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The double layer potential operator on Hardy spaces. (English) Zbl 1524.42027

The author considers mapping of double layer potentials defined on \(\mathbb{R}^n\). The operators are \[ T_Af(x) = \int_{\mathbb{R}^n} \frac{ A(x) - A(y) } {((x - y)^2 + (A(x) - A(y))^2 )^{\frac{n+1}{2} }} f(y) \, dy, \] \[ T_kf(x) = \int_{\mathbb{R}^n} \frac{ x_k - y_k } {((x - y)^2 + (A(x) - A(y))^2 )^{\frac{n+1}{2} }} f(y) \, dy \] and they are used in the study of the operator \[ S_Af(x) = \int_{\mathbb{R}^n} \frac{ A(x) - A(y) - \nabla A(y) \cdot (x - y) } {((x - y)^2 + (A(x) - A(y))^2 )^{\frac{n+1}{2} }} f(y) \, dy, \] which arises is the study of electromagnetics.
M. A. M. Murray [Trans. Am. Math. Soc. 289, 497–518 (1985; Zbl 0574.42012)] proved that \(T_A\), \(T_k \) are bounded operators on \(L^p(\mathbb{R}^n)\) if \(\nabla A \in L^{\infty}(\mathbb{R}^n)^n\). J. Alvarez and M. Milman [J. Math. Anal. Appl. 118, 63–79 (1986; Zbl 0596.42006)] had proved \(H^p\) boundedness of singular integral operators under the condition \(T1 = 0\), related to the fact that functions in \(H^1\) have mean \(0\), but \(T_A 1 \ne 0\) and it is known that \(T_A\) is not bounded on \(H^p\). This leaves the study of whether they map \(H^p(\mathbb{R}^n)\) into the local Hardy space \(h^p(\mathbb{R}^n)\). The author proves this for \( n=1\), and the major result of this paper is
Theorem 1.3: Let \(0 < \alpha < 1\). If \(\nabla A \in L^{\infty}(\mathbb{R}^n)^n \cap \Lambda_{\alpha}(\mathbb{R}^n)\), then if \(\frac{n}{n+ \alpha} < p \leq 1\), \(T_A, T_k\) map \(H^p(\mathbb{R}^n)\) into \(h^p(R^n)\).
Here \(\Lambda_{\alpha}(\mathbb{R}^n)\) are the Lipschitz functions of order \(\alpha\). Since the author shows that the Calderon operator \[ C_Af(x) = \int \frac{A(x) - A(y)}{(x - y)^2} f(y) \, dy \] does not map \(H^p(\mathbb{R})\) into \(h^p(\mathbb{R})\) for \(1/2 <p < \frac{1}{\alpha + 1}\), even though \(A^{\prime} \in L^{\infty}(\mathbb{R}) \cap \Lambda_{\alpha}(\mathbb{R})\), he thinks the results are optimal, but does not have a proof.
The proof uses Hardy spaces based on Clifford algebras introduced for the purpose, but slightly different from the Clifford algebra Hardy spaces introduced by J. E. Gilbert and M. A. M. Murray [Rev. Mat. Iberoam. 4, No. 2, 253–289 (1988; Zbl 0711.35089)] and Z. Wu [in: Clifford algebras in analysis and related topics. Based on a conference, Fayetteville, AR, USA, April 8–10, 1993. Boca Raton, FL: CRC Press. 217–238 (1996; Zbl 0868.30045)].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
15A66 Clifford algebras, spinors
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