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Komla Domelevo

Publications and source records attributed to Komla Domelevo.

At least 19 recordsLinked to original sources
arXiv2026

Dimension-free bounds for Riesz transforms on the Hamming cube via a Bellman function

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(\Omega;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(\Omega)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $\Omega=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $\Omega=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $\Omega$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

math.FA↗
arXiv2026

The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W]_{\mathbf{A}_2}^{3/2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3/2$ power persists if we replace the classical matrix $A_2$ characteristic by the "fattened", larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.

math.CA↗
arXiv2026

A twosided linear estimate and a dyadic reduction of the UMD Conjecture

We define a time faithful dyadic shift operator of complexity one, that is an antisymmetric antiinvolution. We show that the Hilbert transform with values in a Banach space is $L^p$ bounded if and only if the dyadic shift is -- with a linear two sided norm dependence. The results reduce the famous UMD conjecture to a pair of simple dyadic operators.

math.FA↗
arXiv2024

Dimension-free estimates for low degree functions on the Hamming cube

The main result of this paper are dimension-free $L^p$ inequalities, $1 2,$ $\varepsilon>0,$ and $θ=θ(\varepsilon,p)\in (0,1)$ satisfying \[ \frac{1}{p}=\fracθ{p+\varepsilon}+\frac{1-θ}{2} \] we obtain, for any function $f:\{-1,1\}^n\to \mathbb{C}$ whose spectrum is bounded from above by $d,$ the Bernstein-Markov type inequalities \[\|Δ^k f\|_{p} \le C(p,\varepsilon)^k \,d^k\, \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad k\in \mathbb{N}.\] Analogous inequalities are also proved for $p\in (1,2)$ with $p-\varepsilon$ replacing $p+\varepsilon.$ As a corollary, if $f$ is Boolean-valued or $f\colon \{-1,1\}^n\to \{-1,0,1\},$ we obtain the bounds \[\|Δ^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\qquad k\in \mathbb{N}.\] At the endpoint $p=\infty$ we provide counterexamples for which a linear growth in $d$ does not suffice when $k=1$. We also obtain a counterpart of this result on tail spaces. Namely, for $p>2$ we prove that any function $f:\{-1,1\}^n\to \mathbb{C}$ whose spectrum is bounded from below by $d$ satisfies the upper bound on the decay of the heat semigroup $$ \|e^{-tΔ}f\|_{p} \le \exp(-c(p,\varepsilon) td) \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad t>0,$$ and an analogous estimate for $p\in (1,2).$ The constants $c(p,\varepsilon)$ and $C(p,\varepsilon)$ depend only on $p$ and $\varepsilon$; crucially, they are independent of the dimension $n$.

math.FA↗
arXiv2023

Boundedness of Journé operators with matrix weights

We develop a biparameter theory for matrix weights and provide various biparameter matrix-weighted bounds for Journé operators as well as other central operators under the assumption of the product matrix Muckenhoupt condition. In particular, we provide a complete theory for biparameter Journé operator bounds on matrix-weighted $L^2$ spaces. We also achieve bounds in the general case of matrix-weighted $L^p$ spaces, for $1 < p < \infty$ for paraproduct-free Journé operators. Finally, we expose an open problem involving a matrix-weighted Fefferman--Stein inequality, on which our methods rely in the general setting of matrix-weighted bounds for arbitrary Journé operators and $p \neq 2.$

math.CA↗
arXiv2023

The dyadic Riesz vector I

We derive a dyadic model operator for the Riesz vector. We show linear lower $L^p$ bounds for $1 < p < \infty$ between this model operator and the Riesz vector, when applied to functions with values in Banach spaces. By a lower bound we mean that the boundedness of the Riesz vector implies the boundedness of the dydic Riesz vector.

math.FA↗
arXiv2023

The dyadic Riesz vector II

We derive a dyadic model operator for the Riesz vector. We show linear upper $L^p$ bounds for $1 < p < \infty$ between this model operator and the Riesz vector, when applied to functions with values in Banach spaces. By an upper bound we mean that the boundedness of the dyadic Riesz vector implies the boundedness of the Riesz vector. The same holds for single dyadic Riesz transforms and their continuous counterparts. The linear dependence is with constant one.

math.FA↗
arXiv2023

Failure of the matrix weighted bilinear Carleson embedding theorem

We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. Indeed, a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding. We show that any improvement of a recent matrix weighted bilinear embedding, featuring a scalar Carleson sequence and inner products instead of norms must fail. In particular, replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products. Any formulation using norms, even in the presence of a scalar Carleson sequence must fail. As a positive result, we prove the so called matrix weighted redundancy condition in full generality.

math.CA↗
arXiv2023

Continuous sparse domination and dimensionless weighted estimates for the Bakry Riesz vector

We present a fundamentally new proof of the dimensionless Lp boundedness of the Bakry Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion than previous arguments, namely that of some new dimensionless weighted estimates with optimal exponent. Part of the importance of this task lies in the novelty of the techniques: we develop the self similarity argument known as sparse domination in the setting of uniformly integrable cadlag Hilbert space valued martingales and extend the domination to a process with infinite memory. We provide a range of optimal weighted estimates and weak type estimates for these stochastic processes. Previous geometric Riesz transform estimates relied on Bellman functions and did not provide this range of weighted estimates. The development of sparse domination in this probabilistic setting and its use for high dimensional problems is new.

math.PR↗
arXiv2023

Dyadic lower little BMO estimates

We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case as well as with Bloom weights. Moreover, we address the flexibility of one of our methods.

math.CA↗
arXiv2019

$H^\infty$ calculus for submarkovian semigroups on weighted $L^2$ spaces

Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $σ$-finite measure space $(Ω,μ)$. We prove that its negative generator $A$ has a bounded $H^\infty(Σ_θ)$ calculus on the weighted space $L^2(Ω,wdμ)$ as long as the weight $w : Ω\to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(Ω)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $θ> \fracπ{2}$ is admissible in the above $H^\infty$ calculus, and for some semigroups also certain $θ= θ_w < \fracπ{2}$ depending on the size of $Q^A_2(w)$. The norm of the $H^\infty(Σ_θ)$ calculus is linear in the $Q^A_2$ characteristic for $θ> \fracπ{2}$. We also discuss negative results on angles $θ< \fracπ{2}$. Namely we show that there is a markovian semigroup on a probability space and a $Q^A_2$ weight $w$ without Hörmander functional calculus on $L^2(Ω,w dμ)$.

math.CA↗
arXiv2019

Continuous-time sparse domination

We develop the self similarity argument known as sparse domination in an abstract martingale setting, using a continuous time parameter. With this method, we prove a sharp weighted L^p estimate for the maximal operator Y^* of Y with respect to X. Here Y and X are uniformly integrable càdllàg Hilbert space valued martingales and Y differentially subordinate to X via the square bracket process. We also present a second, very simple proof of the special case Y=X. In this generality, notably including processes with jumps, the special case Y = X addresses a question raised in the late 70s by Bonami--Lépingle.

math.PR↗
arXiv2018

Dimensionless $L^p$ estimates for the Riesz vector on manifolds

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.

math.PR↗
arXiv2017

Existence of travelling waves and high activation energy limits for a onedimensional thermo-diffusive lean spray flame model

We provide a mathematical analysis of a thermo-diffusive combustion model of lean spray flames, for which we prove the existence of travelling waves. In the high activation energy singular limit we show the existence of two distinct combustion regimes with a sharp transition -- the diffusion limited regime and the vaporisation controlled regime. The latter is specific to spray flames with slow enough vaporisation. We give a complete characterisation of these regimes, including explicit velocities, profiles, and upper estimate of the size of the internal combustion layer. Our model is on the one hand simple enough to allow for explicit asymptotic limits and on the other hand rich enough to capture some particular aspects of spray combustion. Finally, we briefly discuss the cases where the vaporisation is infinitely fast, or where the spray is polydisperse.

math.CA↗
arXiv2017

Discrete Hilbert Transform a la Gundy-Varopoulos

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting.

math.PR↗