Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves
The main contents of this chapter are closely related with harmonic analysis and operator theory. Let \(\gamma \) denote a Lipschitz graph on the complex plane \(\mathbb C\) .
- PDF / 658,625 Bytes
- 42 Pages / 439.37 x 666.142 pts Page_size
- 55 Downloads / 187 Views
Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves
The main contents of this chapter are closely related with harmonic analysis and operator theory. Let γ denote a Lipschitz graph on the complex plane C: γ = x + ig(x) ∈ C : x ∈ R , where g is a Lipschitz function satisfying g ∞ N < ∞. We will prove the L p boundedness of certain singular convolution integral operators on γ . The main results of this chapter are based on the theory of Fourier multipliers and the H ∞ -functional calculus of type ω operators on the curve γ which are established by A. McIntosh and T. Qian in [1]. Roughly speaking, the type ω operators can be represented as b(Dγ ), where Dγ is the differential operator on γ , and b is a bounded holomorphic function defined on some sector Sν0 , ν > tan−1 N . With the additional assumption g being bounded, A. McIntosh and T. Qian studied a class of generalized Fourier multipliers on γ , see [2, 3] for the related results. For the boundedness of singular convolution integrals, there exist several different methods. In this chapter, we apply the method introduced by A. McIntosh and T. Qian. The proof depends on the quadratic estimates of the type ω operators on sectors. Precisely, we first prove that the quadratic estimates of the type ω operators are equivalent to the inverse quadratic estimates of the dual operators (see Theorem 1.2.1). Then we prove, if an operator T satisfies the quadratic estimates and the related inverse quadratic estimates, then for a bounded holomorphic function b, the holomorphic functional calculus b(T ) is bounded, see Theorem 1.2.3.
© 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_1
1
2
1 Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves
1.1 Convolutions and Differentiation on Lipschitz Graphs In this section, denote by C and R the complex number field and the real number field, respectively. We use γ to denote the following Lipschitz graph : γ = x + ig(x) ∈ C, where g is a Lipschitz function satisfying g ∞ N < ∞ .
We will use the following complex-valued function spaces. Definition 1.1.1 (i) Let 1 p ∞. L p (γ ) denotes the space consisting of all equivalent classes of functions: u : γ → C which are measurable for the measure |dz| and satisfy u p =
γ
|u(z)| p |dz|
1/ p
< ∞, 1 p < ∞
and u∞ = ess-sup|u(z)| < ∞, where “ess-sup” denotes the essential supremum. (ii) Denote by C0 (γ ) the space of all continuous functions on γ which converge to 0 at infinity. The norm of C0 (γ ) is defined by u∞ = max |u(z)|. z∈γ
For 1 p ∞, let p = p/( p − 1). Define the pairing between L p (γ ) and L (γ ) as follows: p
u, v =
u(z)v(z)dz. γ
It can be proved that for 1 < p < ∞, (L p (γ ), L p (γ )) is a dual pair of Banach spaces. For p = 1, (L 1 (γ ), C0 (γ )) is a dual pair of Banach spaces. Here u p = sup |u, v|, v ∈ L p (γ ), v p = 1 and
u
Data Loading...
