Abstract
In this letter we present a dual weight version of a localized Parseval identity found by Coifman and Steinerberger for the finite Hilbert transform.
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1 Introduction
All the functions considered in this letter are real-valued. Recall that the Hilbert transform H of a function f on \({{\mathbb {R}}}\) (in proper function spaces) is defined by
where “p.v." stands for the principal value. This singular integral operator has a local version, say on the interval \(I=(-1,1)\), and is given by
It is called the finite Hilbert transform and arises naturally in applied science. In particular, the resolution of the following airfoil equation from aerodynamics,
involves the inversion of \(H_I\) in proper function spaces. The airfoil equation is approached by Tricomi in [7] via establishing some convolution theorems for \(H_I\); these convolution identities are motivated by his earlier study on mixed type equations. A byproduct from [7] is the following arcsine distribution uniqueness.
Theorem 1.1
(Tricomi 1951) Let \(f(x)(1-x^2)^{\frac{1}{4}}\in L^2_I=L^2(-1,1)\). If H(f) vanishes identically on \(I=(-1,1)\), then for some real-valued constant c,
Here, \(\chi _{I}\) is the indicator function of I.
Remark 1.2
For an application of this to Erdös-Turán inequality, see [1].
Recently, this uniqueness result is revisited by Coifman and Steinerberger in [2]. They further observed the following localized Parseval identity for \(H_I\).
Theorem 1.3
(Coifman and Steinerberger 2019) Let \(f(x)(1-x^2)^{\frac{1}{4}}\in L^2_I\). If the mean value of \(f(x)(1-x^2)^{\frac{1}{2}}\) on I is 0, then
This complements the global \(L^2\)-isometry for the standard Hilbert transform H:
Two proofs for the localized Parseval identity (1.1) are offered in [2]: one is by Chebychev orthogonal expansion (see e.g. [6]), another by working on the unit circle and using the formula of conjugate functions as carried out in [3]. In this letter we point out that the identity (1.1) admits the following dual weight version.
Theorem 1.4
Let \(f(x)(1-x^2)^{-\frac{1}{4}}\in L^2_I\). Then
Remark 1.5
Such a dual weight mechanism is indeed quite common in boundary value problems, see for example Rosén [5] on Euclidean upper half-space.
2 Proof of Theorem 1.4
Our proof of (1.2) adapts the Chebychev orthogonal expansion arguments in [2]. Consider f so that \(\frac{f(x)}{\sqrt{1-x^2}}\) is a polynomial. For some \(N\in {{\mathbb {N}}}\) we can write
where \(\{U_k\}\) denotes the family of Chebychev polynomials of the second kind. Thereby,
Furthermore, we have
After using the crucial formulae (see for example [4, p. 187])
where \(\{T_k\}\) denotes the family of Chebychev polynomials of the first kind, we get
This proves the identity (1.2) for f such that \(\frac{f(x)}{\sqrt{1-x^2}}\) is a polynomial. Note that the subspace of such functions is dense in \(L^2\left( I,(1-x^2)^{-\frac{1}{2}}dx\right) \), so \(H_I\) extends to an isometry \(\widetilde{H_I}\) on \(L^2\left( I,(1-x^2)^{-\frac{1}{2}}dx\right) \). Moreover, this extension agrees with \(H_I\) since \(L^2\left( I,(1-x^2)^{-\frac{1}{2}}dx\right) \) embeds into \(L^2_I\). This finishes the proof of Theorem 1.4.
References
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Communicated by Stefan Steinerberger.
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Research of the author is partially supported by the Provincial HEI and NSF grants of Jiangsu (nos. 17KJD110005 and BK20180725) and the National NSF grant of China (no. 11801274). The author would like to thank Dr. Bo Xia (USTC) for long-standing encouragements and helpful communications, and to thank sincerely the referee for kind suggestions on the convergence issue.
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Huang, Y.C. On a Localized Parseval Identity for the Finite Hilbert Transform. J Fourier Anal Appl 28, 83 (2022). https://doi.org/10.1007/s00041-022-09974-y
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DOI: https://doi.org/10.1007/s00041-022-09974-y