Summary of Research Papers

In this article, the core results of the author’s papers are summarized. The author’s work has been carried out mainly on the finite field analogues of classical problems in harmonic analysis, geometric measure theory, and combinatorics. In particular, the author has investigated the boundedness of extension operators and the Erdős-Falconer type distance problems in the finite field setting. The author has also studied the connection between these two topics. 1. Recent developments in extension problems for finite fields Let V be a subset of R, d ≥ 2, and d a positive measure supported on V . Then, one may ask that for which values of p and r does the estimate ∥f̂d ∥Lr(Rd) ≤ Cp,r,d∥f∥Lp(V,d ) for all f ∈ L(V, d ) hold? This problem is known as the extension theorem which has not been completely solved in higher dimensions. Mockenhaupt and Tao ([8]) first addressed and studied the finite field analogue of the extension problem. Let us review what is the extension problem in the finite field setting. We denote by (Fq , dx) a d-dimensional vector space over the finite field Fq, where we endow the space with a normalized counting measure dx. On the other hand, (Fq , dm) denotes the dual space with a counting measure dm. Let (V, d ) be an algebraic variety in (Fq , dx), where we endow the variety V with a normalized surface measure d . Namely, we have the following formula d (x) = ∣Fq ∣ ∣V ∣ V (x), where ∣V ∣ denotes a cardinality of V ⊂ Fq and we identify the set V with the characteristic function V on V. For each 1 ≤ p, r ≤ ∞ we define R∗(p → r) to be the smallest constant such that the extension estimate ∥f̂d ∥Lr(Fdq ,dm) ≤ R ∗(p→ r)∥f∥Lp(V,d ) holds for all functions f on V. Notice that R∗(p → r) is always a finite number and it may depend on the underlying finite field Fq. The extension problem in finite fields is to determine possibly small exponents, 1 ≤ p, r ≤ ∞ such that R∗(p→ r) ≤ Cp,r,d <∞, where the main point is that the constant Cp,r,d is independent of the size of the underlying finite field Fq. In [8], Mockenhaupt and Tao developed useful tools to study the extension problem in finite fields for various algebraic varieties V , but their work was mostly restricted