By Shanzhen Lu

This booklet typically offers with the Bochner-Riesz technique of a number of Fourier essential and sequence on Euclidean areas. It goals to provide a systematical creation to the basic theories of the Bochner-Riesz capability and critical achievements attained within the final 50 years. For the Bochner-Riesz technique of a number of Fourier imperative, it contains the Fefferman theorem which negates the Disc multiplier conjecture, the well-known Carleson-Sjolin theorem, and Carbery-Rubio de Francia-Vega's paintings on nearly in all places convergence of the Bochner-Riesz skill under the severe index. For the Bochner-Riesz technique of a number of Fourier sequence, it contains the idea and alertness of a category of functionality area generated through blocks, that is heavily concerning nearly in all places convergence of the Bochner-Riesz capacity. moreover, the ebook additionally introduce a little analysis effects on approximation of capabilities via the Bochner-Riesz capability.

Readership: Graduate scholars and researchers in mathematics.

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16), it is obvious that t ≥ tm ≥ t − mζ. Since ζ= and mτ = t−ξ h+1 h−β (t − ξ) < t − ξ, h+1 we have ξ < tm < t. 3 Convergence and the opposite results 29 and t t−τ dt1 · · · tm−1 tm−1 −τ Aβ (tm ) − Aβ (t) dtm = O U β (t) τ m , α where β = h − m. 16) is O U β (t) τ α−β , then α we have τ α−β Aβ (t) = O(W (t)) + τ α−β O U β (t) . α By noticing τ= 1 (t − ξ) h+1 and 1 α W (t) V (t) t−ξ = , we get Aβ (t) = O U β (t) . α This is included in case (ii) when β ∈ Z+ . Now assume that β is not an integer and let β = [β] + γ, 0 < γ < 1.

26) is valid almost everywhere on Rn . 27) is the Fourier series of the function F = Fα , where Fα has the same singular 1 and Fα ∈ C(Q\{0}). properties at x = 0 as |x|n−α 32 C1. An introduction to multiple Fourier series Proof. Let η ∈ C ∞ (Rn ) satisfying 1 , if |x| ≥ 1 , 0 , if |x| < 12 . η(x) = Deﬁne G(x) = η(x) . |x|α If x = 0, we have G(x) = 1 1 + (η(x) − 1) α . 28) α |x| |y| where γα = 1 Γ( n−α 2 ) . α n/2 Γ(α/2) 2 π On the other hand, (η(x) − 1) 1 |x|α is an integrable function supported on {x : |x| < 1}, whose normal Fourier transform denoted by b1 , and b1 ∈ C0∞ (Rn ), where C0∞ (Rn ) refers to a class of inﬁnitely diﬀerentiable functions whose derivatives of any order all vanish at inﬁnity.

2 Suppose that the longer edge of the rectangle Rj is parallel to the vector vj . We denote by Rj two rectangles which have communal edge with the shorter edge of Rj and the same size as Rj . Let fj = HRj , then we have |Tj fj (x)| ≥ C > 0, when x ∈ Rj , where C is some positive constant. 1. Proof. 1. 56 C2. Bochner-Riesz means of multiple Fourier integral Let Rj = [0, a] × [0, b], and fj (y)eix·y dy Tj fj (x) = Lj = Lj ∞ = Rj e−iy·z dz eix·y dy 1 eix1 y1 − ei(x1 −a)y1 dy1 iy1 0 +∞ 1 × eix2 y2 − ei(x2 −b)y2 dy2 .