The convergence of the Bochner-Riesz means at the critical index

Lung-Kee Chen, Dashan Fan

Type: Article

Publication Date: 1996-01-01

Citations: 11

DOI: https://doi.org/10.1090/s0002-9939-96-03333-3

Abstract

In this paper, we study the pointwise convergence of the Bochner-Riesz means at the critical index on the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L log Superscript plus Baseline upper L left-parenthesis upper Q Subscript n Baseline right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:msup> <mml:mi>log</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>⁡</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L\log ^+ L (Q_n).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We weaken the hypothesis, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-turned-comma-quotation-mark x"> <mml:semantics> <mml:mrow> <mml:mo>“</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">“ x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Lebesgue point", which is required on some research results by instead considering the convergence of averages of the function over balls when the radials of the balls approach to 0.

Locations

Proceedings of the American Mathematical Society - View - PDF

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Works Cited by This (5)

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+	Introduction to Fourier Analysis on Euclidean Spaces.	1971	Elias M. Stein Guido Weiss
+ PDF Chat	Summation of multiple Fourier series by spherical means	1936	Salomon Bochner
+	New Theorems on the Convergence of Fourier Series	1954	R. Salem
+	On Certain Exponential Sums Arising In Multiple Fourier Series	1961	Elias M. Stein
+	Lectures on Bochner-Riesz Means	1987	Katherine Davis Yang-Chun Chang