By Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis
The current quantity develops the speculation of integration in Banach areas, martingales and UMD areas, and culminates in a therapy of the Hilbert remodel, Littlewood-Paley thought and the vector-valued Mihlin multiplier theorem.
Over the previous fifteen years, influenced by way of regularity difficulties in evolution equations, there was great development within the research of Banach space-valued services and tactics.
The contents of this wide and strong toolbox were in general scattered round in examine papers and lecture notes. accumulating this assorted physique of fabric right into a unified and available presentation fills a spot within the latest literature. The critical viewers that we have got in brain comprises researchers who want and use research in Banach areas as a device for learning difficulties in partial differential equations, harmonic research, and stochastic research. Self-contained and delivering whole proofs, this paintings is obtainable to graduate scholars and researchers with a history in sensible research or similar areas.
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Additional info for Analysis in Banach Spaces : Volume I: Martingales and Littlewood-Paley Theory
Step 1 – The strong µ-measurability allows us to make two reductions. 15 we may assume that µ is σ-finite. Secondly, we take advantage of the fact that f takes µ-almost all of its values in a separable closed subspace of Lp (T ; X). 29 below will show that such a subspace is always contained in a subspace of the form Lp (T, B , ν|B ; Y ) for some sub-σ-algebra B ⊆ B, with ν|B σ-finite, and a separable closed subspace Y of X. Admitting this result for the moment, it follows that there is no loss of generality in assuming that also ν is σ-finite.
Hence, for almost all t ∈ T , ˆ ˆ F dµ (t) = lim Fn dµ (t) n→∞ S S ˆ ˆ = lim fn (s, t) dµ(s) = f (s, t) dµ(s). 2 Integration 27 Step 3 – If µ is σ-finite, we apply the above to the functions F (j) = 1T (j) F , where (T (j) )j 1 is a disjoint decomposition of T by measurable sets of finite measure, and piece together the resulting functions fj . Step 4 – It remains to prove the uniqueness assertion. Suppose g is as stated in the proposition. Since g(s, ·) = F (s) in Lp (T ; X) for almost all s ∈ S, for almost all s ∈ S we have g(s, t) = f (s, t) for almost all t ∈ T .
Note that τ (X ∗ , X)- A f dµ ∈ X ∗ for all A∈A. 33 it is´ natural to ask for conditions which guarantee that the integrals τ (X, X ∗ )- A f dµ belong to X (rather than just to X ∗∗ ) for all A ∈ A . 39. This prompts the following definition. 35. A weakly µ-integrable function f : S → X is called Pettis integrable if the adjoint Tf∗ of the operator Tf : x∗ → f, x∗ maps L∞ (S) into X. 2 Integration 31 Here, L∞ (S) is identified isometrically with a norming closed subspace of the dual (L1 (S))∗ .