Two-weight norm inequalities for potential type and maximal operators in a metric space

Abstract

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hytönen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type.2010 Mathematics Subject Classification. 42B25 (30L99, 47B38).

Two weight inequalities for fractional integral operators and commutators

“…Here we give a new proof based on ideas of Hytönen [42] and Lacey, et al [53]. For a related proof that avoids duality and is closer in spirit to the proof of Theorem 4.4, see Kairema [45].…”

On some bilinear problems on weighted Hardy‐Sobolev spaces

“…It is shown in 10 (Theorem 7.4 and Lemma 7.9) the following result, which is an extension of the Euclidean result of Sawyer's (13) to homogeneous spaces. We reproduce it for the case p = 2 and the Lebesgue measure d σ:…”

“…Theorem 3.1 (10) Let 0 < γ < n , and ν and η be two positive Borel measures locally finite on S n . Then the following assertions are equivalent: \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Vert M_\gamma f\Vert _{L^2(\eta )} \lesssim \Vert f\Vert _{L^2(\nu )}$\end{document} holds for any f ∈ L 2 ( d ν). σ < <ν and the testing condition holds for all dyadic cubes \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$Q\in {\mathcal D}_j$\end{document}, j = 1, …, K . …”

Two-weighted estimates for positive operators and Doob maximal operators on filtered measure spaces

“…The smaller quantities come in the form of A r constants for large r or A ∞ constants. The dyadic discretization technique is also valid for (linear) positive operators (see [29,30,24,25,50]) and the (fractional) maximal operator (see [4,43,31,29,17,21]).…”