New PDF release: An introduction to functional analysis
By Mischa Cotlar; Roberto Cignoli
This textbook emphasizes these subject matters proper and essential to the examine of research and chance idea. the 1st 5 chapters take care of summary dimension and integration. bankruptcy 6, on differentiation, incorporates a therapy of adjustments of variables in Rd
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Cp,m (e) Case (ii) is treated in the same way. 6. 34) where m > l and u ∈ C0∞ , are equivalent. Proof. The estimate C0 ≤ c C is obvious. We show that C0 ≥ c C. )D2α (−∆)−l u. 2 we get |D2α (−∆)−l u|p dµ ≤ C0 Dα (−∆)−l u p m wp ≤ c C0 u p m−l . 34) holds with C ≤ c C0 . 7. The best constants C0 and C in the inequalities (|∇l u|p + |u|p )dµ ≤ C0 u |u|p dµ ≤ C u where m > l and u ∈ C0∞ , are equivalent. 36) 24 1 Trace Inequalities for Functions in Sobolev Spaces Proof. The estimate C0 ≤ c C is obvious.
48) have been obtained independently of each other. 48) has its own range of applications. 48) has not been obtained so far. 31) to hold, which was obtained by Verbitsky [Ver1]: for every dyadic cube P0 in Rn P ⊆P0 µ(P ) (mesn P )1−m/n p mesn P ≤ C µ(P0 ), where the sum is taken over all dyadic cubes P contained in P0 and the constant does not depend on P0 . 29). 49) if there exists t > 1 such that Br (x) [g(y)]t dy ≤ c rn−mpt . 3 Estimate for the Lq -Norm with respect to an Arbitrary Measure 29 (see [SW]), where Ml f is the fractional maximal function deﬁned by Ml f (x) = sup rl−n r>0 Br (x) |f (y)|dy.
Since 1/p > 1 − k/n, we have 1/p > 1/q + 1/r. By H¨older’s inequality γα u Lp,unif ≤ c γα Lr,unif u Lq,unif and by Sobolev’s imbedding theorem γα u Lp,unif ≤ c γα Wpk−m ,unif u m . Wp,unif This means that γα ∈ M (Wpm → Lp ). 1. 1. Note that by Sobolev’s imbedding theorem −k Wp−m ,unif ⊂ Wp,unif , k ≥ m, if and only if either n ≤ (k − m)p or k−m 2−p ≥ . n p n > (k − m)p, Under these conditions, M (Wpm → Wp−k ) is isomorphic to Wp−m ,unif if kp > n. Analogously, if m ≥ k, mp > n and either n ≤ (m − k)p or p−2 m−k ≥ , n p n > (m − k)p , −k .
An introduction to functional analysis by Mischa Cotlar; Roberto Cignoli