Abstract
Suppose that a measurable 2π-periodic essentially bounded function (the kernel) κλ =κλ(x) is given for any realλ≥1. We consider the following linear convolution operator inL p:
Uniform boundedness of the family of operators {Κλ}λ≥1 is studied. Conditions on the variable exponentp=p(x) and on the kernel κλ that ensure the uniform boundedness of the operator family {Κλ}λ≥1 inL p are obtained. The condition on the exponentp=p(x) is given in its final form.
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I. I. Sharapudinov, “On the topology of the spaceL p(t)([0, 1]),”Mat. Zametki [Math. Notes],26, No. 4, 613–632 (1979).
I. I. Sharapudinov, “The basis property of the Haar system inL p(t)([0, 1]) and the principle of localization in mean,”Mat. Sb. [Math. USSR-Sb.],130, No. 2, 275–283 (1986).
A. Zygmund,Trigonometric Series, Cambridge Univ. Press, Cambridge (1959).
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Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 291–302, February, 1996.
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Sharapudinov, I.I. Uniform boundedness inL p (p=p(x)) of some families of convolution operators. Math Notes 59, 205–212 (1996). https://doi.org/10.1007/BF02310962
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DOI: https://doi.org/10.1007/BF02310962