Uniform Bounds for the Bilinear Hilbert Transforms, II
نویسندگان
چکیده
منابع مشابه
Uniform Bounds for the Bilinear Hilbert Transforms
It is shown that the bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. ∫ R f(x− αt)g(x− βt) dt t map Lp1(R) × Lp2(R) → Lp(R) uniformly in the real parameters α, β when 2 < p1, p2 < ∞ and 1 < p = p1p2 p1+p2 < 2. Combining this result with the main result in [9], we deduce that the operators H1,α map L2(R)×L∞(R) → L2(R) uniformly in the real parameter α ∈ [0, 1]. This completes a program initiated...
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We continue the investigation initiated in [8] of uniform L bounds for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. ∫ R f(x − αt)g(x − βt) dt t . In this work we show that Hα,β map L1(R) × L2(R) into L(R) uniformly in the real parameters α, β satisfying | β − 1| ≥ c > 0 when 1 < p1, p2 < 2 and 2 3 < p = p1p2 p1+p2 < ∞. As a corollary we obtain L × L∞ → L uniform bounds in the ...
متن کاملUniform Bounds for the Bilinear Hilbert Transforms, I
It is shown that the bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. Z R f(x− αt)g(x− βt) dt t map L1(R)×L2(R)→ L(R) uniformly in the real parameters α, β when 2 < p1, p2 <∞ and 1 < p = p1p2 p1+p2 < 2. Combining this result with the main result in [9], it follows that the operators H1,α map L (R) × L∞(R) → L(R) uniformly in the real parameter α ∈ [0, 1], as conjectured by A. Calderón.
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Abstract. We prove old and new L bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on ...
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C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in [27] that the standard bilinear and bi-parameter Hilbert transform does not satisfy any L estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by Tm(f1, f2)(x) := ∫ R m(ξ, η)f̂1(ξ1, η1)f̂2(ξ2, η2)e 1122dξdη satisfies any L estimates, where the symbol m satisfies |∂ ξ ∂ ηm(ξ, η)| . 1 dist(ξ,Γ1...
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ژورنال
عنوان ژورنال: Revista Matemática Iberoamericana
سال: 2006
ISSN: 0213-2230
DOI: 10.4171/rmi/483