By Marnaghan F. D., Wintner A.
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Such that MΩ u(x) = lim utj (x). j→∞ This implies that |u(x)| − MΩ u(x) = lim |u(x)| − utj (x) j→∞ c dist(x, ∂Ω)MΩ |Du|(x). By the maximal function theorem, we conclude that |u(x)| − MΩ u(x) dist(x, ∂Ω) p dx (MΩ |Du|(x))p dx c Ω Ω |Du(x)|p dx. c Ω This implies that |u(x)| − MΩ u(x) ∈ Lp (Ω). 10 we conclude that |u| − MΩ u ∈ W01,p (Ω). 14. We observe that the maximal operator preserves nonnegative superharmonic functions; see . ) Suppose that u : Ω → [0, ∞] is a measurable function which is not identically ∞ on any component of Ω.
1) B(x,r) for 1 p < ∞, 0 λ n, and B(x, r) a ball in Rn (or contained in n some subdomain Ω ⊂ R ). 2) where γ = 1 − λ/p (see [31, Chapt. 7]). , he attempted to prove the corresponding Sobolev Inequality for functions u whose derivatives belong to Lp;λ . Since Lp;n = Lp , one would hope that such a result would simply become (∗), m = 1. But in his paper, he was only able to achieve the corresponding weak type estimate. And since there is no Marcinkiewicz Interpolation Theorem for the Morrey spaces, the result remained somewhat unsatisfying.
Imbedding theorems and the spectrum of a pseudodiﬀerential operator. Sib. Mat. Zh. : English translation: Sib. Math. J. 18, 758-769 (1977). — Note of Ed. My Love Aﬀair with the Sobolev Inequality 21 (3) Recently, I have been looking at certain hyperbolic and/or retarded potentials of the type f (x − s, t − Φ(s)) |s|α−m ds, Rm t > 0. In this setting, solutions to the 3-dimensional wave equation can be treated when Φ(s) = |s| and m = 3, α = 2. The corresponding Sobolev inequalities here are usually called Strichartz inequalities.
A Canonical Form for Real Matrices under Orthogonal Transformations by Marnaghan F. D., Wintner A.