### A class of commutators for multilinear fractional integrals in nonhomogeneous spaces.

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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $ma{x}_{\omega \in}\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$. We study the stability of the APDP by c₀-, ${\ell}_{\infty}$- and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely ${L}_{\infty}(\mu ,X)$ and C(K,X), where X has the APDP.

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.

Let L = -Δ + V be a Schrödinger operator in ${\mathbb{R}}^{d}$ and $H{\xb9}_{L}\left({\mathbb{R}}^{d}\right)$ be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by ${T}^{\pm}(f,g)\left(x\right)=\left(T\u2081f\right)\left(x\right)\left(T\u2082g\right)\left(x\right)\pm \left(T\u2082f\right)\left(x\right)\left(T\u2081g\right)\left(x\right)$, where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from ${L}^{p}\left({\mathbb{R}}^{d}\right)\times {L}^{q}\left({\mathbb{R}}^{d}\right)$ to $H{\xb9}_{L}\left({\mathbb{R}}^{d}\right)$ for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

We give a unified treatment of procedures for complexifying real Banach spaces. These include several approaches used in the past. We obtain best possible results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. These estimates provide generalizations and show sharpness of previously obtained inequalities.

In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples...

Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {\u0302}_{n,\pi}E\right)$ (resp. $\Delta \left(\otimes {\u0302}_{n,s,\pi}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {\u0302}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {\u0302}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to ${E}_{\left[n\right]}$, the completion of the n-concavification of...

We show that a Banach space X is an ℒ₁-space (respectively, an ${\mathcal{L}}_{\infty}$-space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space ${}_{wb}{(}^{m}X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an ${\mathcal{L}}_{\infty}$-space.

We study the following problem: Given a homogeneous polynomial from a sublattice of a Banach lattice to a Banach lattice, under which additional hypotheses does this polynomial factorize through ${L}_{p}$-spaces involving multiplication operators? We prove that under some lattice convexity and concavity hypotheses, for polynomials certain vector-valued norm inequalities and weighted norm inequalities are equivalent. We combine these results and prove a factorization theorem for positive homogeneous polynomials...

In recent papers, the Right and the Strong* topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability)....

The space of the fully absolutely (r;r1,...,rn)-summing n-linear mappings between Banach spaces is introduced along with a natural (quasi-)norm on it. If r,rk C [1,+infinite], k=1,...,n, this space is characterized as the topological dual of a space of virtually nuclear mappings. Other examples and properties are considered and a relationship with a topological tensor product is stablished. For Hilbert spaces and r = r1 = ... = rn C [2,+infinite[ this space is isomorphic to the space of the Hilbert-Schmidt...

Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.