arXiv subjects
Publications and source records attributed to Árpád Bényi.
We establish transference results for multilinear Fourier multiplier operators from Euclidean spaces to product spaces of the form $\mathbb R^{d_1}\times\mathbb T^{d_2}$. We prove strong and weak-type transference principles that preserve the corresponding Euclidean operator bounds. As applications, we obtain bounds for certain multilinear operators, as well as for some multiplier operators arising naturally in PDEs.
We introduce Carleson analogs of the Bourgain--Brezis--Mironescu spaces $B$ and $B_0$ by measuring mean oscillation over upper Carleson tents. For these spaces, denoted by $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$, we prove two types of structural results. First, we show that they contain several natural classes of functions, including BMO/VMO--Carleson spaces, tent-space potential classes, and fractional Sobolev classes. Second, motivated by Zhu's structural theorem for BMO spaces induced by the Bergman metric, we establish decompositions of $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$ into bounded-oscillation and bounded-average components. We then revisit the Bourgain--Brezis--Mironescu rigidity phenomenon in the Carleson setting. Although the direct rigidity statement fails for $B_{\mathcal C,0}^p$, we introduce a natural $B_{\mathcal C}^p$-trace and prove that the rigidity theorem survives at the level of traces.
Motivated by questions on the ranges of commutators of dilated floor functions and one posed in Problem 27327 from Gazeta Matematică, we investigate the precise ranges of certain generalized polynomials depending on a real parameter and defined via the floor function. Our analysis requires non-trivial tools, including Kronecker's approximation theorem. The results highlight sharp distinctions between irrational parameters and sub-unitary and supra-unitary rational parameters. We also propose several conjectures for the irrational and supra-unitary rational cases, supported by extensive computations in Wolfram Mathematica.
We prove a symbolic calculus for a class of pseudodifferential operators, and discuss its applications to $L^2$-compactness via a compact version of the $T(1)$ theorem.
In this paper, we provide the maximal boundedness range (up to end-points) for the Bilinear Hilbert-Carleson operator along curves in the (purely) non-zero curvature setting. More precisely, we show that the operator $$ BHC_{[\vec{a},\vecα]}(f_1,f_2)(x) := \sup_{λ\in\mathbb{R}} \left|\,p.v.\, \int_{\mathbb{R}} f_1(x - a_1 t^{α_1}) \,f_2(x - a_2 t^{α_2}) \,e^{i\,λ\,a_3 \,t^{α_3}} \,\frac{dt}{t}\right|$$ obeys the bounds $$\|BHC_{[\vec{a},\vecα]} (f_1,f_2)\|_{L^r} \lesssim_{\vec{a} \,\vecα,r,p_1,p_2} \|f_1\|_{L^{p_1}}\,\|f_2\|_{L^{p_2}}$$ whenever $\vec{a}=(a_1,a_2,a_3),\,\vecα=(α_1,α_2,α_3)\in (\mathbb{R}\setminus\{0\})^3$ with $\vecα$ having pairwise distinct coordinates and for any Hölder range $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r}$ with $1<p_1,p_2<\infty$ and $\frac{1}{2}<r<\infty$. This result is achieved via the Rank II LGC method introduced in arXiv:2308.10706.
We prove a compact $T(1)$ theorem, involving quantitative estimates, analogous to the quantitative classical $T(1)$ theorem due to Stein. We also discuss the $C_c^\infty$-to-$CMO$ mapping properties of non-compact Calderón-Zygmund operators.
We discuss the fractional Leibniz rule for periodic functions on the $d$-dimensional torus, including the endpoint cases. As an application, we present a product estimate, involving distributions of negative regularities.
We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples. We then prove compactness of bilinear paraproducts with CMO symbols by combining one of the properties of compact bilinear operators thus obtained with vanishing Carleson measure estimates and interpolation of bilinear compactness.
We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the Hörmander classes $BS^ω_{1, 0}$. In particular, we investigate their relation to discrete analogues of the bilinear Calderón-Zygmund singular integral operators and show compactness of their commutators.
We indicate how to construct a family of modulation function spaces that have a scaling symmetry. We also illustrate the behavior of the Schrödinger multiplier on such function spaces.
We consider the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2]. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.
In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón--Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.
We present a unified method to obtain weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones.
We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} = \frac{d-2}{2}$. More precisely, given a function on $\mathbb R^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove `conditional' almost sure global well-posedness for $d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.
In this work, we state and prove versions of the linear and bilinear $T(b)$ theorems involving quantitative estimates, analogous to the quantitative linear $T(1)$ theorem due to Stein.
We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.
Commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.