Singular Integral Operators on Closed Lipschitz Curves
In Chap. 1 , we state a theory of convolution singular integral operators and Fourier multipliers on infinite Lipschitz curves. A natural question is whether there exists an analogy on closed Lipschitz curves. In this chapter, we establish such a theory
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Singular Integral Operators on Closed Lipschitz Curves
In Chap. 1, we state a theory of convolution singular integral operators and Fourier multipliers on infinite Lipschitz curves. A natural question is whether there exists an analogy on closed Lipschitz curves. In this chapter, we establish such a theory for starlike Lipschitz curves. A curve is called a starlike Lipschitz curve if the curve has the following parameterization: γ = {exp(i z) : z ∈ γ }, where γ = x + ig(x) : g ∈ L ∞ ([−π, π ]), g(−π ) = g(π ) . It can be proved that the starlike Lipschitz curves defined using such parameterization are the same as those defined as star-shaped and Lipschitz in the ordinary sense. In the same pattern as in the infinite Lipschitz graph case, we can define Fourier series of L 2 functions on γ . The question can now be specified into the following two: The first, what kind of holomorphic kernels give rise to L 2 -bounded operators on starlike Lipschitz curves γ ? The second, is there a corresponding Fourier multiplier theory? In other words, what complex number sequences act as L p -bounded Fourier multipliers on the curves? It should be pointed out that these questions are not trivial even for the case p = 2, as the Plancherel theorem does not hold in this case. However, on the other hand, the case p = 2 is essential, as the boundedness for 1 < p < ∞ can be deduced from the L 2 theory using the standard Calderón-Zygmund techniques.
© Springer Nature Singapore Pte Ltd. and Science Press 2019 T. Qian and P. Li, Singular Integrals and Fourier Theory on Lipschitz Boundaries, https://doi.org/10.1007/978-981-13-6500-3_2
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2 Singular Integral Operators on Closed Lipschitz Curves
2.1 Preliminaries Let γ be a Lipschitz curve defined on the interval [−π, π ] with the parameterization γ (x) = x + ig(x), g : [−π, π ] → R, where R denotes the real number field, g(−π ) = g(π ), g ∈ L ∞ ([−π, π ]) with g ∞ = N . Denote by pγ the 2π -periodic extension of γ to −∞ < x < ∞, and by γ the closed curve γ = exp(i z) : z ∈ γ = exp(i(x + ig(x))) : −π x π . We call γ the starlike Lipschitz curve associated with γ . to denote the functions defined on pγ , γ and We use f , F and F γ , respectively. on ∈ L 2 ( γ ), the nth coefficient of F γ is defined as For F 1 F γ (n) = 2πi
z −n F(z) γ
dz . z
In the case of no confusion, we will sometimes suppress the subscript and write F(n). Set σ = exp(− max g(x)), τ = exp(− min g(x)). γ ): We consider the following dense subclass of L 2 ( : F(z) is holomorphic in σ − η < |z| < τ + η for some η > 0 . A( γ ) = F(z) Without loss of generality, we assume that min g(x) < 0 and max g(x) > 0. In the case, the domains of the functions in A( γ ) contain the unit circle T, and by Cauchy’s γ ), by the Laurent series, theorem, we know F γ (n) = F T (n). If F and G belong to A( we can obtain the inverse Fourier transform formula = F(z)
∞
n F γ (n)z ,
(2.1)
n=−∞
is defined. We apply Cauchy’s theorem to get t
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