Open research questions in Advanced Harmonic Analysis Research

What the literature leaves open

• The gap is the lack of sparse bounds for rough pseudodifferential operators. The paper identifies a need for an alternative proof of known sparse bounds for pseudodifferential operators with symbols in S^0_{1,δ} for δ < 1.

  Some more sparse bounds for rough and smooth pseudodifferential operators · 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 need to establish boundedness properties for sublinear operators on generalized weighted Morrey spaces. The lack of understanding of regularity properties of solutions to partial differential equations.

  TWO-WEIGHTED ESTIMATES FOR SOME SUBLINEAR OPERATORS ON GENERALIZED WEIGHTED MORREY SPACES AND APPLICATIONS · 2026 · DOI

• The gap is the lack of local Calderón--Zygmund estimates in the singular case. The gap is the lack of gradient regularity results for the singular case.

  Gradient estimates for singular elliptic measure data problems with double phase · 2026

• 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

• 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 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

• 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

• The problem of determining all pairs of (p, q) ∈ [1, ∞]2 for which the estimate holds was not fully solved. The study of mass concentration of x → EΣf(x) near lines in R^2 was not fully understood.

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

• The need for a characterization of fine continuity in Newton-Sobolev spaces. The lack of a sufficient condition for fine continuity at a point in terms of the convergence of a Wiener-type integral involving its minimal p-weak upper gradient.

  A Wiener-Type Criterion for Fine Continuity in Newton-Sobolev Spaces · 2026 · DOI

• The symmetry assumption is crucial for the results to hold. The paper only considers domains with symmetry of order 4. The numerical observations are limited to specific examples.

  Bounds for higher Steklov and mixed Steklov Neumann eigenvalues on domains with holes · 2026 · DOI

• Further study of the properties of crystalline measures and Fourier quasicrystals is needed. Further study of the applications of the results is needed.

  Growth of masses of crystalline measures · 2026 · DOI

• The understanding of the properties of crystalline measures and Fourier quasicrystals is incomplete. The problem of determining the conditions for a crystalline measure to be a Fourier quasicrystal is not fully addressed.

  Growth of masses of crystalline measures · 2026 · DOI

• The paper suggests further study of the properties of symmetric spaces and the generalized Lorentz space. The paper proposes the investigation of the applications of the results to various fields.

  Estimates of the order of approximation of functions of several variables in the generalized Lorentz space · 2026 · DOI

• The paper addresses the need for new tools and results in the study of the Nikol'skii--Besov class.

  Estimates of the order of approximation of functions of several variables in the generalized Lorentz space · 2026 · DOI

• Future research can focus on extending the results to more general classes of variational problems. The authors suggest studying the regularity of minimizers for other types of problems. Further research can explore the applications of the paper's findings to other fields.

  Continuity estimates for variable growth variational problems in the Heisenberg group · 2026 · DOI

• To explore applications to other fields such as physics and engineering.

  TWO-WEIGHTED ESTIMATES FOR SOME SUBLINEAR OPERATORS ON GENERALIZED WEIGHTED MORREY SPACES AND APPLICATIONS · 2026 · DOI

• 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

• 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 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

• 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

• To extend the results to domains without symmetry of order 4. To apply the findings to various domains with holes. To further analyze the properties of the Dirichlet to Neumann operator and its eigenvalues.

  Bounds for higher Steklov and mixed Steklov Neumann eigenvalues on domains with holes · 2026 · DOI

Most-cited papers in Advanced Harmonic Analysis Research

• Fractional Sobolev spaces with variable exponents and fractional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math>-Laplacians · Electronic journal of qualitative theory of differential equations · 2017 · 124 citations
• Failures of the reduction principle in an Ellsberg-type problem · Theory and Decision · 1992 · 51 citations
• Lp-norm spherical distribution · Journal of Statistical Planning and Inference · 1997 · 48 citations
• Mollification in Strongly Lipschitz Domains with Application to Continuous and Discrete De Rham Complexes · Computational Methods in Applied Mathematics · 2015 · 47 citations
• Asymptotically optimal estimation of smooth functionals for interval censoring, part 1 · Statistica Neerlandica · 1996 · 21 citations
• Degree and Sobolev spaces · Topological Methods in Nonlinear Analysis · 1999 · 20 citations
• Functorality and heat content asymptotics for operators of Laplace type · Topological Methods in Nonlinear Analysis · 1993 · 20 citations
• Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator · Publications de l Institut Mathematique · 2012 · 19 citations
• Hardy-Sobolev inequalities with remainder terms · Topological Methods in Nonlinear Analysis · 2002 · 19 citations
• Homogeneous Herz spaces with variable exponents and regularity results · Electronic journal of qualitative theory of differential equations · 2018 · 19 citations

Most recent work

• Matrix-weighted Besov–Triebel–Lizorkin spaces of optimal scale: Real-variable characterizations, invariance on integrable index, and Sobolev-type embedding · Journal of Differential Equations · 2026
• Calderón-Zygmund estimates for double phase problems with matrix weights · Calculus of Variations and Partial Differential Equations · 2026
• Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane · Transactions of the American Mathematical Society · 2026
• A new variational approach for the numerical location of Hardy Z-function zeros on the real line · Journal of Computational and Applied Mathematics · 2026
• Carleson perturbations of locally Lipschitz elliptic operators · Proceedings of the American Mathematical Society · 2026
• Weak type (1,1) bounds for singular integrals with rough kernels on block spaces · Journal of Mathematical Analysis and Applications · 2026
• Sharp weak-type estimate for local lifted Hardy–Littlewood maximal operators with applications to generators of linear operator families and Hardy(–Sobolev) spaces · Advances in Mathematics · 2026
• Sharp estimates for the heat kernel on exterior planar domains · Journal of Differential Equations · 2026
• Matrix-weighted estimates beyond Calderón–Zygmund theory · Mathematische Annalen · 2026
• Two-ends Furstenberg estimates in the plane · Mathematische Annalen · 2026

Related topics in Mathematics