Endpoint bounds for an analytic family of Hilbert transforms
نویسندگان
چکیده
منابع مشابه
Endpoint Bounds for an Analytic Family of Hilbert Transforms
In R2, we consider an analytic family of operators Hz , z ∈ C, whose convolution kernel is obtained by taking −z − 1 derivatives of arclength measure on the parabola (t, t2) in a homogeneous way, defined in such a way so that H−1 be the standard parabolic Hilbert transform. For a fixed z, we study the set of p for which Hz is bounded on Lp(R2) and for the critical z that captures the degree of ...
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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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In R2, we consider an analytic family of fractional integrals , whose convolution kernel is obtained by taking some transverse derivatives of arclength measure on the parabola (t, t2) multiplied by |t|γ , and doing so in a homogeneous way. We determine the exact range of p, q for which the analytic family maps Lp to Lq . We also resolve a similar issue on the Heisenberg group.
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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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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 ...
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 1991
ISSN: 0012-7094
DOI: 10.1215/s0012-7094-91-06202-2