RETRACTED: A priori estimates for orthogonal polynomials

Andrés Pereira

doi:10.58301/hv36hg37

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Meridian Journal of Mathematical Sciences

Retraction6 November 2026

The editors have retracted this article because the reported findings could not be reproduced and the underlying data were not available for inspection. The corresponding author did not agree to the retraction.

Expression of Concern6 November 2026

The editors wish to alert readers that concerns have been raised about the reliability of the data presented in Figure 3. The editors have contacted the authors’ institution and are awaiting the outcome of its inquiry. This notice will be updated when further information is available.

ArticleRetracted16 May 2023

RETRACTED: A priori estimates for orthogonal polynomials

Andrés Pereira^1* (unauthenticated)

¹Department of Biomedical Sciences, Pacific Rim University, Australia

* Corresponding author.

Meridian J. Math. Sci. 2023, 9(2), 290-294; https://doi.org/10.58301/hv36hg37

Abstract

Despite considerable progress in recent years, several fundamental questions about harmonic maps remain open. In contrast to earlier studies, our approach accounts for heterogeneity and avoids the usual simplifying assumptions. We prove an optimal convergence rate for the proposed scheme. Together, these results contribute to a more complete picture and open the way to further applications.

Keywords:stabilityspectral theoryasymptotic analysisfinite element methodorthogonal polynomials

Introduction

Let $Ω \subset R^{d}$ be a bounded domain with Lipschitz boundary and let $p \in (1, \infty)$ . We consider the boundary value problem1

\begin{aligned} - div (| \nabla u |^{p - 2} \nabla u) & = f in Ω, \\ u & = 0 on \partial Ω . \end{aligned}

(1)

Existence of weak solutions in $W_{0}^{1, p} (Ω)$ follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.

Main result

Theorem 1. Suppose $f \in L^{q} (Ω)$ with $q > d / p$ . Then there is a constant $C = C (d, p, q)$ such that

‖ u ‖_{L^{\infty} (Ω)} \leq C | Ω |^{\frac{1}{d} - \frac{1}{p} + \frac{1}{q}} ‖ f ‖_{L^{q} (Ω)}^{\frac{1}{p - 1}} .

(2)

The proof combines a truncation argument with the Sobolev inequality $‖ v ‖_{p^{*}} \leq S ‖ \nabla v ‖_{p}$ . A second estimate, for the gradient, is given by

\int_{Ω} | \nabla u |^{p} d x; \leq; \sum_{k = 1}^{\infty} 2^{- k} λ_{k}^{- 1} ‖ f ‖_{q}^{p^{'}}, p^{'} = \frac{p}{p - 1} .

(3)

Numerical illustration

Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.

Table 1 Computed values of the constant $C (d, p, q)$

d	p	q	C (computed)	C (bound)
2	2	2	0.081	0.125
2	3	4	0.147	0.210
3	2	3	0.052	0.083
3	4	5	0.209	0.341

**Fig. 1** Error versus mesh size on a logarithmic scale; the dashed line has slope 2.

Discussion

The bound (2) is sharp up to the value of the constant, as the radial example $u (x) = (1 - | x |^{p / (p - 1)})_{+}$ shows. Extensions to systems and to variable exponents will be treated elsewhere.

Here the divergence and gradient are taken in the sense of distributions.

References

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Declarations

Funding

The authors acknowledge support from the Wellcome Trust (grant 215801/Z/19/Z) and the National Institute for Health and Care Research.

Competing Interests

None.

Data Availability

All data supporting the findings are available in the supplementary material and in a public repository.

Copyright and License

License: https://creativecommons.org/publicdomain/zero/1.0/

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