolev space $H^{-s}(Σ)$ of negative index. We determine the complete range of indices $(k, s, p)$ for which such a bound exists. $\bullet$ Among f…
We prove that the degenerate trilinear operator $C_3^{-1,1,1}$ given by the formula \begin{eqnarray*} C_3^{-1,1,1}(f_1, f_2, f_3)(x)=\int_{x_1 < x_2 < x_3} \hat{f_1}(x_1) \hat{f_2}(x_2) \hat{f_3}(x_3) e^{2πi x (-x_1 + x_2 + x_3)} dx_1dx_2 dx_3 \end{eqnarray*} satisfies the new estimates \begin{eqnarray*} ||C_3^{-1,1,1}(f_1, f_2, f_3)||_{\frac{1}{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}} \lesssim_{p_1, p_2, p_3} ||\hat{f}_1||_{p^\prime_1} ||f_2||_{p_2}||f_3||_{p_3} \end{eqnarray*} for all $f_1 \in L^{p_1}(\mathbb{R}): \hat{f}_1 \in L^{p_1^\prime}(\mathbb{R}) , f_2 \in L^{p_2}(\mathbb{R})$, an…
A locally integrable function $m(ξ,η)$ defined on $\mathbb R^n\times \mathbb R^n$ is said to be a bilinear multiplier on $\mathbb R^n$ of type $(p_1,p_2, p_3)$ if $$ B_m(f,g)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n}\hat f(ξ)\hat g(η)m(ξ,η)e^{2πi(<ξ+η,x>} dξdη$$ defines a bounded bilinear operator from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) $ to $L^{p_3}(\mathbb R^n)$. The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where $m(ξ,η)= M(ξ-η)$ for a given $M$ defined on $\ma…
We consider, for a class of functions $φ: \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hatμ$ of the Borel measure on $\mathbb{R}^{4}$ defined by \[ μ(E) = \int_{U} χ_{E}(x, φ(x)) \, dx \] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = \{ (t^{α_1}, t^{α_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hatμ$, for the case where the set of nonelliptic points of $φ$ is a curve in $\bar{U} \setminus \{ {\bf 0} \}$. From this estimate we obtain a restriction theo…
In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of…
We study inequalities of the form \begin{equation*} ρ( \lvert \hat{f} \rvert) \leq C σ(f) < \infty, \end{equation*} with $f \in L_{1}(\mathbb{R}^n)$, the Lebesgue-integrable functions on $\mathbb{R}^n$ and \begin{equation*} \hat{f}(ξ) := \int_{\mathbb{R}^n} f(x) \, e^{- 2 πi ξ\cdot x} dx, \ \ \ ξ\in \mathbb{R}^n. \end{equation*} The functionals $ρ$ and $σ$ are so-called rearrangement-invariant (r.i.) norms on $M_{+}(\mathbb{R}^n)$, the nonnegative measurable functions on $\mathbb{R}^n$. Results first proved in the general context of r.i. spaces are then both specialized and expanded on in the …
For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hatω^A$ of a (Radon) measure $ω$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $μ$ of finite energy, concentrated on $A$. For quite general signed $ω$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hatω^A$ does exist, and it maintains the basic features of …
We prove that for any singular measure $μ$ on $\mathbb{R}^n$ it is possible to cover $μ$-almost every point with $n$ families of Lipschitz slabs of arbitrarily small total width. More precisely, up to a rotation, for every $δ>0$ there are $n$ countable families of $1$-Lipschitz functions $\{f_i^1\}_{i\in\mathbb{N}},\ldots, \{f_i^n\}_{i\in\mathbb{N}},$ $f_i^j:\{x_j=0\}\subset\mathbb{R}^n\to\mathbb{R}$, and $n$ sequences of positive real numbers $\{\varepsilon_i^1\}_{i\in\mathbb{N}},\ldots, \{\varepsilon_i^n\}_{i\in\mathbb{N}}$ such that, denoting $\hat x_j$ the orthogonal projection of the poin…
