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Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves

100%
Qian T.
Studia Mathematica
|
1997
|
tom 123
|
nr 3
195-216
EN
The paper presents a theory of Fourier transforms of bounded holomorphic functions defined in sectors. The theory is then used to study singular integral operators on star-shaped Lipschitz curves, which extends the result of Coifman-McIntosh-Meyer on the $L^2$-boundedness of the Cauchy integral operator on Lipschitz curves. The operator theory has a counterpart in Fourier multiplier theory, as well as a counterpart in functional calculus of the differential operator 1/i d/dz on the curves.
2
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Hilbert transforms and the Cauchy integral in euclidean space

51%
Axelsson A., Kou K., Qian T.
Studia Mathematica
|
2009
|
tom 193
|
nr 2
161-187
EN
We generalize the notions of harmonic conjugate functions and Hilbert transforms to higher-dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These harmonic conjugates are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.
3
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Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves

51%
Gaudry G., Qian T., Wang S.
Colloquium Mathematicum
|
1996
|
tom 70
|
nr 1
133-150
EN
The aim of this paper is to study singular integrals T generated by holomorphic kernels 𝛷 defined on a natural neighbourhood of the set ${zζ^{-1}: z, ζ ∈ 𝛤, z ≠ ζ}$, where 𝛤 is a star-shaped Lipschitz curve, $𝛤 ={ exp(iz) : z = x+iA(x), A' ∈ L^{∞}[-π,π], A(-π ) =A(π)}$. Under suitable conditions on F and z, the operators are given by (1) $TF(z)= p.v. ∫_𝛤 𝛷(zη^{-1})F(η)(dη/η).$ We identify a class of kernels of the stated type that give rise to bounded operators on $L^{2} (𝛤,|d𝛤|)$. We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.
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