Malabika Pramanik
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Graduate Student Supervision
Doctoral Student Supervision
Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.
In this thesis, we study two topics in Euclidean harmonic analysis. Thefirst one is the configurations contained in fractal-like sets in the Euclideanspace. The other is decoupling for various geometric objects in the Euclideanspace.In the study of Euclidean configurations, we first discuss the background,address their subtleties and do a simple survey on this subject. Then weproceed to the proof of my main result, which demonstrates the topologicalproperty of a set containing a similar copy of sequences converging to zero.In the study of decoupling, we first formulate a general decoupling inequality and discuss some general upper and lower bound estimates Thenwe move on to decoupling for manifolds in Euclidean space, and in particular curves in the plane. We then state a classical result by Bourgain andDemeter and use it to prove a decoupling inequality that works uniformlyfor all polynomials up to a certain degree, generalising an earlier result ofBiswas et al. in the plane.
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We discuss four different problems. The first, a joint work with Malabika Pramanik, concerns large subsets of ℝn that do not contain various types of configurations. We show that a collection of v points satisfying a continuously differentiable v-variate equation in ℝ can be avoided by a set of Hausdorff dimension 1/(v-1) and Minkowski dimension 1. The second problem concerns large subsets of vector spaces over non-archimedean local fields that do not contain configurations. Results analogous to the real-variable cases are obtained in this setting. The third problem is the construction of measure-zero Besicovitch-type sets in Kn for non-archimedean local fields K. This construction is based on a Euclidean construction of Wisewell and an earlier construction of Sawyer. The fourth problem, a joint work with Kyle Hambrook, is the construction of an explicit Salem set in ℚp. This set is based on a Euclidean construction of Kaufman. Supplementary materials available at: http://hdl.handle.net/2429/69960
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In this dissertation we study the maximal directional Hilbert transform operatorassociated with a set U of directions in the n-dimensional Euclideanspace. This operator shall be denoted by ℋU. We discuss in detail theproof of the (p; p)-weak unboundedness of ℋU in all dimensions n ≥ 2 andall Lebesgue exponents 1
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The main focus of this document is the small ball inequality. The small ball inequality is a functional inequality concerning the lower bound of the supremum norm of a linear combination of Haar functions supported on dyadic rectangles of a fixed volume. The sharp lower bound in this inequality, as yet unproven, is of considerable interest due to the inequality's numerous applications. We prove the optimal lower bound in this inequality under mild assumptions on the coefficients of a linear combination of Haar functions, and further investigate the lower bounds under more general assumptions on the coefficients. We also obtain lower bounds of such linear combinations of Haar functions in alternative function spaces such as exponential Orlicz spaces.
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Given a Cantor-type subset Ω of a smooth curve in ℝ(d+1), we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in ℝ(d+1), and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz [4], [3]. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(ℝ(d+1)) for all 1 ≤ p
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We prove a result which adds to the study of continuous analogues of Szemerédi-type problems. Let E ⊆ ℝⁿ be a Lebesgue-null set of Hausdorff dimension α, k, m be integers satisfying a suitable relationship, and {B₁,…, Bk} be n × (m − n) matrices. We prove that if the set of matrices Bi are non-degenerate in a particular sense, α is sufficiently close to n, and if E supports a probability measure satisfying certain dimensionality and Fourier decay conditions, then E contains a k-point configuration of the form {x + B₁y,…,x + Bky}. In particular, geometric configurations such as collinear triples, triangles, and parallelograms are contained in sets satisfying the above conditions.
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Master's Student Supervision
Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.
We give an overview of Hilbert transforms, followed by new results concerning maximal directional Hilbert transforms. Historically, the Hilbert transform motivated the development of many tools in harmonic analysis, such as interpolation theorems and more general singular integrals. Over time, variants of the Hilbert transform were studied as prototypical examples of singular integrals and maximal directional operators. In our research, we are especially concerned with maximal directional Hilbert transforms. After rigorously constructing the Hilbert transform and directional Hilbert transforms, we proceed to define the maximal directional Hilbert transforms. We then prove general L² mapping estimates for maximal directional Hilbert transforms, followed by specific examples which sharpen these estimates. Finally, we prove sharp L²(R²) to L²(R²) estimates for a large class of maximal directional Hilbert transforms.
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Fractal uncertainty principles (FUPs) in harmonic analysis quantify the extent to which a function and its Fourier transform can be simultaneously localized near a fractal set. We investigate the formulation of such principles for ellipsephic sets, discrete Cantor-like sets consisting of integers in a given base with digits in a specified alphabet. We employ a combination of theoretical and numerical methods to find and support our results.To wit, we resolve a conjecture of Dyatlov and Jin by constructing a sequence of base-alphabet pairs whose FUP exponents converge to the basic exponent and whose dimensions converge to δ for any given δ ∊ (½, 1), thereby confirming that the improvement over the basic exponent may be arbitrarily small for all δ ∊ (0, 1). Furthermore, using the theory of prolate matrices, we show that the exponents β₁ of the same sequence decay subexponentially in the base.In addition, we explore extensions of our work to higher-order ellipsephic sets using blocking strategies and tensor power approximations. We also discuss the connection between discrete spectral sets and base-alphabet pairs achieving the maximal FUP exponent.
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The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as consideringnew strategies to avoid ‘rough patterns’. There are several problems that fit intothe framework of rough pattern avoidance. For instance, we prove that for any set X with lower Minkowski dimension s, there exists a set Y with Hausdorff dimension 1 − s such that for any rational numbers a₁, ..., aN, a₁Y + ··· + aNY is disjoint from X, or intersects solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension 1/2 not containing the vertices of any isosceles triangle.
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The middle-third Cantor set is one of the most fundamental examples of self-similar fractal sets introduced by the German mathematician George Cantor in the late 1800s. Many questions about this set remain unanswered. In this thesis, we study the mapping property of a measure associated with the middle-third Cantor set. Specifically, we study whether the Cantor measure is Lebesgue improving through partly theoretical and partly numerical methods.
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