Sharp estimates for the Bochner-Riesz operator of negative order in $\mathbf {R}^2$

Abstract

Abstract. The Bochner-Riesz operator T α on R n of order α is defined bywhere denotes the Fourier transform and r α + = r α if r > 0, andTo be more precise, we prove that for 0 < δ < 3/2 the estimateWe also obtain some weak-type results for T α .The Bochner-Riesz operator T α on R n , n ≥ 2, of order α is a multiplier operator defined bywhere denotes the Fourier transform and r For our present purposes the following equivalent definition is more convenient:

Restrictions of higher derivatives of the Fourier transform

“…Application of the Littlewood-Paley inequality and homogeneity considerations (see [24]) reduce (35) to the case where the spectrum of f lies in a small neighborhood of a point on Γ k,l . So, by the results of [2], the inequality 35 Note that the Fourier transform of the function (∂ k 1 − σ∂ l 2 )f vanishes on Γ k,l , which is a smooth convex curve in the plane (with, possibly, a singularity at zero).…”

“…Surface measure conditions. Let χ be a smooth function of one variable supported in [1,2] such thatχ (k) (0) = 1. Consider the functions f n defined aŝ…”

“…Let K be a compact subset of Σ and let k be a natural number. We consider as a model problem the inequality (2)…”

“…If one is determined not to weaken the L 2 (K) norm in (2), it is necessary to consider f belonging to an a priori narrower space than L p (R d ). We introduce the main character.…”

“…It will turn out (see Theorem 1.6 below) that for a certain range of p and k, the space Σ L k p contains precisely the functions f ∈ L p whose Fourier transform vanishes on Σ to order k − 1. We take advantage of the additional structure of the domain to formulate a second adaptation of inequality (2), this time with the trace of ∇ kf still belonging to L 2 loc (Σ): (9) (…”