Related papers: Optimal constants for a mixed Littlewood type ineq…

We consider the $L^p$ Hardy inequality involving the distance to the boundary for a domain in the $n$-dimensional Euclidean space. We study the dependence on $p$ of the corresponding best constant and we prove monotonicity, continuity and…

The $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$, if $d\geq2$.…

In this paper, we study the form of the constant $C$ in the Bernstein--Nikolskii inequalities $\|f^{(s)}\|_q \lesssim C(s, p, q)\left\|f\right\|_p,\,0<p<q \leq\infty$, for trigonometric polynomials and entire functions of exponential type.…

Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and $X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n}, \ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there exists an…

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2\pi}\gamma^pw^2\le C\int_0^{2\pi}\gamma^qw'^2 \] where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such…

We study the boundedness problem for maximal operators $\mathbb{M}$ associated to averages along families of finite type curves in the plane, defined by $$\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \,…

In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s,p}$. The case $p = 2$ of their result implies the nonlocal isoperimetric inequality \[ \frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge…

Recently, Gilmer proved the first constant lower bound for the union-closed sets conjecture via an information-theoretic argument. The heart of the argument is an entropic inequality involving the OR function of two i.i.d.\ binary vectors,…

This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the…

This paper focuses on optimal constants and optimizers of the second order Caffarelli-Kohn-Nirenberg inequalities. Firstly, we aim to study optimal constants and optimizers for the following second order Caffarelli-Kohn-Nirenberg inequality…

The Coulomb potential at an interior ion in a finite crystal of size $p$ is given by a linear superposition of contributions from displacement vectors ${\mathbf r}=(x,y,z)$ to its neighbors. This additive structure underlies universal…

In this paper, we prove that for x\in(0,{\pi}/2) (cos p_0x)^{1/p_0}<((sin x)/x)<(cos(x/3))^3 with the best constants p_0=0.347307245464... and 1/3. Moreover, if p\in (0,1/3] then the double inequality {\beta}_{p}(cos px)^{1/p}<((sin…

If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2<p\ne p_c=\tfrac{2(n+1)}{n-1}$, the critical exponent. Since improved…

We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of $L^p$ bounds between different partial derivatives $\partial^\beta u$ of $u\in C^\infty_c(\mathbb{R}^n,X)$. In…

This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates…

We discuss the Lieb-Thirring inequality for periodic systems, which has the same optimal constant as the original inequality for finite systems. This allows us to formulate a new conjecture about the value of its best constant. To…

In this work we improve the sharp Hardy inequality in the case $p>n$ by adding an optimal weighted Hoelder semi-norm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for $p>n$…

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_p^2$ spaces, $2<p\leq\infty$…

Let $D$ be a positive nonsquare integer, $p$ a prime number with $p \nmid D$, and $0< \sigma < 0.847$. We show that if the equation $x^2+D=p^n$ has a huge solution $(x_0,n_0)_{(p,\sigma)}$, then there exists an effectively computable…

In this paper, we establish the best constant in the G-N inequality for the mixed local and nonlocal Laplacian. In our problem, classical methods cannot apply directly since regularity results for the operator under study seem to be highly…