Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.08192

An interpretation of the BMO boundedness of Hörmander--Mikhlin multipliers

Tao Mei
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a pointwise domination of Poisson variances for Hörmander--Mikhlin multipliers on the real line. The estimate holds in operator order for operator-valued functions and gives a direct Fourier-side proof of BMO boundedness. The same argument gives an algebraic proof of the one-dimensional Schur multiplier estimate of Conde-Alonso, González-Pérez, Parcet and Tablate ( Annals of Mathematics, 2023), formulated here using Poisson BMO. The endpoint estimates are proved for $L^2$ functions and Hilbert--Schmidt operators; interpolation gives the corresponding $L^p$ and Schatten estimates.

Explore related subjects

math.CAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tao Mei. 2026-11-05. An interpretation of the BMO boundedness of Hörmander--Mikhlin multipliers. https://arxiv.org/abs/2611.08192

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Factorized $q$-difference equations and their fundamental solutions

We construct fundamental solutions of a class of factorized $q$-difference equations in terms of Jacobi theta functions and Zwegers' $μ$-functions. To obtain a formal series solution $f_0$ for a factorized $q$-difference equation, we present a $q$-analogue of the fundamental relation (often called the Wronski-type relation) between the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. We also provide some formulas for the image of the $q$-Borel summation of the formal series solution $f_0$. As an application, we give another proof of the expressions due to S. Y. Kang and S. P. Zwegers of the universal mock theta functions and higher level Appell functions in terms of $μ$-functions.

math.CA↗
arXiv2026

Multilinear extrapolation on Banach function spaces

We solve an open problem on the multilinear Rubio de Francia extrapolation on Banach function spaces. The main result states that, under the assumption of the boundedness of the multisublinear Hardy-Littlewood maximal operator, the multivariable boundedness on Banach function spaces can be extrapolated from some weighted Lebesgue space, including the cases where some exponents may be infinite. Based on this result, we also establish multilinear extrapolation of boundedness on rearrangement invariant Banach function spaces, and multilinear extrapolation of compactness on general Banach function spaces. Beyond that, we give characterizations of the boundedness of the multisublinear Hardy-Littlewood maximal operator on both general Banach function spaces and certain specific function spaces. All results do not require any convexity or concavity conditions on Banach function spaces. The main ingredients in the proof are a multilinear version of the Rubio de Francia algorithm, a stopping-time argument, convex analysis, and a compactness criterion on Banach function spaces. As applications of extrapolation theorems, we obtain multilinear extrapolation for the boundedness and compactness of commutators on Banach function spaces. These results can be further applied to specific operators, which cover both the full range case including multilinear Calderón-Zygmund operators, multilinear pseudo-differential operators, and bilinear paraproducts, and the limited range case including the bilinear Hilbert transform, multilinear rough singular integrals, and bilinear oscillatory Fourier multipliers.

math.CA↗
arXiv2026

Another Application of the Alternating Map: Time-and-Band Limiting

Castillo's dual equations for the alternating map reduce polynomial time-and-band limiting to two scalar problems. We determine the concentration multiplicities at a single cutoff, with sharp finite support and moment requirements. When the scalar problems admit irreducible tridiagonal commutants, we classify every real symmetric pentadiagonal commutant of the simple concentration matrix and construct one with simple spectrum. Its half-bandwidth is minimal for $N\ge3$ in polynomial degree order. Laguerre, Jacobi and Meixner input give applications; the constructions of Castro and Grünbaum follow by specialization. We give spectral-gap and rounding bounds, and certify small concentration values and deficits for noninteger Jacobi parameters.

math.CA↗