Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if T is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, thenHere Ap stands for the bi-parameter weights in R n × R m and bmo(ν) is a suitable weighted little BMO space. We also simplify the proof of the known first order case.2010 Mathematics Subject Classification. 42B20.
We develop a wide general theory of bilinear bi-parameter singular integrals T . First, we prove a dyadic representation theorem starting from T 1 assumptions and apply it to show many estimates, including L p × L q → L r estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like [b1, T ]1 and [b2, [b1, T ]1]2, where b1 and b2 are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.
We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral Tn in R n and a bounded singular integral Tm in R m we prove thatwhere p ∈ (1, ∞), µ, λ ∈ Ap and ν := µ 1/p λ −1/p is the Bloom weight. Here T 1 n is Tn acting on the first variable, T 2 m is Tm acting on the second variable, Ap stands for the bi-parameter weights of R n × R m and BMO prod (ν) is a weighted product BMO space.2010 Mathematics Subject Classification. 42B20.
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.
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