Related papers: Geometric conditions for bounded point evaluations…

We give some necessary conditions and sufficient conditions for the compactness of the embedding of Sobolev spaces $W^{1,p}(\Omega,w) \to L^p(\Omega,w),$ where $w$ is some weight on a domain $\Omega \subset \Real^n$.

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

The well known result of Bourgain and Kwapie\'n states that the projection $P_{\leq m}$ onto the subspace of the Hilbert space $L^2\left(\Omega^\infty\right)$ spanned by functions dependent on at most $m$ variables is bounded in $L^p$ with…

In this paper, by introducing some parameters, we define and study certain $p$-adic Hardy-Littlewood-P\'{o}lya-type integral operators acting on $p$-adic weighted Lebesgue spaces. We completely characterize $L^{q}-L^{r}$ boundedness of…

We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in $L^r$-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We…

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a…

Let $L^{m,p}(\mathbb{R}^n)$ be the homogeneous Sobolev space for $p \in (n,\infty)$, $\mu$ be a Borel regular measure on $\mathbb{R}^n$, and $L^{m,p}(\mathbb{R}^n) + L^p(d\mu)$ be the space of Borel measurable functions with finite seminorm…

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $\Omega \subset \mathbb{R}^N$ whose boundary $\partial\Omega$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to…

Let $f: S^1\to S^1$ be a $C^3$ homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of $f$, which are beau in the sense…

Let $P$ and $Q$ be polynomials in one variable over an algebraically closed field $k$ of characteristic zero. Let $f$ and $g$ be elements of a function field $\K$ over $k$ such that $P(f)=Q(g).$ We give conditions on $P$ and $Q$ such that…

Let $X$ be a compact subset of the complex plane. It is shown that if a point $x_0$ admits a bounded point derivation on $R^p(X)$, the closure of rational function with poles off $X$ in the $L^p(dA)$ norm, for $p >2$ and if $X$ contains an…

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf…

Classical boundary Hardy inequality, that goes back to 1988, states that if $1 < p < \infty, \ ~\Omega$ is bounded Lipschitz domain, then for all $u \in C^{\infty}_{c}(\Omega)$, $$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx…

This work investigates the Sobolev regularity of solutions to perturbed fractional 1-Laplace equations. Under the assumption that weak solutions are locally bounded, we establish that the regularity properties are analogous to those…

Supplying the missing necessary conditions, we complete the characterisation of the $L^p\to L^q$ boundedness of commutators $[b,T]$ of pointwise multiplication and Calder\'on-Zygmund operators, for arbitrary pairs of $1<p,q<\infty$ and…

Let $U$ be a bounded open subset of the complex plane. Let $0<\alpha<1$ and let $A_{\alpha}(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $\alpha$ on the complex plane, are analytic on $U$ and are such…

Let $p(\cdot):\ \mathbb{R}^n\to(0,\infty]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. Let $H_A^{p(\cdot)}(\mathbb{R}^n)$ be the variable…

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are…

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partial\Omega)$ on a strong Lipschitz boundary $\partial\Omega$, i.e., $\Omega$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and…

We introduce a generalized inverse Gaussian setting and consider the maximal operator associated with the natural analogue of a nonsymmetric Ornstein--Uhlenbeck semigroup. We prove that it is bounded on $L^{p}$ when $p\in (1,\infty]$ and…