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We provide $L^1$ estimates for a class of transport equations containing singular integral operators. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is…

We prove a contractive Hardy-Littlewood type inequality for functions from $H^p(\mathbb{T})$, $0 < p \le 2$ which is sharp in the first two Taylor coefficients and asymptotically at infinity.

In this paper, we establish several inequalities for different convex mappings that are connected with the Riemann-Liouville fractional integrals. Our results have some relationships with certain integral inequalities in the literature.

We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A…

A companion of Ostrowski like inequality for mappings whose second derivatives belong to $L^{\infty}$ spaces is established. Applications to composite quadrature rules, and to probability density functions are also given.

We prove a reverse Lieb-Thirring inequality with a sharp constant for the matrix Schr\"odinger equation on the half-line.

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calder\'on's and Zygmund's theory of first order…

An "exact discretization" of the Schroedinger operator is considered and its direct and inverse scattering problems are solved. It is shown that a differential-difference nonlinear evolution equation depending on two arbitrary constants can…

We prove new sharp $L^p$, logarithmic, and weak-type inequalities for martingales under the assumption of differentially subordination. The $L^p$ estimates are "Fyenman-Kac" type versions of Burkholder's celebrated martingale transform…

We present in this note a lower bound for the Calabi functional in a given K\"ahler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A…

For any two real-valued continuous-path martingales $X=\{X_t\}_{t\geq 0}$ and $Y=\{Y_t\}_{t\geq 0}$, with $X$ and $Y$ being orthogonal and $Y$ being differentially subordinate to $X$, we obtain sharp $L^p$ inequalities for martingales of…

We study local and global higher integrability properties for quasiminimizers of a class of double-phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure…

We establish a quantitative weighted inequality for the bilinear rough singular integral, where the bound is controlled by the cube of the weight constant.

We prove estimates for the sharp constants in fractional Poincar\'e-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local…

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} \Delta(|x|^{-\gamma}\Delta u)=|x|^\gamma u^{\frac{N+4+3\gamma}{N-4-\gamma}},\quad u\geq 0 \quad…

We prove weighted $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators in higher dimensions. The vector-valued extensions of these inequalities are also given.

The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Lieb's sharp form of…

In this paper, we prove a Skoda type division theorem with sharp $L^2$-estimate. Furthermore, using this estimate, we provide new characterizations of plurisubharmonic functions. We also explain that the sharp $L^2$-division theorem leads…

In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic…