Properties and applications of Lorentz--Muckenhoupt classes

Abstract: In this paper, through the introduction of Lorentz--Muckenhoupt classes, we systematically investigate the boundedness of maximal operators on multiplier weighted Lorentz spaces. As applications, we give the characterization of the commutators of fractional integrals, which yields a partial answer to an open question proposed by D. Cruz-Uribe. Second, Hardy inequalities in Lorentz spaces are established, with the critical case $p=d$, in which the classical Hardy inequality fails. Finally, we apply the Lorentz estimates to fractional Schrödinger equations with singular potentials.

• The paper introduces Lorentz–Muckenhoupt classes, establishes duality, interpolation, monotonicity, and power-weight criteria, and identifies them as a natural framework for multiplier-weighted Lorentz spaces.
• The paper characterizes fractional maximal-operator bounds in several Lorentz-index ranges, proves the sharp result for r=p and q≤s≤∞, and identifies unresolved endpoint cases and quantitative exponent gaps.
• The paper applies the theory to BMO characterizations of fractional commutators, critical weak-Lorentz Hardy inequalities, and mild solutions for fractional Schrödinger equations with small critical Hardy potentials.

Overview and motivation

This paper by Wang and Zhang introduces the Lorentz–Muckenhoupt weight classes $A_{(p,r),(q,s)}$, defined for $1<p\le q<\infty$ and $1\le r,s\le\infty$ by

$$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$

where the norms are normalized on cubes. These classes govern the boundedness of the fractional maximal operator $M_\alpha$ between multiplier-weighted Lorentz spaces $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$, a setting distinct from the classical measure-weighted Lorentz spaces $L^{p,r}(W\,dx)$ studied, e.g., by Chung, Hunt, and Kurtz. The paper demonstrates that the two constructions genuinely differ when $r\ne p$: explicit examples show neither space contains the other in general. The associate-space identity $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$ and real interpolation identity $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$ are recorded as basic tools.

The motivation is twofold: multiplier weak-type estimates have recently reappeared in connection with matrix weights and sharp quantitative bounds (Sweeting), and their endpoint classes contain critical multipliers such as $1<p\le q<\infty$0 that lie outside the classical $1<p\le q<\infty$1 and $1<p\le q<\infty$2 classes. This forces the development of a genuinely Lorentz-based weight theory rather than an extrapolation from Lebesgue-scale results.

Structural properties of the classes

The paper first develops a two-weight class $1<p\le q<\infty$3 and proves that the norm of the fractional averaging operator $1<p\le q<\infty$4 equals, up to constants, the two-weight characteristic; this yields necessity of the one-weight condition for any boundedness of $1<p\le q<\infty$5 or $1<p\le q<\infty$6. Several structural results follow:

• Duality: $1<p\le q<\infty$7.
• Monotonicity in both secondary indices.
• Interpolation: if $1<p\le q<\infty$8, then $1<p\le q<\infty$9 with the expected quasi-norm bound.
• Comparison with $1\le r,s\le\infty$0: always $1\le r,s\le\infty$1, and equality holds precisely when $1\le r,s\le\infty$2 and $1\le r,s\le\infty$3.

A complete classification of power weights is obtained: $1\le r,s\le\infty$4 if and only if $1\le r,s\le\infty$5 (or $1\le r,s\le\infty$6 with $1\le r,s\le\infty$7) and $1\le r,s\le\infty$8 (or $1\le r,s\le\infty$9 with $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$0). Consequently, $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$1 — the weak secondary index strictly enlarges the admissible weights to include critical powers whose $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$2-th power is not even locally integrable.

Boundedness of maximal operators: a case analysis

The central contribution is a detailed map of when $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$3 holds, with $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$4. The results are sharply range-dependent:

Range	Conclusion
$$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$5	fails even for $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$6
$$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$7	iff $$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$8
$$[w]_{A_{(p,r),(q,s)}}:=\sup_Q \|w\|_{L^{q,s}(Q)}\,\|w^{-1}\|_{L^{p',r'}(Q)},$$9, $M_\alpha$0	iff $M_\alpha$1
$M_\alpha$2, $M_\alpha$3	$M_\alpha$4 necessary but not sufficient
$M_\alpha$5, $M_\alpha$6; $M_\alpha$7; $M_\alpha$8, $M_\alpha$9	characterization open

The main new positive result is the case $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$0, $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$1: writing $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$2, one has

$L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$3

with quantitative bound $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$4. The proof combines a disjoint-support lemma for Lorentz norms, a reverse-measure inequality for $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$5 derived from Sweeting's multiplier-class estimates, and a sparse-domination decomposition of the dyadic maximal operator. A mixed estimate retaining the reverse-measure constant $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$6 is also established.

On the quantitative side, a power-weight example shows any uniform exponent must satisfy $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$7, while the theorem gives $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$8. At $L_w^{p,r}=\{f: fw\in L^{p,r}(\mathbb R^d)\}$9 these coincide and reproduce the sharp Lacey–Moen–Pérez–Torres exponent $L^{p,r}(W\,dx)$0; the gap for $L^{p,r}(W\,dx)$1 is left as an open problem, connected to recent work of Lerner–Li–Ombrosi–Rivera-Ríos on sharp Muckenhoupt–Wheeden inequalities. At the endpoint $L^{p,r}(W\,dx)$2, the sufficiency of $L^{p,r}(W\,dx)$3 reduces to the classical Muckenhoupt–Wheeden problem, which remains unresolved.

Commutators and a partial answer to Cruz-Uribe's question

The first application concerns fractional commutators $L^{p,r}(W\,dx)$4. The paper proves a one-weight Lorentz characterization: for $L^{p,r}(W\,dx)$5 with $L^{p,r}(W\,dx)$6,

$L^{p,r}(W\,dx)$7

Necessity uses a Fourier-series argument on carefully placed cube pairs together with the weighted test-product estimate; sufficiency follows from the weighted Lebesgue commutator bounds plus off-diagonal extrapolation in Lorentz form. Notably, taking $L^{p,r}(W\,dx)$8 gives $L^{p,r}(W\,dx)$9 iff $r\ne p$0, where the target is the sharp Sobolev–Lorentz scale associated with the Riesz potential — strictly finer than the Lebesgue target $r\ne p$1.

Combining this with a Köthe-duality identity ($r\ne p$2 for $r\ne p$3) yields a partial answer to an open question posed by D. Cruz-Uribe in 2017: if there exist $r\ne p$4 and $r\ne p$5 such that

$r\ne p$6

for all nonnegative $r\ne p$7, then $r\ne p$8 with $r\ne p$9. The authors emphasize this requires only a single fixed $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$0 (with $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$1 ranging over $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$2), in contrast to extrapolation-based approaches requiring all admissible pairs. The result covers the difficult regime $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$3; the full two-weight question remains only partially answered.

Critical Hardy inequalities in Lorentz spaces

A Lorentz-refined Stein–Weiss theorem is proved by interpolating classical off-diagonal estimates at nearby exponents: under $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$4, $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$5, $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$6, and $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$7,

$(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$8

This yields the endpoint replacement for the Hardy inequality, which fails at $(L_w^{a,b})'=L_{w^{-1}}^{a',b'}$9 in the classical form:

$(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$0

Two sharpness statements are proved. First, the right-hand side cannot be replaced by $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$1 for any $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$2 — demonstrated via logarithmic radial test functions $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$3 normalized so that $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$4 while $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$5. Second, for the weighted critical inequality $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$6, the power balance $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$7 is shown to be necessary by a dilation argument. These are genuine endpoint results: the critical case is captured exactly at the weak Lorentz level and fails one degree of integrability away.

Fractional Schrödinger equations with singular potentials

The Stein–Weiss estimates are applied to

$(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$8

Writing $(L_w^{a_0},L_w^{a_1})_{\theta,b}=L_w^{a,b}$9, $1<p\le q<\infty$00, and $1<p\le q<\infty$01, the paper proves that if $1<p\le q<\infty$02 with sufficiently small norm, then for every source with $1<p\le q<\infty$03 there exists a unique mild solution $1<p\le q<\infty$04 satisfying

$1<p\le q<\infty$05

The proof is a Banach fixed-point argument in which the smallness of $1<p\le q<\infty$06 makes the perturbation term contractive. As a corollary, the critical Hardy potential $1<p\le q<\infty$07 is admitted for $1<p\le q<\infty$08 below an explicit smallness threshold, since $1<p\le q<\infty$09 exactly matches the required class. The restriction to small potentials is inherent to the perturbative method; no global-in-$1<p\le q<\infty$10 statement is claimed.

Limitations and open questions

Several gaps are stated explicitly. For $1<p\le q<\infty$11, complete characterizations of $1<p\le q<\infty$12 remain open in three ranges: $1<p\le q<\infty$13 with $1<p\le q<\infty$14 (where $1<p\le q<\infty$15 is necessary but only $1<p\le q<\infty$16 is known sufficient, and the inclusion is strict); $1<p\le q<\infty$17 with $1<p\le q<\infty$18; and $1<p\le q<\infty$19 with $1<p\le q<\infty$20. The optimal power $1<p\le q<\infty$21 in the quantitative bound lies in $1<p\le q<\infty$22 but is pinned down only at $1<p\le q<\infty$23. At $1<p\le q<\infty$24, sufficiency of $1<p\le q<\infty$25 contains the classical Muckenhoupt–Wheeden problem as a special case. The commutator result answers Cruz-Uribe's question only partially, since it assumes a one-weight condition with a specific Lorentz-dual structure rather than arbitrary pairs $1<p\le q<\infty$26. Finally, the Schrödinger existence theorem requires $1<p\le q<\infty$27 to be small, leaving the large-potential regime untreated.

Conclusion

The paper establishes Lorentz–Muckenhoupt classes as the natural one-weight framework for maximal operators on multiplier-weighted Lorentz spaces, proving duality, monotonicity, interpolation, a full power-weight classification, and a nearly complete boundedness theory whose principal positive novelty is the case $1<p\le q<\infty$28, $1<p\le q<\infty$29. The applications — a Lorentz-level BMO characterization of fractional commutators giving a partial answer to Cruz-Uribe's question, exact critical Hardy inequalities at $1<p\le q<\infty$30 with proven failure at every $1<p\le q<\infty$31, and mild solvability of fractional Schrödinger equations with critical Hardy potentials — demonstrate that the refined weight classes carry analytic content beyond the classical $1<p\le q<\infty$32 theory.