The convergence of the Bochner-Riesz means at the critical index

In this paper, we study the pointwise convergence of the BochnerRiesz means at the critical index on the space L log L(Qn). We weaken the hypothesis, “x is a Lebesgue point”, which is required on some research results by instead considering the convergence of averages of the function over balls when the radials of the balls approach to 0. Let Qn denote the n-dimensional cube {x ∈ R| |xi| ≤ π, i = 1, 2, · · · , n}, B(x, r) denote the n-dimensional closed ball with center x and radius r, and |E| denote the Lebesgue measure of a measurable set E ⊂ R. Definition 1. Suppose f ∈ L(Qn). If lim r→0 |B(x0, r) ∩Qn| ∫ B(x0,r)∩Qn |f(x)− f(x0)|dx = 0, then x0 is called a Lebesgue point of f. If f ∈ L(Qn), the partial sum of the Bochner-Riesz mean of its Fourier series at critical index (n− 1)/2 is defined by SR(f, x) = ∑ |l|≤R (1− |l| 2 R2 )ale il·x where ∑ ale il·x is the Fourier series of f. In 1954 [4], Salem studied the difference between the partial sum operator of Fourier series and the Bochner-Riesz mean of an integrable function on [−π, π] where the function f satisfies the “Salem condition” ∫ h 0 (f(x+ t)− f(x− t))dt = o(|h|/ log |h|−1) (h→ 0) (A) uniformly for x ∈ [−π, π]. On the other hand, it is well known that the BochnerRiesz mean at critical index converges on the Banach space L(Qn) for 1 0 such that r{Fα(x0, r + 2h) + Fα(x0, r) − 2Fα(x0, r + h)} = o(|h|/ log |h|−1) (h→ 0) (B) uniformly for r, h ≤ r ≤ r0, then f is said to satisfy the α-Salem condition at x0. Here is the main theorem of this paper. Theorem. Let f ∈ L log L(Qn), n ≥ 2. Suppose f satisfies the α-Salem condition at x0 for some α ≥ 0. Then lim R→∞ SR(f, x0) = f(x0) provided ∫ |y|≤t (f(y)− f(x0))dy = o(t) (t→ 0) (C) and |f(x0)| <∞. Remark. 1. The (n − 1)-Salem condition is equivalent to the Salem condition in Definition 2. 2. If x0 is a Lebesgue point of f , then (C) is satisfied. 3. The hypothesis (C) is equivalent to ∫ t 0 τfx0(τ)dτ = o(t ) (t→ 0). (D) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2720 LUNG-KEE CHEN AND DASHAN FAN Lemma 1. Let k be any positive real number. Then ∫ k 0 {(r + h+ τ)fx(r + h+ τ)− (r + h− τ)fx(r + h− τ)}dτ = Fα(x, r + h+ k) + Fα(x, r + h− k)− 2Fα(x, r + h). Proof. The above integral is equal to ∫ r+h+k r+h τfx(τ)dτ − ∫ r+h r+h−k τfx(τ)dτ = ∫ r+h+k 0 τfx(τ)dτ − 2 ∫ r+h 0 τfx(τ)dτ + ∫ r+h−k 0 τfx(τ)dτ. Lemma 1 is proved. Lemma 2. Suppose f ∈ L(Qn) and f satisfies (C) at x0. Then for any fixed positive real number N and 0 < t ≤ Nπ/R, one has ∫ t 0 τfx0(τ) cos(Rτ)dτ = o(R −α−1) and ∫ t 0 τfx0(τ) sin(Rτ)dτ = o(R −α−1) as R→∞. Proof. We will prove the first equality since both equalities are similar. Write ∫ t 0 τfx0(τ) cos(Rτ)dτ = ∫ t 0 τfx0(τ)τ α−n+1 cos(Rτ)dτ. Then Lemma 2 is proved by applying integration by parts and the hypothesis that f satisfies (C) at x0 and (D) in the remark. Lemma 3. Let f ∈ L(Qn). Suppose f satisfies (B) and (C) at x0. Then ∫ t 0 τfx0(τ) cos(Rτ)dτ = o(R −α−1) + (logR)−1o(tα+1) and ∫ t 0 τfx0(τ) sin(Rτ)dτ = o(R −α−1) + (logR)−1o(tα+1) for π/R ≤ t as t→ 0. Proof. By Lemma 2, without loss of generality, we may assume t ≥ 4π/R and we only prove the first equality. Let us choose and fix r0 such that 0 < r0 ≤ 2π/R and R(t− r0)/π − 1 is even. Denote m = R(t− r0)/π − 1 and decompose the integral I = ∫ t 0 τfx0(τ) cos(Rτ)dτ = ∫ r0 0 · · ·+ m ∑ j=1 ∫ r0+(j+1)π/R r0+jπ/R · · · . By Lemma 2, the first integral on the right-hand side of the above equality is ∫ r0 0 τfx0(τ) cos(Rτ)dτ = o(R −α−1). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CONVERGENCE OF THE BOCHNER-RIESZ MEANS 2721