arXiv subjects

Komla Domelevo

Publications and source records attributed to Komla Domelevo.

26 records · Page 2Linked to original sources

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↗

arXiv2017

Various sharp estimates for semi-discrete Riesz transforms of the second order

We give several sharp estimates for a class of combinations of second order Riesz transforms on Lie groups ${G}={G}_{x} \times {G}_{y}$ that are multiply connected, composed of a discrete abelian component ${G}_{x}$ and a connected component ${G}_{y}$ endowed with a biinvariant measure. These estimates include new sharp $L^p$ estimates via Choi type constants, depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal $L^q \to L^p$ estimate as well. It was shown recently by Arcozzi, Domelevo and Petermichl that such second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The proofs of our theorems combine this stochastic integral representation with a number of deep estimates for pairs of martingales under strong differential subordination by Choi, Banuelos and Osekowski. When two continuous directions are available, sharpness is shown via the laminates technique. We show that sharpness is preserved in the discrete case using Lax-Richtmyer theorem.

math.CA↗

arXiv2016

A linear dimensionless bound for the weighted Riesz vector

We show that the norm of the vector of Riesz transforms as operator in the weighted Lebesgue space L^2(w) is bounded by a constant multiple of the first power of the Poisson-A_2 characteristic of w. The bound is free of dimension. Our argument requires an extension of Wittwer's linear estimate for martingale transforms to the vector valued setting with scalar weights, for which we indicate a proof. Extensions to L^p(w) for 1 1, the Poisson-A_2 class is properly included in the classical A_2 class.

math.CA↗

arXiv2016

Differential subordination under change of law

We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales are adapted, uniformly integrable, and cadlag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer, where homogeneity was heavily used. Recent progress by Thiele-Treil-Volberg and Lacey, independently, resloved the so-called non-homogenous case of discrete in time filtrations with two completely different proofs. The general case for continuous-in-time filtrations remained open and is adressed here. As a by-product, we give the needed explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps.

math.PR↗

arXiv2015

Second order Riesz transforms on multiply-connected Lie groups and processes with jumps

We study a class of combinations of second order Riesz transforms on Lie groups that are multiply connected, composed of a discrete abelian component and a compact connected component. We prove sharp $L^{p}$ estimates for these operators, therefore generalising previous results. We construct stochastic integrals with jump components adapted to functions defined on our semi-discrete set. We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Ito integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes.

math.PR↗

arXiv2015

Sharp $L^p$ estimates for discrete second order {R}iesz transforms

We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if one considers all multipliers of the form $\sum_i σ_{i} R_{i} R^{\ast}_{i}$ with $| σ_{i} | \leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\sum_{i} σ_{i} R_{i} R^{\ast}_{i}$ with $0 \leqslant σ_{i} \leqslant 1$. These are the first known precise $L^{p} $ estimates for discrete Calderón-Zygmund operators.

math.CA↗