Open research questions in Advanced Harmonic Analysis Research

What the literature leaves open

• The lack of explicit global formulas for the kernel of the wave operator on compact manifolds. The need for precise kernel estimates using the Hadamard parametrix.

  Hardy Estimates about Wave Operator on Compact Manifolds · 2026 · DOI

• The paper notes that the study of commutators with variable exponents is challenging due to the complexity of the operators. The paper identifies the need to develop new techniques to prove the boundedness of the operators.

  Boundedness of Multilinear Commutators on Herz Spaces with Variable Exponents · 2026 · DOI

• There is a gap in the literature regarding the characterization of the minimizer of nonlocal and anisotropic Riesz energies. The paper fills this gap by providing a characterization of the minimizer in any space dimension.

  Nonlocal Anisotropic Riesz Interactions with a Physical Confinement · 2026 · DOI

• Prior work has focused on specific cases, but a general framework is needed. The heat kernel upper bounds are not well understood for general metric measure spaces.

  Spectral Bounds and Heat Kernel Upper Estimates for Dirichlet Forms · 2026 · DOI

• Future research can apply the authors' techniques to other problems in harmonic analysis. The paper suggests that further study of the weighted theory for Calderón-Zygmund operators is needed. The authors' results can be used to study other types of operators and their applications.

  Necessary Conditions for Weighted Estimates of Multilinear Multipliers and Pseudo-Differential Operators · 2026 · DOI

• The paper identifies a gap in the literature regarding the sharp necessary conditions for weighted estimates. The authors note that the prior work of Li and Sun, Kurtz and Wheeden, and Chanillo and Torchinsky does not provide the sharp necessary conditions.

  Necessary Conditions for Weighted Estimates of Multilinear Multipliers and Pseudo-Differential Operators · 2026 · DOI

• Future research can focus on extending the results to unbounded domains. The paper suggests studying the properties of the optimal domains in more detail. Future research can explore the applications of the paper's findings to specific shape optimization problems.

  Optimal Domains for the Cheeger Inequality · 2026 · DOI

• The paper identifies a gap in the understanding of the relationship between spectral and geometric properties of domains. The paper addresses the lack of characterization of the maximal and minimal values of F p,q.

  Optimal Domains for the Cheeger Inequality · 2026 · DOI

• Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state $\varrho(4,-\tfrac12)$ is two-copy distillable, where $\varrho(d,α)=(I+αF)/(d^2+αd)$.

  On the two-copy distillability of Werner states and a new partial trace inequality · 2026

• We show that no summation process satisfying Brezis's natural axioms can recover the degree universally for $α=1/3$, thereby resolving an open problem by Brezis.

  The Winding Number at the Critical Hölder Exponent 1/3: Failure of Universal Fourier Summation · 2026

• In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater's condition), a setting in which existing results are limited.

  Radial-type error bounds for semidefinite feasibility problems without strict feasibility: qualitative estimates and asymptotic tightness · 2026

• Further study of the critical exponent p_c = 2(n-1)/(n-2). Extension of the results to higher dimensions. Application of the new method to related problems.

  Maximal cone multipliers: sharp $$L^p$$ estimates in $$\mathbb {R}^3$$ and beyond · 2026 · DOI

• The existing literature lacks sharp L^p bounds for the maximal cone multiplier operator. The critical exponent p_c = 2(n-1)/(!!!!!!!!!!!!!!!!!!n-2) remains an open case.

  Maximal cone multipliers: sharp $$L^p$$ estimates in $$\mathbb {R}^3$$ and beyond · 2026 · DOI

• The paper does not provide a comprehensive survey of prior work on the topic. The results are limited to n ≥ 2. The paper does not discuss potential applications of the results.

  Spherical Maximal Functions and Hardy Spaces for Fourier Integral Operators · 2026 · DOI

• The lack of sharp bounds for spherical maximal functions and Hardy spaces for Fourier integral operators. The need for improved bounds for these functions to study wave equations.

  Spherical Maximal Functions and Hardy Spaces for Fourier Integral Operators · 2026 · DOI

• The existing generalizations of Bôcher's theorem are limited to specific types of operators. There is a need for a new approach to the generalization of Bôcher's theorem that applies to a wider class of operators.

  Bôcher type theorem for elliptic equations with drift-perturbed Lévy operators · 2026 · DOI

• There is a gap in the existing literature, as previous studies have focused on specific types of QVIs. There is a need for a unified analysis for a class of coercive elliptic quasi-variational inequalities.

  Subsolutions, Discrete Regularity and \(L^\infty\)- Error Estimates for a Class of Elliptic QVIs: A Unified Analysis · 2026 · DOI

• The study of Strichartz-type estimates for the wave equation on compact manifolds. The development of kernel estimates for other oscillatory integral operators on manifolds.

  Hardy Estimates about Wave Operator on Compact Manifolds · 2026 · DOI

• The paper identifies the gap in the current understanding of the pointwise convergence of the Boussinesq operator. The paper identifies the need for a new understanding of the pointwise convergence of the Boussinesq operator in higher dimensions.

  A Characterization of Boussinesq Maximal Estimates in Higher Dimensions · 2026 · DOI

• The paper identifies a gap in the literature on commutators with variable exponents. The paper notes that previous work has focused on commutators with constant exponents.

  Boundedness of Multilinear Commutators on Herz Spaces with Variable Exponents · 2026 · DOI

• The characterization of WCMO~q,λ(R^n) was not obtained via the boundedness of commutators of the fractional Hardy operator and its dual operator on weak mixed Lebesgue spaces. The boundedness of commutators of Hardy type operators was not derived on mixed λ-central Morrey space B~q,λ(R^n).

  Some Characterizations of the Mixed CMO Type Spaces via Commutators of Hardy Type Operators · 2026 · DOI

• The lack of mixed radial-angular estimates for Hausdorff type operators. The need to establish boundedness and other properties of Hausdorff operators on various function spaces.

  Mixed Radial-Angular Estimates for Some Hausdorff Type Operators · 2026 · DOI

• Future research could apply the results to problems in engineering and the sciences. Future research could extend the theory to more general classes of partial differential equations. Future research could develop new numerical methods based on the weighted summability results.

  Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces · 2026 · DOI

• Prior work has focused on compact parameter domains, but the paper identifies a gap in the consideration of unbounded parameter domains. The paper identifies a gap in the application of the theory of holomorphic functions between Hilbert spaces to the study of sparsity of Wiener polynomial chaos expansions.

  Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces · 2026 · DOI

• To further study the mass concentration of x → EΣf(x) near lines in R^2. To characterize the pairs (p, q) for which the estimates ∥EΣf ∥Lq(Ω) ≤ C∥f ∥Lp(Σ) and (R(|EΣf|q))q ≤ C∥f ∥Lp(Σ) hold in more general cases.

  Fourier Extension Estimates on a Strip in $$\mathbb {R}^2$$ · 2026 · DOI

Most-cited papers in Advanced Harmonic Analysis Research

• Compactness of Riemann–Liouville fractional integral operators · Electronic journal of qualitative theory of differential equations · 2020 · 13 citations
• Multiplicity theorems involving functions with non-convex range · Studia Universitatis Babes-Bolyai Matematica · 2023 · 5 citations
• Polynomial estimates for solutions of parametric elliptic equations on complete manifolds · Studia Universitatis Babes-Bolyai Matematica · 2022 · 5 citations
• Generalized $c$-almost periodic type functions in ${\mathbb{R}}^{n}$ · Archivum Mathematicum · 2021 · 3 citations
• Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations · Studia Universitatis Babes-Bolyai Matematica · 2022 · 3 citations
• A class of functionals possessing multiple global minima · Studia Universitatis Babes-Bolyai Matematica · 2021 · 3 citations
• Sobolev inequality with non-uniformly degenerating gradient · Electronic journal of qualitative theory of differential equations · 2022 · 3 citations
• Discrete iterated Hardy-type inequalities with three weights · Journal of Mathematics Mechanics and Computer Science · 2020 · 3 citations
• Korn’s Inequality and Eigenproblems for the Lamé Operator · Computational Methods in Applied Mathematics · 2022 · 3 citations
• On Weights Which Admit Harmonic Bergman Kernel and Minimal Solutions of Laplace’s Equation · Annales Mathematicae Silesianae · 2022 · 2 citations

Most recent work

• Some more sparse bounds for rough and smooth pseudodifferential operators · Proceedings of the Royal Society of Edinburgh: Section A Mathematics · 2026
• On extension of non-tangential maximal Calderón–Zygmund type singular integrals · manuscripta mathematica · 2026
• The Fourier restriction and Kakeya conjectures: some recent developments · Sugaku Expositions · 2026
• Maximal cone multipliers: sharp $$L^p$$ estimates in $$\mathbb {R}^3$$ and beyond · Mathematische Annalen · 2026
• Young type inequality and Hausdorff–Young type inequality for convolution operator associated with fractional Fourier transform of $$\alpha $$-order · Fractional Calculus and Applied Analysis · 2026
• TWO-WEIGHTED ESTIMATES FOR SOME SUBLINEAR OPERATORS ON GENERALIZED WEIGHTED MORREY SPACES AND APPLICATIONS · Journal of Mathematical Sciences · 2026
• Spherical Maximal Functions and Hardy Spaces for Fourier Integral Operators · The Journal of Geometric Analysis · 2026
• Bôcher type theorem for elliptic equations with drift-perturbed Lévy operators · Journal de Mathématiques Pures et Appliquées · 2026
• Subsolutions, Discrete Regularity and \(L^\infty\)- Error Estimates for a Class of Elliptic QVIs: A Unified Analysis · International Journal of Analysis and Applications · 2026
• Hardy type inequalities in weighted Herz-Morrey-Orlicz spaces · Analysis and Mathematical Physics · 2026

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