Related papers: Bellman function technique for multilinear estimat…
We prove new results for multi-parameter singular integrals. For example, we prove that bi-parameter singular integrals in $\mathbb{R}^{n+m}$ satisfying natural $T1$ type conditions map $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ to…
We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove $L^p$ boundedness of…
We prove $L^p$ estimates for the shifted bilinear Hilbert transform, with a polylogarithmic bound in the size of the shift. As applications, we obtain $r$-variation estimates for bilinear ergodic averages in the sharp range $r > 2$, a sharp…
We provide a quantitative two weight estimate for the dyadic paraproduct $\pi_b$ under certain conditions on a pair of weights $(u;v)$ and $b$ in $Carl_{u,v}$, a new class of functions that we show coincides with BMO when $u = v \in A^d_2$.…
The classical $L^2$ estimate for the $\overline{\partial}$ operators is a basic tool in complex analysis of several variables. Naturally, it is expected to extend this estimate to infinite dimensional complex analysis, but this is a…
By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,\mu_\phi)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the…
The main purpose of this short note is to present an adaptation of the multilinear Bellman function technique from [4] to the time-frequency analysis. Demeter and Thiele introduced the two-dimensional bilinear Hilbert transform in [3] and…
We establish new $p$-estimates for the norm of the generalized Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;\Lambda)\to L^p(\R^n;\Lambda)}\leq n(p^{*}-1)$…
We establish multilinear $L^p$ bounds for a class of maximal multilinear averages of functions on one variable, reproving and generalizing the bilinear maximal function bounds of Lacey. As an application we obtain almost everywhere…
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work…
We derive explicit bounds for two general classes of $L$-functions, improving and generalizing earlier known estimates. These bounds can be used, for example, to apply Turing's method for determining the number of zeros up to a given…
We consider a general class of non-linear Bellman equations. These open up a design space of algorithms that have interesting properties, which has two potential advantages. First, we can perhaps better model natural phenomena. For…
The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well studied objects of harmonic analysis. We investigate $L^p$ bounds for a dyadic model of this form in the particular…
This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction…
We prove some weighted $L_p$ estimates for generalized harmonic extensions in the half-space.
Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can…
We develop a theory of Lp spaces based on outer measures rather than measures. This theory includes the classical Lp theory on measure spaces as special case. It also covers parts of potential theory and Carleson embedding theorems. The…
This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…
We establish the pseudo-differential variant of the $L^{p}$ estimates for multi-linear and multi-parameter Coifman-Meyer multiplier operators proved by C. Muscalu, J. Pipher, T. Tao and C. Thiele in \cite{MPTT1,MPTT2}.
The authors use steepest descent ideas to obtain a priori $L^p$ estimates for solutions of Riemann-Hilbert Problems. Such estimates play a crucial role, in particular, in analyzing the long-time behavior of solutions of the perturbed…