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We say that f is non-decreasing on R if f (x′ , y ′ ) − f (x, y ′ ) − f (x′ , y) + f (x, y) ≥ 0 for every subrectangle [x, x′ ] × [y, y ′ ] of R. As in the one-dimensional case, it is easy to prove the following. 13. If f is HKBV on a rectangle R, then f is the difference of two non-decreasing functions on R. The following two-dimensional integration by parts formula can be proved by iterating the one-dimensional formula. 14. Let f be continuous and let h be HKBV on the rectangle [x1 , x2 ] × [y1 , y2 ].

Then MH (X, Y ) is embedded in X ′ in the sense that every operator of this type is of the form Sϕ : f → f, Th ϕ h∈H ′ for some ϕ ∈ X . (ii) If in addition, X and Y are Banach spaces, then the embedding is continuous. Proof. (i) Let T ∈ MH (X, Y ). By assumption, the map f → (T f )[0] is continuous, and thus (T f )[0] = f, ϕ for some ϕ ∈ X ′ . Furthermore, we have for all h ∈ H that (T f )[h] = (T−h T f )[0] = (T T−h f )[0] = T−h f, ϕ = f, Th ϕ = (Sϕ f )[h] , so T = Sϕ as claimed. 6 (Representation Theorem for MH X(G), ℓ∞ (H) ).

Pn ) an n-tuple with pi ∈ [1, ∞]. Then LP (X) is the Banach space of measurable functions on (X, Σ, µ) = ( Xi , Σi , µi ) with norm f LP (X) = ··· = Xn f ··· Lp1 (X1 ) Lp2 (X2 ) ··· f (x1 , . . , xn ) X1 Lpn (Xn ) p1 1/pn p2 /p1 dµ1 · · · dµn with the usual conventions for pi = ∞. If for the quotient G/H there exists a measurable fundamental domain D, we can write G = H × D as measure spaces and consider the mixed-norm spaces Lp1 ,p2 (H×D) of measurable functions on G. 3 Semidiscrete Multipliers 33 We can avoid making use of D by the following construction.

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A 3-color Theorem on Plane Graphs without 5-circuits by Xu B.G.

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