Vol. 76, No. 1, 1978

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Potential operators and equimeasurability

Ata Nuri Al-Hussaini

Vol. 76 (1978), No. 1, 1–7
Abstract

W. Rudin proved the following.

Theorem 1.1. Assume 0 < p < , p2,4,6, . Let n be a positive integer. If fi Lp(μ), gi Lp(ν) for 1 i n and

∫                          ∫
|1+ z f + ⋅⋅⋅+ z f |pdμ =   |1+ z g + ⋅⋅⋅+ z g|pdν
X     1 1        n n       Y     1 1       n n

for all (z1,,zn) Cn, then (f1,,fn) and (g1,,gn) are equimeasurable. Here as usual Lp(μ) and Lp(ν) stand for p-th power integrable functions defined on finite measure spaces (X,X,μ) and (Y,Y,ν) respectively. 𝒞 is the field of complex numbers.

The purpose of this paper is to provide a new setting for Rudin’s result by recasting it and its extension to real valued functions into the framework of the theory of potential operators as formulated by K. Yosida.

Mathematical Subject Classification 2000
Primary: 60J45
Secondary: 46E30
Milestones
Received: 20 May 1977
Published: 1 May 1978
Authors
Ata Nuri Al-Hussaini