Related papers: Fractional Trudinger-Moser type inequalities with …

In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm $\left(\int_{\Omega}F^{N}(\nabla u)\;\mathrm{d}x\right)^{\frac{1}{N}}$ in the Sobolev-type space $D^{N,q}(\mathbb{R}^{N})$,…

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the…

We study the fractional Laplacian $(-\Delta)^{\sigma/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^\gamma$, $\gamma>0$, where $L$ is…

We obtain a characterization of the weighted inequalities for the Riesz transforms on weighted local Morrey spaces. The condition is sufficient for the boundedness on the same spaces of all Calder\'on-Zygmund operators suitably defined on…

The paper deals about Hardy-type inequalities associated with the following higher order Poincar\'e inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}}…

In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u-\mu\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (\alpha)-2}u}{|x|^{\alpha}} + f(x,u), && \mbox{in} \ \Omega,\\ &u=0, && \mbox{in} \…

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we…

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary,…

Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\…

In a previous work (Int. Math. Res. Notices 13 (2010) 2394-2426), Adimurthi-Yang proved a singular Trudinger-Moser inequality in the entire Euclidean space $\mathbb{R}^N$ $(N\geq 2)$. Precisely, if $0\leq \beta<1$ and $0<\gamma\leq1-\beta$,…

In this work, we focus on a recent variant of the Trudinger-Moser-Onofri inequality introduced by S. Y. Alice Chang and Changfeng Gui \cite{CG-2023}: \begin{align*} \alpha\int_{\mathbb{S}^2}|\nabla_{\mathbb{S}^2}u|^2 {\rm d}\omega+2…

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth…

We obtain a global fractional Calder\'on-Zygmund regularity theory for the fractional Poisson problem. More precisely, for $\Omega \subset \mathbb{R}^N$, $N \geq 2$, a bounded domain with boundary $\partial \Omega$ of class $C^2$, $s \in…

In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of $\mathcal{\dot{C}}^\eta$, for $\eta\in(0,1)$.

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{\mu,\nu}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality,…

We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f \Gamma(f)^2$ for some difference operator $\Gamma$, and show how it implies two-level…

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in…

While the critical nonlinearity $\int |u|^{2^*}$ for the Sobolev space $H^1$ in dimension $N>2$ lacks weak continuity at any point, Trudinger-Moser nonlinearity $\int e^{4\pi u^2}$ in dimension $N=2$ is weakly continuous at any point except…

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the…

The local maximal inequality for the Schr\"{o}dinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to…