Inequalities for Fourier transforms

Max Jodeit; Alberto Torchinsky

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On the real line, let the Fourier transform of k_n be k'_n(ξ) = k'(ξ-n) where k'(ξ) is a smooth compactly supported function. Consider the bilinear operators S_n(f, g)(x) = ∫ f(x+y)g(x-y)k_n(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove that Σ^∞ _n=-∞ ||S_n(f,g)||₂ ^2...