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Mahdi Hormozi

Publications and source records attributed to Mahdi Hormozi.

At least 19 recordsLinked to original sources
arXiv2026

Dimension-free estimates for full discrete maximal functions associated with Euclidean balls and spheres

We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.

math.CA↗
arXiv2026

Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions

For any $1 < p < q < \infty$, we investigate fixed-time hypercontractive bounds from $L^p$ to $L^q$ of Poisson semigroups associated with the Ornstein--Uhlenbeck, Laguerre and Jacobi operators. We prove that, in the Ornstein--Uhlenbeck and Laguerre cases, the Poisson semigroups fail to be $L^p \to L^q$ bounded for any fixed $t > 0$. In contrast, for Jacobi operators with $\alpha, \beta \ge -1/2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^\infty$. More generally, we study Bernstein subordinations of these semigroups and show that fixed-time hypercontractivity is not stable under subordination. The analysis relies on quantitative $L^q$-estimates for the corresponding orthogonal polynomial eigenfunctions, together with a bilinear test with the exponential family.

math.CA↗
arXiv2026

The Gundy-Stein decomposition with explicit constants

Let $(\mathcal F_n)_{n\ge 1}$ be a filtration and let $f\ge0$ belong to $L^1(\mathcal F_\infty)$. For the martingale $f_n=\mathbb E[f\mid \mathcal F_n]$ and each $\lambda>0$ we prove a Gundy--Stein decomposition \[ f=g+h+k \] with explicit numerical constants. In the positive closed case the three parts satisfy explicit bounds, and the bounded part is bounded above by $\lambda$. We also prove a one-parameter form for the bounded part and two-point sharpness results, including a joint sharpness statement for arbitrary decompositions under the condition $0\le k\le \lambda$. We also obtain an exact four-term refinement of the decomposition, separating the bounded term into a stopped part and a conditional expectation term. As applications we obtain an explicit weak-type $(1,1)$ estimate for truncated martingale multipliers and a John--Nirenberg inequality for martingale $\mathrm{BMO}$ on atomic $\alpha$-regular filtrations.

math.PR↗
arXiv2023

On the weak boundedness of multilinear Littlewood--Paley functions

In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author's paper \cite{SXY} significantly. In particular, the new proof for \cite[Theorem 1.1]{SXY} is quite short and, unlike the original proof, does not rely on the properties of the "Marcinkiewicz function". This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewood--Paley operators by well-known techniques. In other words, we relax the log-Dini condition in the pointwise bound to the classical Dini condition. %$\int\limits_{0}^{1} \frac{φ(t)}{t}dt<\infty$. This solves an open problem (see e.g. \cite[pp. 37--38]{CY}). Our method can be applied to the multilinear case.

math.CA↗
arXiv2020

The modulus of $p$-variation and its applications

In this note, we introduce the notion of modulus of $p$-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain $K$-functionals. To be more specific, let $ν$ be a nondecreasing concave sequence of positive real numbers and $1\leq p<\infty$. Using our new tool, we first define a Banach space, denoted $V_p[ν]$, that is intermediate between the Wiener class $BV_p$ and $L^\infty$, and prove that it satisfies a Helly-type selection principle. We also prove that the Peetre $K$-functional for the couple $(L^\infty,BV_p)$ can be expressed in terms of the modulus of $p$-variation. Next, we obtain equivalent sharp conditions for the uniform convergence of the Fourier series of all functions in each of the classes $V_p[ν]$ and $H^ω\cap V_p[ν]$, where $ω$ is a modulus of continuity and $H^ω$ denotes its associated Lipschitz class. Finally, we establish optimal embeddings into $V_p[ν]$ of various spaces of functions of generalized bounded variation. As a by-product of these latter results, we infer embedding results for certain symmetric sequence spaces.

math.FA↗
arXiv2020

Weak and strong type estimates for the multilinear Littlewood-Paley operators

Let $S_α$ be the multilinear square function defined on the cone with aperture $α\geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_α$. We first obtain a sharp weighted estimate in terms of aperture $α$ and $\vec{w} \in A_{\vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer's conjecture, for which a Coifman-Fefferman inequality with the precise $A_{\infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_λ$ function. Some results are new even in the linear case.

math.FA↗
arXiv2017

Relations between Schramm spaces and generalized Wiener classes

We give necessary and sufficient conditions for the embeddings $Λ\text{BV}^{(p)}\subseteq Γ\text{BV}^{(q_n\uparrow q)}$ and $Φ\text{BV}\subseteq\text{BV}^{(q_n\uparrow q)}$. As a consequence, a number of results in the literature, including a fundamental theorem of Perlman and Waterman, are simultaneously extended.

math.FA↗
arXiv2016

New bounds for bilinear Calderón-Zygmund operators and applications

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón--Zygmund operators and their commutators with $BMO$ functions, square functions and multilinear Fourier multipliers.

math.CA↗
arXiv2015

$A_p$-$A_\infty$ estimates for general multilinear sparse operators

In this paper, we study the $A_p$-$A_\infty$ estimates for a class of multilinear dyadic positive operators. As applications, the $A_p$-$A_\infty$ estimates for different operators e.g. multilinear square functions and multilinear Fourier multipliers can be deduced very easily.

math.CA↗
arXiv2015

Orthogonal bases of Brauer symmetry classes of tensors for groups having cyclic support on non-linear Brauer characters

This paper provides some properties of Brauer symmetry classes of tensors. We derive a dimension formula for the orbital subspaces in the Brauer symmetry classes of tensors corresponding to the irreducible Brauer characters of the groups having cyclic groups support on non-linear Brauer characters. Using the derived formula, we investigate the necessary and sufficient condition for the existence of the o-basis of Dicyclic groups, Semi-dihedral groups and also reinvestigate those things on Dihedral groups. Some criteria for the non-vanishing elements in the Brauer symmetry classes of tensors associated to those groups are also included.

math.GR↗
arXiv2015

On General multilinear square function with non-smooth kernels

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$ T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}. $$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given.

math.CA↗
arXiv2015

Weighted bounds for multilinear operators with non-smooth kernels

Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds for certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schrödinger operators on $\mathbb{R}^n$. Our results are new even in the linear case.

math.CA↗
arXiv2015

Weighted bounds for multilinear square functions

Let $\vec{P}=(p_1,\dotsc,p_m)$ with $1<p_1,\dotsc,p_m<\infty$, $1/p_1+\dotsb+1/p_m=1/p$ and $\vec{w}=(w_1,\dotsc,w_m)\in A_{\vec{P}}$. In this paper, we investigate the weighted bounds with dependence on aperture $α$ for multilinear square functions $S_{α,ψ}(\vec{f})$. We show that $$ \|S_{α,ψ}(\vec{f})\|_{L^p(ν_{\vec{w}})} \leq C_{n,m,ψ,\vec{P}}~ α^{mn}[\vec{w}]_{A_{\vec{P}}}^{\max(\frac{1}{2},\tfrac{p_1'}{p},\dotsc,\tfrac{p_m'}{p})} \prod_{i=1}^m \|f_i\|_{L^{p_i}(w_i)}. $$ This result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calderón--Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.

math.CA↗
arXiv2015

Orthogonal bases of Brauer relative symmetric polynomials for certain groups

In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Semi-Dihedral groups and Dihedral groups. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Semi-Dihedral groups.

math.CV↗
arXiv2015

Orthogonal bases of Brauer relative symmetric polynomials for the the Dicyclic group

In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Dicyclic group. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Dicyclic group.

math.CV↗