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Behavior in $ L^\infty $ of convolution transforms with dilated kernels

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  • Assuming that $ K(x) $ is in $ L^1( {\mathbb R}) $, $ K_t(x) = t^{-1} K(x/t) $, and $ f(x) $ is in $ L^\infty( {\mathbb R}) $, we study the behavior of the convolution $ K_t*f(x) $ as the parameter $ t $ tends to $ \infty $. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of $ x $. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.

    Mathematics Subject Classification: Primary: 42A85, 42B99, 44A35; Secondary: 40A30.

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  • [1] M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Mathematics Studies, 46. Notas de Matemtica [Mathematical Notes], 75. North-Holland Publishing Co., Amsterdam-New York, 1981.
    [2] F. John, Partial Differential Equations, Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4684-9333-7.
    [3] B. F. Logan, Limits in $L^p$ of convolution transforms with kernels $aK(at)$, $a \to 0$, SIAM J. Math. Anal., 10 (1979), 733-740.  doi: 10.1137/0510068.
    [4] W. R. Madych, Limits of dilated convolution transforms, SIAM J. Math. Anal., 16 (1985), 551-558.  doi: 10.1137/0516041.
    [5] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
    [6] F. Zo, A note on approximation of the identity, Studia Math., 55 (1976), 111-122.  doi: 10.4064/sm-55-2-111-122.
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