On two-weight norm inequalities for positive dyadic operators

Let $\sigma$ and $\omega$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(f\sigma) \rVert_{L^q(\omega)}\leq C \lVert f \rVert_{L^p(\sigma)}, \quad \text{for all} \, \, f \in L^p(\sigma), $$ for both the positive summation operators $T=T_\lambda(\cdot \sigma)$ and positive maximal operators $T=M_\lambda(\cdot \sigma)$. Here, for a family $\{\lambda_Q\}$ of non-negative reals indexed by the dyadic cubes $Q$, these operators are defined by $$ T_\lambda(f\sigma):=\sum_Q \lambda_Q \langle f\rangle^\sigma_Q 1_Q \quad\text{ and } \quad M_\lambda(f\sigma):=\sup_Q \lambda_Q \langle f\rangle^\sigma_Q 1_Q, $$ where $\langle f\rangle^\sigma_Q:=\frac{1}{\sigma(Q)} \int_Q |f| d \sigma.$ We obtain new characterizations of the two-weight norm inequalities in the following cases: 1. For $T=T_\lambda(\cdot\sigma)$ in the subrange $q<p$. Under the additional assumption that $\sigma$ satisfies the $A_\infty$ condition with respect to $\omega$, 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 $T=M_\lambda(\cdot \sigma)$ 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 $T=T_\lambda(\cdot\sigma)$ in the subrange $1<q<p$. We characterize the inequality for summation operators by means of related inequalities for maximal operators $T=M_\lambda(\cdot \sigma)$. This maximal-type characterization is an alternative to the known potential-type characterization.

The uppercase letters P, Q, R, S are reserved for dyadic cubes. The indexing 'Q ∈ D' is abbreviated as 'Q' in the indexing of summations, and omitted in the indexing of families (and similarly for the cubes P, R, S).
The standing assumption is that p ∈ (1, ∞), q ∈ (0, ∞) and q < p. Hence only further restrictions on the exponents are mentioned.

Introduction
Let σ and ω be locally finite Borel measures on R d , and let λ = {λ Q } Q∈D be a sequence of non-negative reals indexed by the dyadic cubes Q ∈ D. We study the two-weight inequalities in the range of the exponents 0 < q < p and p > 1. It is a standing assumption throughout this article that the exponents are in this range and hence only further restrictions on the exponents are mentioned. (Here, T is either the dyadic summation operator T λ ( ⋅ σ), or the dyadic maximal operator M λ ( ⋅ σ), both of which are defined in Notation.) This range of exponents appeared recently in applications to nonlinear elliptic PDE [2], [24], [33]; in this case is the dyadic Riesz potential, a discrete analogue of the classical Riesz potential is the dyadic fractional maximal operator (see [1], [5], [25], [31]).
Nevertheless, this range of exponents is still insufficiently understood, especially in the range 0 < q < 1 < p for the summation operator. (The characterization in the case 1 < q < p was completed recently by Tanaka [27,Theorem 1.3].) For the two-weight inequality (1.1) in its full generality, the known sufficient and necessary conditions are complicated. The conditions that characterize the inequality for maximal operators in this range are, in essence, conditions that are required to hold uniformly over all linearizations of maximal operators (see [31, Theorem 2] by Verbitsky, [12,Theorem 7.8] by Hänninen, and [13, Theorem 5.2]) by Hänninen, Hytönen, and Li). These conditions are described in Subsection 2.1. Similarly, the conditions that characterize the inequality for summation operators in the subrange 0 < q < 1 are required to hold over all possible factorizations (see [14, Theorem 1.1. and Theorem 1.2] by the authors). Although these characterizations provide us with alternative viewpoints at these inequalities and offer an alternative starting point for their study, such conditions are difficult to verify in applications.
To ameliorate this problem in the case of summation operators, we introduced earlier a scale of conditions that depends on an integrability parameter and showed that, on this scale, the sufficiency and necessity conditions are separated by a certain integrability gap (see [14,Theorem 1.3] by the authors). In this article, we now introduce an analogous scale of conditions for maximal operators and show that, on this scale, the sufficiency and necessity conditions are separated only by an arbitrarily small integrability gap (see Proposition 2.1 for the precise statement).
Under the additional assumption that the measures σ and ω satisfy the A ∞ condition with respect to each other, simple conditions for both summation and maximal operators are known in many ranges of exponents p and q. In this article, we complete this picture by addressing the remaining case: the case of summation operators and the range p ∈ (1, ∞), q ∈ (0, ∞), and q < p (see Proposition 2.2 for the precise statement). The proof is based on a characterization of multipliers of Carleson coefficients (see Proposition 3.7 for the precise statement).
Although the summation operator and supremum operator can both be viewed on the scale of vector-valued operators the characterizations of them, both the statements and the proofs, seem to be very different from each other and, to the best of the authors' knowledge, no explicit connections between the inequalities for summation and maximal operators are known.
In this article, we find that, in the range q ∈ (1, ∞), the inequality for summation operators can be characterized in terms of inequalities for related maximal operators (see Proposition 2.3 for the precise statement). This maximal-type condition can also be regarded as an alternative to the known potential-type condition (see [5, Theorem A] by Cascante, Ortega, and Verbitsky, and [27, Theorem 1.3] by Tanaka). The known potential-type condition is described in Subsection 2.3. Next, we present in more detail each of our results and how they are related to the earlier results in the literature.

Statements of results
2.1. Scale of conditions for maximal operators. Let 0 < q < p < ∞ and p > 1. We study the two-weight norm inequality , for all f ∈ L p (σ).
In the general case, the known sufficient and necessary conditions are complicated and difficult to apply, whereas only in the limited particular cases simpler and more easily applicable conditions are known. For general measures σ and ω and coefficients λ, the following complicated conditions are known: • For every collection Q of dyadic cubes, we define the auxiliary function λ Q by Inequality (2.1) holds if and only if there exists a constant C > 0 such that for all collections Q of dyadic cubes. This characterization was obtained by Verbitsky [31,Theorem 2].
• Inequality (2.1) holds if and only if there exists a constant C > 0 such that This characterization was observed by Hänninen [12,Theorem 7.8], and a variant of it by Hänninen, Hytönen, and Li [13, Theorem 5.2]. For particular measures σ and ω, or for particular coefficients λ, from these conditions the following simpler conditions follow: • Assume that the coefficients λ satisfy for all dyadic cubes Q. This is an analogue of the so called dyadic logarithmic bounded oscillation condition (DLBO) for summation operators (see, for example, [4]). Then inequality (2.1) holds if and only if there exists a constant C > 0 such that • Assume that the measures σ and ω satisfy the A ∞ condition with respect to each other and have no point masses. Then, by Corollary 3.9, for each collection for all dyadic cubes Q, and conversely. From combining this with the condition (2.3) it follows that the two-weight norm inequality holds if and only if there exists a constant C > 0 such that In this paper, we introduce a scale of simple conditions that depend on an integrability parameter, and prove that the necessity and sufficiency on this scale are separated only by an arbitrarily small integrability gap. For each integrability parameter γ ∈ (−∞, ∞), we define the localized auxiliary quantity Λ sup γ,Q by Our result reads as follows: , and q < p. The following assertions hold: (ii) (Necessary condition) Let ǫ > 0 be an arbitrarily small positive real. We have Remark. Our condition (2.4) is sufficient in the general case. In addition, it is also necessary in the particular case where sup R∶R⊆Q λ R 1 R ≂ λ Q , and also in the particular case where σ and ω are A ∞ measures with respect to each other. Thus, our condition includes the above-listed earlier particular cases in which simple conditions were known. Furthermore, our sufficient condition is close to being necessary even in the general case, since the sufficient condition (2.4) becomes necessary once the integrability parameter q in the quantity Λ sup q,Q is lowered by an arbitrarily small ǫ > 0.

Characterization for summation operators under the A ∞ assumption.
In the case where the measure σ satisfies the A ∞ condition with respect to ω, the two-weight norm inequality can be characterized by simple integral conditions. In this work, we use the Fujii-Wilson A ∞ condition. Since the Coifman-Fefferman A ∞ condition is also used in related earlier work, such as [31], we recall both of these conditions and their relations. The conditions are as follows: (1) (Fujii-Wilson) A measure σ is said to satisfy the dyadic Fujii-Wilson A ∞ condition with respect to a measure ω if there exists a constant C such that, for every dyadic cube Q, we have The least such constant C is called the Fujii-Wilson A ∞ characteristic and denoted by [σ] A∞(ω) . (2) (Coifman-Fefferman) A measure σ is said to satisfy the dyadic Coifman-Fefferman A ∞ condition with respect to a measure ω if there exist α, β ∈ (0, 1) such that for every dyadic cube and every subset E ⊆ Q we have that We observe, by contraposition and by taking complement, that the Coifman-Fefferman condition is symmetric in the measures σ and ω. Some relations between the conditions are as follows: • For non-doubling measures, the Coifman-Fefferman condition is in general strictly stronger than the Fujii-Wilson condition. For a proof that (2) implies (1), see, for example, [10, Proof of Lemma 2.5]. To see that measures may satisfy the Fujii-Wilson condition, but fail to satisfy the Coifman-Fefferman condition, notice that, by the Lebesgue differentiation theorem, the Coifman-Fefferman condition requires that σ is absolutely continuous with respect to ω, whereas the Fujii-Wilson condition does not require this. Accordingly, the case with σ being Lebesgue measure and ω a Dirac measure is an example of measures satisfying (1) but not (2). • Nevertheless, the conditions are equivalent provided both ω and σ are doubling [9, Theorem 1]. Moreover, the doubling properties of the measures were originally assumed in the Coifman-Fefferman condition [7]. Furthermore, because the Coifman-Fefferman condition is symmetric in the measures, in the case of doubling measures, σ satisfies the Fujii-Wilson A ∞ condition with respect to ω if and only if ω satisfies the same condition with respect to σ .
Under the A ∞ assumption, the following simpler (than in the general case) characterizations are known: • The subrange 1 < p ≤ q < ∞ for maximal and summation operators. Hänninen [10, Theorem 1.5] noticed that the two-weight norm inequality for the summation operators is characterized by testing the bilinear estimate against the indicator functions of cubes: A similar characterization holds for the maximal operators as well, and it can be proven, for example, by a parallel stopping cubes argument analogous to the argument appearing in [10]. • The subrange 0 < q < p and p > 1 for maximal operators. Verbitsky [31] proved that the two-weight norm inequality for the maximal operators is characterized by a simple integral condition: In this paper, we address the remaining case: The subrange 0 < q < p and p > 1 for summation operators. For brevity, we write for the integral expression, whose finiteness is sufficient and necessary for inequality (2.5): Proposition 2.2 (Characterization under the A ∞ assumption). Let σ and ω be measures that satisfy the A ∞ condition with respect to each other. Let p ∈ (1, ∞) and q ∈ (0, ∞) be such that q < p. Then we have the following characterization by subranges: • In the subrange q ∈ (0, 1], we have Remark. In the subrange q ∈ (1, ∞), by the L q (ω) − L q ′ (ω) duality, we have where the dual integral expression I * σ,ω,p,q,λ is defined by and is related to the expression I σ,ω,p,q,λ via interchanging λ Q ω(Q) σ(Q) and λ Q , q ′ and p, and ω and σ.

Inequality for summation operators via maximal operators.
In this section, we show that the two-weight norm inequality (2.5) for the summation operator is equivalent to a pair of two-weight norm inequalities for certain related maximal operators: Let {λ Q } be non-negative reals. Then the following assertions are equivalent: (i) Inequality (2.5) holds, that is, for all functions f .
(ii) The following two-weight norm inequalities hold for the related maximal operators: . In this range 1 < q < p, inequality (2.7) for the summation operator can also be characterized by the following two potential-type conditions: and the dual Wolff potential W q λ,ω [σ] is the discrete Wolff potential associated with the adjoint operator , p by q ′ , q by p ′ , respectively, and with σ and ω swapped.) Whereas in the range 1 < q < p the potential-type condition (2.8) is both sufficient and necessary, in the more difficult range 0 < q < 1 and p > 1 no explicit necessary and sufficient condition is known. The authors hope that the connection between the two-weight norm inequality for the summation operator and the two-weight inequalities for the related maximal operators (Proposition 2.3) may be extended from the range 1 < q < ∞ to the range 0 < q < 1, which would be useful in finding a concrete necessary and sufficient condition for summation operators.

Discrete Littlewood-Paley spaces.
We recall the definition of discrete Littlewood-Paley spaces f r,s (µ) for exponents p ∈ (0, +∞], q ∈ R ∖ {0}, and a locally finite Borel measure µ on R d . Essentially this scale of spaces was introduced by Frazier and Jawerth [8] in the case of Lebesgue measure (see [6] in the general case). The discrete Littlewood-Paley norm a f p,q (µ) of a family {a Q } Q∈D of nonnegative reals is defined by cases as follows: • For p ∈ (0, ∞) and q ∈ R ∖ {0}, • For p ∈ (0, ∞) and q = ∞, • For p = ∞ and q ∈ R ∖ {0}, for every family {a Q } Q∈D .
Remark. In particular, in the case p = ∞, q = 1, the dual norm formula reads that the dual estimate holds if and only if the Carleson condition holds, which is a dyadic form of the Carleson imbedding theorem.

Dyadic Hardy-Littlewood maximal inequality.
We recall the dyadic Hardy-Littlewood maximal inequality. The dyadic Hardy-Littlewood maximal operator M µ ( ⋅ ) is defined by

3.4.
Reformulations of the two-weight norm inequalities. We reformulate the two-weight norm inequalities in terms of coefficients in place of functions. These reformulations are used in Subsection 4.3 to pass between the two-weight norm inequality for summation operators and the related inequalities for related maximal operators.

Lemma 3.5 (Reformulations for summation operators).
Let p, q ∈ (1, ∞). Then the following estimates are equivalent: for all functions f .
for all functions f and g.
for all families a and b. (iv) We have for all familiesã andb.
Proof. The equivalence between estimates (i) and (ii) follows from the L q (ω) − L q ′ (ω) duality.
Estimate (ii) implies estimate (iii) via the substitutions f ∶= sup Q a Q 1 Q and g ∶= sup R b R 1 R , and, conversely, estimate (iii) implies estimate (ii) via the substitutions a Q ∶= ⟨f ⟩ σ Q and b R ∶= ⟨g⟩ ω R together with the Hardy-Littlewood maximal inequality. Estimate (iii) implies estimate (iv) via the substitutions a Q ∶= ∑ S⊇QãS and b R ∶= ∑ S⊇RbR . We next check that, conversely, estimate (iv) implies estimateestimate (iii) via the substitutions whereQ andR denote the dyadic parents of the cubes Q and R.
By the monotone converge theorem, we may assume without loss of generality that the the families a and b are supported on finitely many cubes. Now, in the expression appearing on the right-hand side of estimate (iv), by a telescoping summation, we have and, in the expression appearing on the left-hand side of estimate (iv), again by a telescoping summation, we have Combining the inequalities (3.2) and (3.3) for the family a and the same inequalities for the family b with estimate (iv) yields estimate (iii). The proof is complete.
Similarly, using the same substitutions as in the proof of Lemma 3.5, we obtain the following reformulations of the two-weight norm inequality for maximal operators: Lemma 3.6 (Reformulations for maximal operators). Let q ∈ (0, ∞) and p ∈ (1, ∞). Then the following estimates are equivalent: for all functions f .
for all families a. (iii) We have for all familiesã.

Characterization of multipliers between Carleson coefficients.
We characterize the two-weight norm inequality for multipliers of Carleson coefficients. In addition to being interesting in its own right, this characterization is applied to characterize the two-weight norm inequality for summation operators under the A ∞ assumption (see Proposition 2.2). (i) We have Furthermore, the constants in the assertions are comparable.
Proof. The equivalence of assertions (ii) and (iii) follows from the duality in the discrete Littlewood-Paley spaces by using Proposition 3.1. The equivalence of assertions (i) and (ii) can be checked using essentially a standard proof of Sawyer's two-weight norm inequality for maximal operators. For the reader's convenience, we write out the proof. Assertion (ii) implies assertion (i) by substituting the family {a R } with a R = 1 when R ⊆ Q and a R = 0 when R ⊈ Q. Assertion (i) implies assertion (ii) as follows. Let a = {a Q } be a family of non-negative reals. By the monotone convergence theorem, we may assume without loss of generality that the family a is supported on finitely many cubes. We linearize the supremum (which now is a maximum) by writing max for the pairwise disjoint sets E(Q), which can be defined, for example, as fol- . By using this linearization, we have and hence we need to prove the estimate By the dual estimate for the Carleson coefficients (see the remark after Proposition

3.1), this estimate holds if and only if the Carleson condition
holds. This condition holds because, by the assumption, we have The proof of the equivalence of assertions (i) and (ii) is complete.
We recall that the dyadic Fujii-Wilson A ∞ characteristic [σ] A∞(ω) (of a measure σ with respect to a measure ω) is defined by Accordingly, the measure σ is said to satisfy the A ∞ condition with respect to the measure ω if Applying Proposition 3.7 to the family µ ∶= { σ(Q) ω(Q) } of multipliers, we record the following corollary: Assume that the measure µ has no point masses. Under this assumption, the coefficients {b Q } are µ-Carleson, which means (in our normalization) that if and only if they are µ-sparse, which means that there exist pairwise disjoint sets for all dyadic cubes Q.
This equivalence was originally proven by Verbitsky [

4.1.
Scale of conditions for maximal operators. We recall that, for γ ∈ (0, ∞), the auxiliary quantity Λ sup γ,Q is defined by In this section, we prove the following result: (ii) (Necessary condition) Let γ ∈ (0, q). Then we have First, we prove two lemmas, which combined yield the necessary condition. Let p, q ∈ (0, ∞), and γ ∈ (0, q). Then the following assertions are equivalent: (i) We have for every family a.
(ii) We have γ , assertion (ii) implies assertion (i) trivially. We next prove the converse. We substitute the monotonous rearrangement a Q ∶= sup R⊇Q a R into estimate (4.1). Under this substitution, the right-hand side of the estimate remains unchanged, and, by interchanging the order of the suprema, the left-hand side of the estimate becomes By the scaling of the L p norms, and by the Hardy-Littlewood maximal inequality, we estimate this from below as The proof is complete.

Lemma 4.3 (Necessary condition for maximal operators).
We have Proof. By the duality in the Littlewood-Paley spaces, the two-weight norm inequality for maximal operators is equivalent to the estimate By the comparison of the ℓ p norms, we have By the scaling of the discrete Littlewood-Paley norms, and by renamingb ∶= b . Therefore, altogether, we have .
The following duality formula obtained by Verbitsky [32] holds: For every measure µ and every s ∈ (0, 1], we have (For s = 1, this is the usual duality in the discrete Littlewood-Paley spaces.) Applying this formula completes the proof.
Proof of Proposition 4.1. We observe that the necessary condition follows by combining Lemma 4.2 and Lemma 4.3. We next prove the sufficient condition. By the duality in the Littlewood-Paley spaces, the two-weight norm inequality (2.1) for the maximal operator is equivalent to the estimate By the using an equivalent expression (Lemma 3.4), we have By using twice the dual estimate The proof is complete.

Characterization for summation operators under the A ∞ assumption.
Let p ∈ (1, ∞), q ∈ (0, ∞), and q < p. We recall that the integral expression I σ,ω,p,q,λ is defined by In this section, we prove the following result: Proposition 4.4 (Characterization under the A ∞ assumption). Let σ and ω be measures that satisfy the A ∞ condition with respect to each other. Let p ∈ (1, ∞) and q ∈ (0, ∞) be such that q < p. Then we have the following characterization by subranges: • In the subrange q ∈ (0, 1], we have  By the duality (f p q , 1 1−q (σ)) * = f p p−q , 1 q (σ) in the discrete Littlewood-Paley spaces, estimate (4.9) is equivalent to the estimate Next, we consider the range 1 < q < p < ∞. We write the proof only for the estimate as the reverse estimate and the dual estimates (with q ′ and p, ω(Q) σ(Q) λ Q and λ Q , and ω and σ interchanged) can be proven similarly.
The two-weight norm inequality (2.5) is equivalent (see Lemma 3.5 for the proof of this) to the bilinear estimate By the scaling of the Littlewood-Paley norms, and by the A ∞ assumption together with Proposition 3.7, we have  We define the exponent r ∈ (1, ∞) by setting 1 r ′ ∶= 1 p + 1 q ′ . Thus r = pq p−q . By the factorization f p,∞ (σ) ⋅ f q ′ ,∞ (σ) = f r ′ ,∞ (σ), the following assertions hold: • For every a and b, ab f r ′ ,∞ (σ) ≲ p,q a f p,∞ (σ) b f q ′ ,∞ (σ) .
By these assertions, estimate (4.12) is equivalent to the estimate .
By the f r ′ ,∞ (σ) − f r,1 (σ) duality, this estimate is equivalent to the estimate The proof is complete.

Inequality for summation operators via maximal operators.
We recall that the auxiliary quantity Λ Q = Λ sum Q is defined by In this section, we prove the following result: Proposition 4.5 (Characterization for summation operators in terms of maximal operators). Let 1 < q < p < ∞. Then . Proof. By Lemma 3.5, the two-weight norm inequality P λ P ⟨f ⟩ σ P 1 P L q (ω) ≲ p,q C f L p (σ) is equivalent to the estimate (4.13) Since Q ∩ R ⊇ P , by dyadic nestedness, we have either R ⊆ Q or Q ⊆ R. Hence, the summation splits into the cases P ⊆ R ⊆ Q and P ⊆ Q ⊆ R. Therefore, estimate (4.13) is equivalent to the pair of estimates We handle only subestimate (4.14a), as the other subestimate (4.14b) can be handled similarly. By the f q ′ ,1 (ω) − f q,∞ (ω) duality in the discrete Littlewood-Paley spaces, subestimate (4.14a) is equivalent to the estimate sup R Q⊇R a R 1 ω(R) P ⊆R λ P ω(P ) L q (ω) ≲ q Q a Q 1 Q L p (σ) .
By Lemma 3.6, this estimate is equivalent to the two-weight norm inequality sup Q Λ Q ⟨f ⟩ σ Q 1 Q L q (ω) ≲ p,q f L p (σ) .
The proof is complete.