Moser's second one-dimensional inequality for a class of integral operators |
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Authors: | D Walsh |
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Institution: | Department of Mathematics, NUI Maynooth, Co. Kildare, Ireland, IE
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Abstract: | Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? Lp(-¥ ,¥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò0xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò-¥¥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant CpC_p, such that ò-¥¥ expa |F(x)|q-|x|] dx £ Cp\int\limits _{-\infty }^\infty \exp a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||p £ 1||f||_p\le 1 and every a £ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a. |
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