Multilinear singular integrals on non-commutative $$L^p$$ spaces

Abstract

We prove L p bounds for the extensions of standard multilinear Calderón-Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space-in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L p spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.

Banach-Valued Modulation Invariant Carleson Embeddings and Outer-$$L^p$$ Spaces: The Walsh Case

“…For this simplicity we pay the price of having to manipulate R-bounds, which leads to the assumption of the RMF property on the trilinear form . The same technical difficulty occured in [15]; a new method of obtaining these results without RMF was recently given in [14].…”

“…Thus, comparing our result with that of Hytönen, Lacey, and Parissis [23], we see that we obtain the same L p bounds for the tritile operator as they do for the quartile operator when restricted to the reflexive range p u ∈ (1, ∞). 14…”

Multilinear operator-valued Calderón-Zygmund theory

“…Boundedness of p extensions is classically obtained through weighted norm inequalities, more recently in connection with localized techniques such as sparse domination: see [14] and the more recent [6,34,33,38] for a nonexhaustive overview of their interplay. The paper [10] finally established L p bounds for the extensions of n-linear SIOs to tuples of UMD spaces tied by a natural product structure -for example, the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. Before [10], Di Plinio and Y. Ou [11] considered operator-valued bilinear multiplier theorems that apply to certain non-lattice UMD spaces.…”

“…The paper [10] finally established L p bounds for the extensions of n-linear SIOs to tuples of UMD spaces tied by a natural product structure -for example, the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. Before [10], Di Plinio and Y. Ou [11] considered operator-valued bilinear multiplier theorems that apply to certain non-lattice UMD spaces. The results of [11] may be thought of as a first attempt of generalization of Weis' R-bounded multiplier theorem [43]; however, the treatment of [11] relies upon additional assumptions on the triple of Banach spaces involved -some bilinear variants of the RMF conditions appearing in [23].…”

“…The RMF property of a Banach space X, involving L p estimates for a certain analogue of the Hardy-Littlewood maximal operator obtained by replacing uniform bounds with Rbounds, dates back to the work of Hytönen, McIntosh and Portal on the vector-valued Kato square root problem [23]: see also [21,31,32]. The recent multilinear vector-valued (but not operator-valued) setup of [10] avoids the use of RMF assumptions in all linearities, arguing by induction on the multilinearity index. On the other hand, the inductive argument of [10] relies on an abstract assumption modeling the Hölder type structure typical of concrete examples of Banach n-tuples, such as that of non-commutative L p spaces with the exponents p satisfying the natural Hölder relation.…”