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Flat singular integrals in product domains

Ahmad Al-Salman

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Abstract

In this paper, we study singular integrals on product domains with kernels in L(logL)2(Sn?1?Sm?1) supported by surfaces of revolutions. We prove that our operators are bounded on Lp under certain convexity assumption on the surfaces. Also, in this paper we prove that the convexity assumption is not necessary for the L boundedness to hold. Moreover, additional related results are presented. Our condition on the kernel is known to be optimal.

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What this paper is about

In this paper, we study singular integrals on product domains with kernels in L(logL)2(Sn?1?Sm?1) supported by surfaces of revolutions. We prove that our operators are bounded on Lp under certain convexity assumption on the surfaces. Also, in this paper we prove that the convexity assumption is not necessary for the L boundedness to hold. Moreover, additional related results are presented. Our condition on the kernel is known to be optimal.

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Available abstract

In this paper, we study singular integrals on product domains with kernels in L(logL)2(Sn?1?Sm?1) supported by surfaces of revolutions. We prove that our operators are bounded on Lp under certain convexity assumption on the surfaces. Also, in this paper we prove that the convexity assumption is not necessary for the L boundedness to hold. Moreover, additional related results are presented. Our condition on the kernel is known to be optimal.

Key concepts: Mathematics, Convexity, Bounded function, Product (mathematics), Kernel (algebra), Singular integral, Pure mathematics, Mathematical analysis

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