Sparse domination implies vector-valued sparse domination

We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the $${{\,\mathrm{UMD}\,}}$$ UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a $${{\,\mathrm{UMD}\,}}$$ UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calderón-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.

BMO Functions Generated by A X ℝ n Weights on Ball Banach Function Spaces

Let X be a ball Banach function space on ℝ n . We introduce the class of weights A X ℝ n . Assuming that the Hardy-Littlewood maximal function M is bounded on X and X ′ , we obtain that BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A X ℝ n . As a consequence, we have BMO ℝ n = α ln ω : α ≥ 0 , ω ∈ A L p · ℝ n ℝ n , where L p · ℝ n is the variable exponent Lebesgue space. As an application, if a linear operator T is bounded on the weighted ball Banach function space X ω for any ω ∈ A X ℝ n , then the commutator b , T is bounded on X with b ∈ BMO ℝ n .

On boundedness of the Hilbert transform on Marcinkiewicz spaces

We study boundedness properties of the classical (singular) Hilbert transform (Hf)(t) = p.v.1/π \int_R f(s)/(t − s)ds acting on Marcinkiewicz spaces. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Questions involving the H arise therefore from the utilization of complex methods in Fourier analysis, for example. In particular, the H plays the crucial role in questions of norm-convergence of Fourier series and Fourier integrals. We consider the problem of what is the least rearrangement-invariant Banach function space F(R) such that H : Mφ(R) → F(R) is bounded for a fixed Marcinkiewicz space Mφ(R). We also show the existence of optimal rearrangement-invariant Banach function range on Marcinkiewicz spaces. We shall be referring to the space F(R) as the optimal range space for the operator H restricted to the domain Mφ(R) ⊆ Λϕ0(R). Similar constructions have been studied by J.Soria and P.Tradacete for the Hardy and Hardy type operators [1]. We use their ideas to obtain analogues of their some results for the H on Marcinkiewicz spaces.

On solvability of nonlinear problem for some elliptic by Petrovsky system

The article presents the results of study the existence of the solution of nonlinear problem for elliptic by Petrovsky system in unbounded domain. The nonlinear elliptic system is transformed into an equivalent fixed point problem for a suitable nonlinear operator in a convex subset of a Banach function space. Applying the fixed point and embedding theorems the solvability of nonlinear problem for elliptic by Petrovsky system in weighted Sobolev space are established.

Nuclear operators on Banach function spaces

Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

Banach Lattice Structures and Concavifications in Banach Spaces

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

Let [Formula: see text] be a finite measure space and consider a Banach function space [Formula: see text]. Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels [Formula: see text] that define integration maps [Formula: see text] between Köthe–Bochner spaces of Hilbert-valued functions [Formula: see text] We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces [Formula: see text] and [Formula: see text] — where [Formula: see text] — under the assumption of [Formula: see text]-concavity of [Formula: see text] Equivalently, a new kernel obtained by multiplying [Formula: see text] by scalar functions can be given in such a way that the kernel operator is defined from [Formula: see text] to [Formula: see text] in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.

Generalized Lebesgue Points for Hajłasz Functions

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX<∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX<(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.