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- W2022709156 abstract "Let u ( x , t ) be the solution of u tt − Δ x u = 0 with initial conditions u(x, 0) = g(x) and u t (x, 0) = ƒ;(x) . Consider the linear operator T: ƒ; → u(x, t) . (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L p if and only if ¦ p −1 − 2 −1 ¦ = (n − 1) −1 and ∥ Tƒ; ∥ L α P = ∥ƒ;∥ L P with α = 1 −(n − 1) ¦ p −1 − 2 −1 ¦ . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2 k the result holds for ¦ p −1 − 2 −1 ¦ < (n − 1) −1 . (b) If n = 2 k − 1, the result is valid for ¦ p −1 − 2 −1 ¦ ⩽ (n − 1) . This result are sharp in the sense that for p such that ¦ p −1 − 2 −1 ¦ > (n − 1) −1 we prove the existence of ƒ; ϵ L P in such a way that Tƒ; ∉ L P . Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers m(ξ) = ψ(ξ) e i¦ξ¦ ¦ ξ ¦ −b and finally we get that the convolution against the kernel K(x) = ϑ(x)(1 − ¦ x ¦) −1 is bounded in H 1 ." @default.
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- W2022709156 date "1980-03-01" @default.
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- W2022709156 title "Lp estimates for the wave equation" @default.
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- W2022709156 doi "https://doi.org/10.1016/0022-1236(80)90110-x" @default.
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