By Shanzhen Lu
This ebook typically offers with the Bochner-Riesz technique of a number of Fourier imperative and sequence on Euclidean areas. It goals to provide a systematical creation to the basic theories of the Bochner-Riesz capacity and demanding achievements attained within the final 50 years. For the Bochner-Riesz technique of a number of Fourier critical, it comprises the Fefferman theorem which negates the Disc multiplier conjecture, the recognized Carleson-Sjolin theorem, and Carbery-Rubio de Francia-Vega's paintings on nearly in all places convergence of the Bochner-Riesz potential under the severe index. For the Bochner-Riesz technique of a number of Fourier sequence, it comprises the idea and alertness of a category of functionality area generated through blocks, that's heavily with regards to nearly far and wide convergence of the Bochner-Riesz capacity. additionally, the ebook additionally introduce a little analysis effects on approximation of services by means of the Bochner-Riesz potential.
Readership: Graduate scholars and researchers in arithmetic.
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Additional resources for Bochner-Riesz Means on Euclidean Spaces
1 holds. 10), and letting η > 0, R > η1 , we conclude that ηR 2 π α (f ; x) − s = Cn,α BR 1 2 π + Cn,α cos t − n2 π dt t t fx ( ) − s R ∞ ηR t fx ( ) − s R cos t − n2 π dt + o(1), t as R → ∞. Since fx (t)t−1 ∈ L(η, ∞), by the Riemann-Lebesgue theorem, we obtain ∞ fx ηR and ∞ ηR cos t − n2 π dt = o(1) t t R cos t − n2 π dt = o(1), t as R → ∞. Therefore when R → ∞, we immediately have α BR (f ; x) − s = Cn,α 2 π η R−1 (fx (t) − s) cos Rt − n2 π dt + o(1). 2. 1 In the proof, the notation 1 lim A→+∞ 1 .
Chapter 3 will get back to the Bochner-Riesz means of the series and focus on the discussion for the critical index situations. 1 Localization principle and classic results on ﬁxed-point convergence Suppose that f ∈ L(Rn ) and fˆ is the the Fourier transform of f . 2) and the index α0 = n−1 2 is called the critical index. In this section, we mainly consider the case of n−1 n−3 <α≤ . 2 2 Suppose that f is locally integrable on Rn , that is, f is integrable on any bounded set of Rn , and we denote it by f ∈ Lloc (Rn ).
Then we have Rn+1 m(x)fˆ(x)α(x)dx ≤ Tm f Lp (Rn+1 ) α Lq (Rn+1 ) . Let f (x) = h(ξ)g(η) and α(x) = γ(ξ)β(η), where h, γ ∈ S (R) and g, β ∈ S (Rn ). The above inequality can be rewritten as R Rn ≤ Tm p m(ξ, η)ˆ g (η)β(η)dη h(ξ)γ(ξ)dξ h Lp (R) g Lp (Rn ) γ Lq (R) β Set M (ξ) = Rn m(ξ, η)ˆ g (η)β(η)dη and deﬁne the operator TM by TM h(ξ) = M (ξ)h(ξ). Lq (Rn ) . 4 The disc conjecture and Feﬀerman theorem 51 Then we rewrite the above inequality as R TM h(ξ)γ(ξ)dξ ≤ Tm p h g Lp (R) Lp (Rn ) β Lq (Rn ) γ Lq (R) , which implies that TM h Lp (R) ≤ Tm p g Lp (Rn ) β Lq (Rn ) h Lp (R) .