Related papers: An improvement to the John-Nirenberg inequality fo…
We study self-improving properties in the scale of Lebesgue spaces of generalized Poincar\'e inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the…
In this paper, we first prove an interpolation inequality of Ehrling-type, which is an improvement of a special case to the well known Gargliardo-Nirenberg inequality. Then we apply it to study the classical Keller-Segel system…
A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As…
For $p \in (1,N)$ and $\Omega \subseteq \mathbb{R}^N$ open, the Beppo-Levi space $\mathcal{D}^{1,p}_0(\Omega)$ is the completion of $C_c^{\infty}(\Omega)$ with respect to the norm $\left( \int_{\Omega}|\nabla u|^p \right)^ \frac{1}{p}.$…
Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity…
We investigate properties of ($\alpha,\beta$)-harmonic functions. First, we discuss the coefficient estimates for ($\alpha,\beta$)-harmonic functions. In particular, we obtain Heinz's inequality for ($\alpha,\beta$)-harmonic functions,…
The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subset \mathbb R^n$ a bounded domain) with $\int_\Omega |\nabla u|^ndx \le 1$ one has $\int_\Omega (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx \le c…
In this paper we prove the existence of extremal functions for the Adams-Moser-Trudinger inequality on the Sobolev space $H^{m}(\Omega)$, where $\Omega$ is any bounded, smooth, open subset of $\mathbb{R}^{2m}$, $m\ge 1$. Moreover, we extend…
We give a sharp sufficient condition on the distribution function, $|\{x\in \Omega :\,p(x)\leq 1+\lambda\}|$, $\lambda>0$, of the exponent function $p(\cdot): \Omega \to [1,\infty)$ that implies the embedding of the variable Lebesgue space…
Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying…
In this paper we present new embedding results for Musielak-Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of $W^{1,\mathcal{H}}(\Omega)$ into $L^{\mathcal{H}_*}(\Omega)$, where $\mathcal{H}_*$ is the Sobolev…
We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…
The celebrated Heinz inequality asserts that $ 2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, $A,B\in \+$, every unitarily invariant norm $|||\cdot|||$ and $\nu \in…
Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on $\rn$ to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these…
We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit…
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*}…
Let $\Omega\subset \mathbb{R}^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{\alpha,\beta,\lambda,\mu}(\Omega)…
We investigate the best constant $J(n,d)$ such that Jackson's inequality \[ \inf_{\mathrm{deg}(g) \leq d} \|f - g\|_{\infty} \leq J(n,d) \, s(f), \] holds for all functions $f$ on the hypercube $\{0,1\}^n$, where $s(f)$ denotes the…
We use the formalism of the R{\'e}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality…