Related papers: A Herglotz-Nevanlinna function from the optimal di…

In this article, we study the eigenvalue of nonlinear $p-$fractional Hardy operator \begin{align*} (-\Delta)_p^{\alpha}u - \mu \frac{|u|^{p-2}u}{|x|^{p\alpha}} = \lambda V(x) |u|^{p-2}u \; \text{in}\; \Omega, \quad u = 0 \; \mbox{in}\;…

In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By simply choosing weights and specifying $1<p<\infty$, we are able to recover the identity by Krej{\v{c}}i{\v{r}}{\'\i}k-\v{S}tampach…

We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective…

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit…

We investigate the weighted fractional order Hardy inequality $$ \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partial\Omega)^{-\alpha}\text{dist}(y,\partial\Omega)^{-\beta}\,dy\,dx\geq…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert…

Let $n\geqslant 3$ be an integer. For the Bekoll\'e-Bonami weight $\omega$ on the real unit ball $\mathbb{B}_n$, we obtain the following sharp one-weight estimate for the $\mathcal{H}$-harmonic Bergman projection: for $1<p<\infty$ and…

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1 < p < \infty$). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if…

Let $\mathcal{Q}(\varphi):=\int_\Omega \big(|\nabla \varphi|^p+V|\varphi|^p\big)\dnu$ on $\core$, and assume that $\mathcal{Q}\geq 0$. The aim of the paper is to obtain ''as large as possible" nonnegative (optimal) Hardy-type weight $W$…

In this article, we prove a weak type $(p,p)$ maximal inequality, $1<p<\infty$, for weighted averages of a positive Dunford-Schwarz operator $T$ acting on a noncommutative $L_p$-space associated to a semifinite von Neumann algebra…

Let $\Omega$ be a domain in $R^d$ and $d_\Gamma$ the Euclidean distance to the boundary $\Gamma$. We investigate whether the weighted Hardy inequality \[ \|d_\Gamma^{\delta/2-1}\varphi\|_2\leq…

We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our…

We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality…

Let $\omega$ be a B\'ekoll\'e-Bonami weight. We give a complete characterization of the positive measures $\mu$ such that $$\int_{\mathcal H}|M_\omega f(z)|^qd\mu(z)\le C\left(\int_{\mathcal H}|f(z)|^p\omega(z)dV(z)\right)^{q/p}$$ and…

For a general subcritical second-order elliptic operator $P$ in a domain $\Omega \subset \mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - \lambda W$ is…

We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-\Delta_{\Lambda})$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove…

We present an easy proof that $p$--Hardy's inequality implies uniform $p$--fatness of the boundary when $p=n$. The proof works also in metric space setting and demonstrates the self--improving phenomenon of the $p$--fatness. We also explore…

We prove the self-improvement of a pointwise $p$-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.

For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $\mu _ p (\Omega, \varphi)$ of the $p$-Laplacian on a convex domain $\Omega \subset \mathbb{R}^N$ with a power-concave weight $\varphi$. Our…