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Isroil A. Ikromov

Publications and source records attributed to Isroil A. Ikromov.

13 recordsLinked to original sources
arXiv2026

Sharp estimates for the Fourier transform of surface-carried measures and maximal operators associated with hypersurfaces in $\mathbb{R}^4$ with vanishing Gaussian curvature

In this paper, we study problems related to harmonic analysis on hypersurfaces in $\mathbb{R}^4 $ with zero Gaussian curvature and given as graphs of polynomial functions. We derive sharp uniform estimates with respect to the direction of frequencies for the Fourier transform of measures supported on such hypersurfaces. Additionally, we study the $L^p$-boundedness problem of maximal operators associated with hypersurfaces. We determine the exact value of the boundedness exponent in terms of the heights of these hypersurfaces.

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arXiv2025

$L^p$-estimates for FIO-cone multipliers

The classical cone multipliers are Fourier multiplier operators which localize to narrow $1/R$-neighborhoods of the truncated light cone in frequency space. By composing such convolution operators with suitable translation invariant Fourier integral operators (FIOs), we obtain what we call FIO-cone multipliers. We introduce and study classes of such FIO-cone multipliers on $\Bbb R^3$, in which the phase functions of the corresponding FIOs are adapted in a natural way to the geometry of the cone and may even admit singularities at the light cone. By building on methods developed by Guth, Wang and Zhang in their proof of the cone multiplier conjecture in $\Bbb R^3,$ we obtain $L^p$-estimates for FIO-cone multipliers in the range $4/3\le p\le 4$ which are stronger by the factor $R^{-|1/p-1/2|}$ than what a direct application of the method of Seeger, Sogge and Stein for estimating FIOs would give. An important application of our theory is to maximal averages along smooth analytic surfaces in $\Bbb R^3.$ It allows to confirm a conjecture on the the critical Lebesgue exponent for a prototypical surface from a small class of ``exceptional'' surfaces, for which this conjecture had remained open.

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arXiv2024

Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$

In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurface $Σ\subset \mathbb{R}^3$ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the $X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system.

math.AP↗
arXiv2024

Estimates for oscillatory integrals with phase having $D$ type singularities

In this paper, we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integrals is the linear perturbation of a function having $D$ type singularities. We consider estimates for the oscillatory integrals in terms of the Randol's type maximal functions. We obtain a sharp $L^p_{loc}$ estimates for the Randol's maximal functions. Moreover, we investigate the sharp exponent $p$ depending on whether, the phase function has linearly adapted coordinates system or not.

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arXiv2023

On the sharp estimates for convolution operators with oscillatory kernel

In this article, we study the convolution operators $M_k$ with oscillatory kernel, which are related to solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurfaces $Σ\subset \mathbb{R}^3$ of a hyperbolic equation and smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which exactly one of the principal curvatures of the surface $Σ$ does not vanish. Such surfaces exhibit singularities of type $A$ in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal number such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the surface $Σ$.

math.AP↗
arXiv2022

Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part II -- A geometric conjecture and its proof for generic 2-surfaces

In this article, we continue the study of $L^p$-boundedness of the maximal operator $\mathcal M_S$ associated to averages along isotropic dilates of a given, smooth hypersurface $S$ in 3-dimensional Euclidean space. We focus here on small surface-patches near a given point $x^0$ exhibiting singularities of type $\mathcal A$ in the sense of Arnol'd at this point; this is the situation which had yet been left open. Denoting by $p_c$ the minimal Lebesgue exponent such that $\mathcal M_S$ is $L^p$-bounded for $p>p_c,$ we are able to identify $p_c$ for all analytic surfaces of type $\mathcal A$ (with the exception of a small subclass), by means of quantities which can be determined from associated Newton polyhedra. Besides the well-known notion of height at $x^0,$ a new quantity, which we call the effective multiplicity, turns out to play a crucial role here. We also state a conjecture on how the critical exponent $p_c$ might be determined by means of a geometric measure theoretic condition, which measures in some way the order of contact of arbitrary ellipsoids with $S,$ even for hypersurfaces in arbitrary dimension, and show that this conjecture holds indeed true for all classes of 2-hypersurfaces $S$ for which we have gained an essentially complete understanding of $\mathcal M_S$ so far. Our results lead in particular to a proof of a conjecture by Iosevich-Sawyer-Seeger for arbitrary analytic 2-surfaces.

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arXiv2021

Sharp time decay estimates for the discrete Klein-Gordon equation

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$ dispersive decay rate is $|t|^{-3/4}$ for $d=2$, $|t|^{-7/6}$ for $d=3$ and $|t|^{-3/2}\log|t|$ for $d=4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.

math.AP↗
arXiv2010

Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction

Let $S$ be a hypersurface in $\Bbb R^3$ which is the graph of a smooth, finite type function $ϕ,$ and let $μ=ρ\, d\si$ be a surface carried measure on $S,$ where $d\si$ denotes the surface element on $S$ and $ρ$ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform $\hat μ$ of $μ,$ which are sharp except for the case where the principal face of the Newton polyhedron of $ϕ,$ when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp $L^p$-$L^2$ Fourier restriction theorem for $S$ in the case where the original coordinates are adapted to $ϕ.$ This improves on earlier joint work with M. Kempe.

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arXiv2007

On adapted coordinate systems

The notion of an adapted coordinate system, introduced by V.I.Arnol'd, plays an important role in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A.N.Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for a class of analytic functions without multiple components. Varchenko's proof is based on Hironaka's theorem on the resolution of singularities. In this article, we present a new, elementary and concrete approach to these results, which is based on the Puiseux series expansion of roots of the given function. Our method applies to arbitrary real analytic functions, and even extends to arbitrary smooth functions of finite type. Moreover, by avoiding Hironaka's theorem, we can give necessary and sufficient conditions for the adaptedness of a given coordinate system in the smooth, finite type setting.

math.CA↗
arXiv2007

Sharp $L^p$-estimates for maximal operators associated to hypersurfaces in $\bR^3$ for $p>2.$

We study the boundedness problem for maximal operators $\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic, then the associated maximal operator is bounded on $L^p(\RR^3)$ if and only if $p>h(S),$ where $h(S)$ denotes the so-called height of the surface $S.$ For non-analytic finite type $S$ we obtain the same statement with the exception of the exponent $p=h(S).$ Our notion of height $h(S)$ is closely related to A. N. Varchenko's notion of height $h(ϕ)$ for functions $ϕ$ such that $S$ can be locally represented as the graph of $ϕ$ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E.M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on $S$ and the $L^p$-boundedness of the associated maximal operator $\M$, and a conjecture by Iosevich and Sawyer which relates the $L^p$-boundedness of $\M$ to an integrability condition on $S$ for the distance function to tangential hyperplanes, in dimension three. In particular, we also give ess. sharp uniform estimates for the Fourier transform of the surface measure on $S,$ thus extending a result by V.N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V.I. Arnol'd in our context.

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