Related papers: Extremizers and sharp weak-type estimates for posi…

In this note, we present new properties for a sequence arising in some refinements of Carleman's inequality. Our results extend some results of Yang [Approximations for constant e and their applications J. Math. Anal. Appl. 262 (2001)…

We strengthen H\"older's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of Pythagorean theorem for the $L^p$ spaces. Our reasonings rely upon Bellman functions of four variables.

The paper contains the proof of $L^p$-weighted norm inequalities for both, martingales square functions and the classical square functions in harmonic analysis of Littlewood-Paley and Lusin. Furthermore, the bounds are completely explicit…

With motivation from K. D\c{e}bicki and P. Kisowski (2007), in this paper we derive the exact tail asymptotics of $\alpha(t)$-locally stationary Gaussian processes with non-constant variance functions. We show that some certain variance…

We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling…

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds:…

In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a…

For a large class of irreducible shift spaces $X\subset\tA^{\Z^d}$, with $\tA$ a finite alphabet, and for absolutely summable potentials $\Phi$, we prove that equilibrium measures for $\Phi$ are weak Gibbs measures. In particular, for…

This paper presents a new proof of the results regarding the continuity of weighted estimates with respect to the characteristic of the weight. Here we first prove the result in the dyadic case which is "easier" and then by the use of the…

We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of…

We give examples of $L^{1}$-functions that are essentially unbounded on every nonempty open subset of their domains of definition. We obtain such functions as limits of weighted sums of functions with the unboundedly increasing number of…

We define the harmonic Bergman space on locally finite trees with respect to a suitable probabilistic Laplacian and a class of weighted flow measures. We characterise the corresponding Bergman projection and prove that it is bounded on…

In this note, we establish a discrete method to characterize the limiting weak type behaviors of the centered Hardy-Littlewood maximal operator on the positive real axis through testing on Dirac deltas. As an application, we give some new…

Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which is sharp as $\alpha\to0$. We also show that the distributional determinant of $u$ is…

In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the $\mathrm{RCD}^*(K,N)$ condition for $K$ in $\mathbb{R}$ and $N$ in $(2,\infty)$. We show the existence, regularity…

The best constant in the usual Lp norm inequality for the centered Hardy-Littlewood maximal function on R1 is obtained for the class of all ``peak-shaped'' functions. A positive function on the line is called ``peak-shaped'' if it is…

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $\text{Re}…

In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent…

We prove two sharp estimates for the subspace of a standard weighted Bergman space that consists of functions vanishing at a given point (with prescribed multiplicity).

We discuss a topological structure on families of convex functions and then apply it to show the existence of extrimizers for the functional Santal\'{o} inequality with respect to polar transform and its reverse.