Related papers: Endpoint Sobolev continuity of the fractional maxi…

Let $(X,d,\mu)$ is a space of homogeneous type, we establish a new class of fractional-type variable weights $A_{p(\cdot), q(\cdot)}(X)$. Then, we get the new weighted strong-type and weak-type characterizations for fractional maximal…

Let $0 \leq \alpha<n$, $M_{\alpha}$ be the fractional maximal operator, $M^{\sharp}$ be the sharp maximal operator and $b$ be the locally integrable function. Denote by $[b, M_{\alpha}]$ and $[b, M^{\sharp}]$ be the commutators of the…

In this paper, the fractional Hardy-type operator of variable order $\beta(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{\alpha,\lambda}(\mathbb{R}^{n})$ with variable exponent $q_{1}(x)$ into…

We prove continuity properties of higher order commutators of fractional operators on the multilinear setting, between a product of weighted Lebesgue spaces into certain weighted Lipschitz spaces. The considered operators include the…

In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. In particular, we…

We introduce and study the median maximal function \mathcal{M} f, defined in the same manner as the classical Hardy-Littlewood maximal function, only replacing integral averages of f by medians throughout the definition. This change has a…

In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $\pp$ for a fractional maximal operator $M_\alpha$…

Given a regular compact set $E$ in the complex plane, a unit measure $\mu$ supported by $\partial E,$ a triangular point set $\beta := \{\{\beta_{n,k}\}_{k=1}^n\}_{n=1}^{\infty},\beta\subset \partial E$ and a function $f$, holomorphic on…

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces $W^sL_{\varPhi_{x,y}}$ such that the generalized Poincar\'e type inequality and some continuous and compact embedding theorems of…

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p\colon\overline{\Omega}\times \overline{\Omega}\to (1,\infty)$ and $q:\partial \Omega \rightarrow (1,\infty)$ are…

Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of the set $\{\lambda_i^\alpha\}_{i=1}^\infty$ of eigenvalues associated to $X^D_t$,…

We study the mapping behavior of the Marchaud fractional derivative with different extensions in the scale of fractional weighted Sobolev spaces. In particular we show that the $\alpha$--order Riemann--Liouville fractional derivative maps…

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of…

Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $\Omega\subseteq\mathbb{R}^{d}$. We prove that the maximal operator…

For $\alpha >1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-\Delta)^{\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values…

We prove two maximal regularity results in spaces of continuous and H\"older continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^\alpha$. This derivative is intended in the sense…

Pseudo-differential operators of type $(1,1)$ and order $m$ are continuous from $F_p^{s+m,q}$ to $F_p^{s,q}$ if $s>d/\min{(1,p,q)}-d$ for $0<p<\infty$, and from $B_p^{s+m,q}$ to $B_{p}^{s,q}$ if $s>d/\min{(1,p)}-d$ for $0<p\leq\infty$. In…

Let $\Omega \subset \mathbb{R}^d$ be bounded open and connected. Suppose that $W^{1,2}(\Omega) \subset L^r(\Omega)$ for some $r > 2$. Let $A$ be a pure second-order elliptic differential operator with bounded real measurable coefficients on…

We prove that all the composition operators $T_f(g):= f\circ g$, which take the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, are continuous mappings from $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to…

We characterize the class of weights related to the boundedness of variable fractional maximal operator $M_{\beta(\cdot),r(\cdot)}$ on variable Lebesgue spaces. This extend previously known results, including those corresponding to the…