Weighted Riesz Transform Inequalities for General Laguerre Expansions

Definitions and mathematical statement

For an integer $n\geq 1$, write ${\mathbb{R}}_+^n=(0,\infty)^n$ and let $m=(m_1,\ldots,m_n)$ denote the type parameter of the intended Laguerre expansion. Set \[ d\mu_m(r)=\prod_{k=1}^n r_k^{2m_k+1}\,{\,\mathrm{d}} r_k, \qquad w_{m,p}(r)=\prod_{k=1}^n r_k^{m_k(p-2)}. \] The symbol $R_{j,m}$, for $1\leq j\leq n$, denotes the Riesz transform associated with that expansion in the supplied statement. The record does not give its operator definition, normalization, or initial domain, and does not specify the numerical range of the coordinates of an admissible arbitrary type $m$.

Conjecture as supplied, with unresolved scope. For every $p\in(1,\infty)$, does there exist a finite constant $C_p>0$, depending only on $p$, such that for every $n\geq1$, every admissible type $m$ in the intended arbitrary-type Laguerre family, every $j\in\{1,\ldots,n\}$, and every function $f$ in the intended operator domain with finite right-hand side, \[ \int_{{\mathbb{R}}_+^n}|R_{j,m}f(r)|^p w_{m,p}(r)\,{\,\mathrm{d}}\mu_m(r) \leq C_p \int_{{\mathbb{R}}_+^n}|f(r)|^p w_{m,p}(r)\,{\,\mathrm{d}}\mu_m(r)? \] In particular, the constant is required to be independent of $n$ and $m$. The supplied record leaves the admissible type set and function domain unspecified; these must be recovered from the cited formulation before this becomes a fully specified operator inequality. No boundary conditions or operator realization at the coordinate hyperplanes are supplied.

Short English statement

The conjecture asks whether each Laguerre Riesz transform satisfies the displayed weighted $L^p$ inequality for arbitrary Laguerre type, with a constant depending only on $p$, uniformly in dimension and type. The type range and precise operator domain remain unresolved. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001855$, source problem code catalogue entry, titled Weighted Riesz Transform Inequalities for General Laguerre Expansions; collection: Oberwolfach Reports Open Problems.
• Supplied source citation: Real Analysis, Harmonic Analysis and Applications (2012), Oberwolfach Reports, pp. 2045--2111, DOI 10.4171/owr/2011/36. https://doi.org/10.4171/owr/2011/36. The record gives proposed year 2011 and no named proposer. Its background identifies Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied reference for the arbitrary-type Laguerre expansions: Nowak--Stempak, DOI 10.1016/j.aim.2007.04.010. https://doi.org/10.1016/j.aim.2007.04.010. No title or further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the Oberwolfach report explicitly presents the arbitrary-type assertion as a conjecture, and states that no later proof of the full assertion was found. Its listed sources checked are the two DOI links above. This is supplied dated curation, not independently verified literature review; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the measure, weighted inequality, parameter range, dimension-free constant, and arbitrary-type extension. That verification is a supplied source assertion; it has not been independently repeated here.