We investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $\mu ,\lambda \in A_{p,q}$ and α/n + 1/q = 1/p, the norm $\left|\right|[b,I_{\alpha }]:L^p\left({\mu }^p\right)\to L^q\left({\lambda }^q\right)\left|\right|$ is equivalent to the norm of b in the weighted BMO space BMO(ν), where $\nu =\mu {\lambda }^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey,...

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...

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.

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.

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...

We consider the summation equation, for $t\in {[\mu -2,\mu +b]}_{{\mathbb{N}}_{\mu -2}}$, $$\begin{array}{ccc}\hfill y\left(t\right)={\gamma }_1\left(t\right)H_1\left(\sum _{i=1}^n{a}_{i}y\left({\xi }_i\right)\right)& +{\gamma }_2\left(t\right)H_2\left(\sum _{i=1}^m{b}_{i}y\left({\zeta }_i\right)\right)\hfill & \hfill +\lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1))\end{array}$$ in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence...

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.

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.

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.

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 $H_1=H_2$.

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{\alpha ₙ}-{\sum }_{i=1}^{n-1}{a}_i{D}^{{\alpha }_i})x\left(t\right)\in F(t,x\left(\phi \left(t\right)\right))$, a.e. on (0,1), $I^{1-\alpha ₙ}x\left(0\right)=c$, αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

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.

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla ·\left(u\nabla {(-\Delta )}^{-s}u\right),0<s<1$. The problem is posed in $\{x\in {ℝ}^n,t\in ℝ\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{\alpha }$ regularity of such weak solutions. Finally, we extend...

In the paper we study the subject of stability of systems with $h$-differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$-difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$-orders.

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...

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.

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...

We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ $-d/dt(1/2_0{D}_t^{-\sigma }\left(u^{\text{'}}\left(t\right)\right)+1/2_t{D}_T^{-\sigma }\left(u^{\text{'}}\left(t\right)\right))-\lambda \beta \left(t\right)f\left(u\left(t\right)\right)-\mu \gamma \left(t\right)g\left(u\left(t\right)\right)=0$, a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where ${}_0{D}_t^{-\sigma }$ and ${}_t{D}_T^{-\sigma }$ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

We introduce Sobolev spaces $L_{\alpha }^p$ for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in $L^p$ with fractional derivative of order α, $D^{\alpha }f$, as introduced in [2], in $L^p$. We show that for small α, $L_{\alpha }^p$ coincides with the continuous version of the Triebel-Lizorkin space $F_p^{\alpha ,2}$ as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity...

Let ϕ and ψ be functions defined on [0,∞) taking the value zero at zero and with non-negative continuous derivative. Under very mild extra assumptions we find necessary and sufficient conditions for the fractional maximal operator $M_{\Omega }^{\alpha }$, associated to an open bounded set Ω, to be bounded from the Orlicz space $L^{\psi }\left(\Omega \right)$ into $L^{\varphi }\left(\Omega \right)$, 0 ≤ α < n. For functions ϕ of finite upper type these results can be extended to the Hilbert transform f̃ on the one-dimensional torus and to the fractional integral operator...