Timo S. Hanninen

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovski\u{\i}'s factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers. In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved - it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida-Hewitt theory of such measures - the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.

In previous work we established a multilinear duality and factorisation theory for norm inequalities for pointwise weighted geometric means of positive linear operators defined on normed lattices. In this paper we extend the reach of the theory for the first time to the setting of general linear operators defined on normed spaces. The scope of this theory includes multilinear Fourier restriction-type inequalities. We also sharpen our previous theory of positive operators. Our results all share a common theme: estimates on a weighted geometric mean of linear operators can be disentangled into quantitative estimates on each operator separately. The concept of disentanglement recurs throughout the paper. The methods we used in the previous work - principally convex optimisation - relied strongly on positivity. In contrast, in this paper we use a vector-valued reformulation of disentanglement, geometric properties (Rademacher-type) of the underlying normed spaces, and probabilistic considerations related to p-stable random variables.

Let σ and ω be locally finite Borel measures on ℝ^d, and let $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) and $q\in (0,\infty )$ q ∈ ( 0 , ∞ ) . We study the two-weight norm inequality $ \lVert T(f\sigma ) \rVert _{L^{q}(\omega )}\leq C \lVert f \rVert _{L^{p}(\sigma )}, \quad \text {for all} f \in L^{p}(\sigma ), $ ∥ T ( f σ ) ∥ L q ( ω ) ≤ C ∥ f ∥ L p ( σ ) , for all f ∈ L p ( σ ) , for both the positive summation operators T = T_λ(⋅σ) and positive maximal operators T = M_λ(⋅σ). Here, for a family {λ_Q} of non-negative reals indexed by the dyadic cubes Q, these operators are defined by $ T_{\lambda }(f\sigma ):={\sum }_{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q} \quad \text { and } \quad M_{\lambda }(f\sigma ):=\sup _{Q} \lambda _{Q} \langle {f}\rangle ^{\sigma }_{Q} 1_{Q}, $ T λ ( f σ ) : = ∑ Q λ Q 〈 f 〉 Q σ 1 Q and M λ ( f σ ) : = sup Q λ Q 〈 f 〉 Q σ 1 Q , where $\langle {f}\rangle ^{\sigma }_{Q}:=\frac {1}{\sigma (Q)} {\int \limits }_{Q} |f| d \sigma $ 〈 f 〉 Q σ : = 1 σ ( Q ) ∫ Q | f | d σ . We obtain new characterizations of the two-weight norm inequalities in the following cases: (1) For T = T_λ(⋅σ) in the subrange q < p. Under the additional assumption that σ satisfies the $A_{\infty }$ A ∞ condition with respect to ω, we characterize the inequality in terms of a simple integral condition. The proof is based on characterizing the multipliers between certain classes of Carleson measures. (2) For T = M_λ(⋅σ) in the subrange q < p. We introduce a scale of simple conditions that depends on an integrability parameter and show that, on this scale, the sufficiency and necessity are separated only by an arbitrarily small integrability gap. (3) For the summation operators T = T_λ(⋅σ) in the subrange 1 < q < p. We characterize the inequality for summation operators by means of related inequalities for maximal operators T = M_λ(⋅σ). This maximal-type characterization is an alternative to the known potential-type characterization. The subrange of the exponents q < p appeared recently in applications to nonlinear elliptic PDE with $\lambda _{Q} = \sigma (Q) |Q|^{\frac {\alpha }{d}-1}$ λ Q = σ ( Q ) | Q | α d − 1 , α ∈ (0, d). In this important special case T_λ is a discrete analogue of the Riesz potential $I_{\alpha }=(-{\Delta })^{-\frac {\alpha }{2}}$ I α = ( − Δ ) − α 2 , and M_λ is the dyadic fractional maximal operator.

We study the domination of the lattice Hardy–Littlewood maximal operator by sparse operators in the setting of general Banach lattices. We prove that the admissible exponents of the dominating sparse operator are determined by the q q -convexity of the Banach lattice.

We study the domination of the lattice Hardy--Littlewood maximal operator by sparse operators in the setting of general Banach lattices. We prove that the admissible exponents of the dominating sparse operator are determined by the $q$-convexity of the Banach lattice.

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