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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Inequalities in the most simple Sobolev space and convolutions of $ L\sb 2$ functions with weights


Author: Saburou Saitoh
Journal: Proc. Amer. Math. Soc. 118 (1993), 515-520
MSC: Primary 46E35; Secondary 30C40
MathSciNet review: 1134626
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Abstract: For the most simple Sobolev space on $ \mathbb{R}$ composed of real-valued and absolutely continuous functions $ f(x)$ on $ \mathbb{R}$ with finite norms

$\displaystyle {\left\{ {\int_{ - \infty }^\infty {({a^2}{f'}{{(x)}^2} + {b^2}f{{(x)}^2})\,dx} } \right\}^{1/2}}\qquad (a,b > 0),$

we shall apply the theory of reproducing kernels, and derive natural norm inequalities in the space and the related inequalities for convolutions of $ {L_2}$ functions with weights.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1134626-3
PII: S 0002-9939(1993)1134626-3
Keywords: Sobolev space, inequality, reproducing kernel, Green's function, convolution, norm inequality, absolutely continuous functions
Article copyright: © Copyright 1993 American Mathematical Society