We introduce function spaces $B^{p,\lambda }$ with Morrey-Campanato norms, which unify $B^{p,\lambda }$, $CM{O}^{p,\lambda }$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{\alpha }$ on these spaces.

In this paper, we consider the following boundary value problem $$\left\{\begin{array}{c}{D}_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }\left(D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)\right)=f(t,u\left(t\right)),t\in [0,T],\hfill \\ u\left(0\right)=u\left(T\right)=0\hfill \\ D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(0\right)\right)=D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(T\right)\right)=0,\hfill \end{array}\right.$$ where $0<\alpha \le 1$ and $f:[0,T]\times ℝ\to ℝ$ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

We introduce the generalized fractional integrals $I{̃}_{\alpha ,d}$ and prove the strong and weak boundedness of $I{̃}_{\alpha ,d}$ on the central Morrey spaces $B^{p,\lambda }\left(ℝⁿ\right)$. In order to show the boundedness, the generalized λ-central mean oscillation spaces ${\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ and the generalized weak λ-central mean oscillation spaces $W{\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ play an important role.

The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

Given a two-dimensional fractional multiplicative process ${\left(F_t\right)}_{t\in [0,1]}$ determined by two Hurst exponents $H_1$ and $H_2$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_1=H_2$.

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1$ and $g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\gamma }_{fg}\left(G\right)$ is defined as follows: ${\gamma }_{fg}\left(G\right)$ = min|g|:g is an MGDF of G where $\left|g\right|={\sum }_{v\in V}g\left(v\right)$. In this paper we initiate a study of this parameter.

For α ∈ (1,2) we consider the equation ${\partial }_{t}u={\Delta }^{\alpha /2}u+b·\nabla u$, where b is a time-independent, divergence-free singular vector field of the Morrey class $M{₁}^{1-\alpha }$. We show that if the Morrey norm ${\left|\right|b\left|\right|}_{M{₁}^{1-\alpha }}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

We consider a double analytic family of fractional integrals $S_z^{\gamma ,\alpha }$ along the curve ${t\mapsto \left|t\right|}^{\alpha }$, introduced for α = 2 by L. Grafakos in 1993 and defined by $\left(S_z^{\gamma ,\alpha }f\right)(x₁,x₂):=1/\Gamma (z+1/2){\int \int |u-1|}^z\psi (u-1){f(x₁-t,x₂-u|t|}^{\alpha }{\left)du\right|t|}^{\gamma }dt/t$, where ψ is a bump function on ℝ supported near the origin, $f{\in }_c^{\infty }\left(ℝ²\right)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_z^{\gamma ,\alpha }$ maps $L^p\left(ℝ²\right)$ to $L^q\left(ℝ²\right)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{-1+i\theta }^{i\varrho ,\alpha }$ is a product kernel on ℝ², adapted to the curve ${t\mapsto \left|t\right|}^{\alpha }$; as a consequence, we show...

Given a measurable set $A\subset {ℝ}^n$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\&lt;\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

We collect and extend results on the limit of ${\sigma }^{1-k}{(1-\sigma )}^k{\left|v\right|}_{l+\sigma ,p,\Omega }^p$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and ${\left|·\right|}_{l+\sigma ,p,\Omega }$ is the intrinsic seminorm of order l+σ in the Sobolev space $W^{l+\sigma ,p}\left(\Omega \right)$. In general, the above limit is equal to $c{\left[v\right]}^p$, where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\left\{\lambda {\theta }^n\right\}$, $n=0,1,2,\cdots$, where $\theta$ is a Pisot number and $\lambda \in \mathbb{Q}\left(\theta \right)$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence ${\left({\lambda }_n\right)}_{n\ge 0}$ of elements of $\mathbb{Q}\left(\theta \right)$ such that $Card\left(L(\theta ,{\lambda }_n)\right)<Card\left(L(\theta ,{\lambda }_{n+1})\right)$, $\forall$ $n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer a ≥ 2 the sequence of fractional parts ${aⁿ/n}_{n=1}^{\infty }$ is everywhere dense in the interval [0,1]. We prove a similar result for all Pisot numbers and Salem numbers α and show that for each c > 0 and each sufficiently large N, every subinterval of [0,1] of length $c{N}^{-0.475}$ contains at least one fractional part Q(αⁿ)/n, where Q is a nonconstant polynomial in ℤ[z] and n is an integer satisfying 1 ≤ n ≤ N.

The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on ${ℝ}_{0+}^{n+1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $-A_{\pm }=-\psi \left(D_{x^{\text{'}}}\right)\pm \partial /\left(\partial x_{n+1}\right)$, $(x^{\text{'}},x_{n+1})\in {ℝ}_{0+}^{n+1}$, where $\psi \left(D_{x^{\text{'}}}\right)$ is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator $-{(-A_{\pm })}^{\alpha }$ and prove that with this domain it generates an $L_p$-sub-Markovian semigroup.

Let $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{\alpha }$ is bounded from $L^p$ to $L^{p_{\alpha }}$ with $p_{\alpha }=np/(\alpha p-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{\alpha ,Y}$, which is strictly larger than X, and a ’target’ space $T_Y$, which is strictly smaller than Y, under the assumption that $A_{\alpha }$ is bounded from X into Y and the Hardy-Littlewood...

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $$x_{n+1}=\frac{a{x}_{n-3}}{b+c{x}_{n-1}{x}_{n-3}},n=0,1,\cdots$$ where $a,b,c$ are positive real numbers and the initial conditions $x_{-3}$, $x_{-2}$, $x_{-1}$, $x_0$ are real numbers.

Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, ${(X,\varrho ,\mu )}_{d,\theta }$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B_{pq}^s\left(X\right)$ and Triebel-Lizorkin spaces $F_{pq}^s\left(X\right)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional...

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...