Inequalities in the most simple Sobolev space and convolutions of 𝐿₂ functions with weights

Saitoh, Saburou

doi:10.1090/S0002-9939-1993-1134626-3

Inequalities in the most simple Sobolev space and convolutions of $L_ 2$ functions with weights
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Proc. Amer. Math. Soc. 118 (1993), 515-520 Request permission

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 \[ {\left \{ {\int _{ - \infty }^\infty {({a^2}{f’}{{(x)}^2} + {b^2}f{{(x)}^2}) dx} } \right \}^{1/2}}\qquad (a,b > 0),\] 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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