Compactness of Hardy-Steklov operator

Pair of weights u, v is characterized so that the Hardy-Steklov operator is compact between weighted Lebesgue spaces L^p(u) and L^q(v), where 1<p,q<∞, a,b are certain increasing functions and f?0. The compactness of the conjugate operator is also studied.     

Some estimates for Bochner-Riesz operators on the weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces , we obtain some weighted boundedness properties of the Bochner-Riesz operator and the maximal Bochner-Riesz operator on these spaces for α=n(1/p−1/q), 0<p?1 and 1<q<∞.     

Decompositions of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff spaces and their applications

Let p∈(1,∞), q∈[1,∞), s∈R and . In this paper, the authors establish the φ-transform characterizations of Besov-Hausdorff spaces and Triebel-Lizorkin-Hausdorff spaces (q>1); as applications, the authors then establish their embedding properties (which on is also sharp), smooth atomic and molecular decomposition characterizations for suitable τ. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in and (q>1), which generalize the corresponding classical results on homogeneous Besov and Triebel-Lizorkin spaces when p∈(1,∞) and q∈[1,∞) by taking τ=0.     

Coincidence of strict s-numbers of weighted Hardy operators

We establish the equality of all the so-called strict s-numbers of the weighted Hardy operator T:Lp(I)→Lp(I), where 1<p<∞, I=(a,b)⊂R and

     

On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at −∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(⋅)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I−S)/2 is Fredholm on the variable Lebesgue space , then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces and with some exponents ql and qr lying in the segments between the lower and the upper limits of p at −∞ and +∞, respectively.     

On some iterated weighted spaces

It is proved that the Hörmander and spaces (Ω₁⊂Rn, Ω₂⊂Rm open sets, 1?p<∞, ki Beurling-Björck weights, k=k₁⊗k₂) are isomorphic whereas the iterated spaces and are not if 1<p≠q<∞. A similar result for weighted Lp-spaces of entire analytic functions is also obtained. Finally a result on iterated Besov spaces is given: and are not isomorphic when 1<q≠2<∞.     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems

We prove that the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) where 0<α?1, 0<β?1, and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim|x|→∞u(x)=lim|x|→∞v(x)=∞ if and only if the functions p and q satisfy

     

The oscillation of differential transforms - entire Jacobi expansions

Let L=(1−x²)D²−((β−α)−(α+β+2)x)D with , and . Let f∈C^∞[−1,1], , with normalized Jacobi polynomials and the Cn decrease sufficiently fast. Set Lk=L(Lk−1), k?2. Let ρ>1. If the number of sign changes of (Lkf)(x) in (−1,1) is O(k1/(ρ+1)), then f extends to be an entire function of logarithmic order . For Legendre expansions, the result holds with replaced with .     

Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Let −Dω(·,z)D+q be a differential operator in L²(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form

     

A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces

Let s∈R, τ∈[0,∞), p∈(1,∞) and q∈(1,∞]. In this paper, we introduce a new class of function spaces which unify and generalize the Triebel-Lizorkin spaces with both p∈(1,∞) and p=∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel-Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where s∈R, p,q∈[1,∞), max{p,q}>1, , and t^′ denotes the conjugate index of t∈(1,∞); as an application of this, we further introduce certain Hardy-Hausdorff spaces and prove that the dual space of is just when p,q∈(1,∞).     

Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces

We consider the generalized shift operator, associated with the Laplace-Bessel differential operator . The maximal operator Mγ (B-maximal operator) and the Riesz potential (B-Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B-maximal operator Mγ is bounded from the B-Morrey space Lp,λ,γ to Lp,λ,γ for all 1<p<∞ and 0?λ<n+|γ|. We prove that the B-Riesz potential , 0<α<n+|γ| is bounded from the B-Morrey space Lp,λ,γ to Lq,λ,γ if and only if α/(n+|γ|−λ)=1/p−1/q, 1<p<(n+|γ|−λ)/α. Also we prove that the B-Riesz potential is bounded from the B-Morrey space L1,λ,γ to the weak B-Morrey space WLq,λ,γ if and only if α/(n+|γ|−λ)=1−1/q.     

Entropy solutions for a doubly nonlinear elliptic problem with variable exponent

We study the boundary value problem in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in RN (N?3) and is a p(x)-Laplace type operator with p(.):Ω→[1,+∞) a measurable function and b a continuous and nondecreasing function from R→R. We prove the existence and uniqueness of an entropy solution for L¹-data f.     

Analysis of a system of fractional differential equations

We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations , where D^α denotes standard Riemann-Liouville fractional derivative, 0<α<1, and A is a square matrix. The unique solution to this initial value problem turns out to be , where E_α denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system , , 0<α<1, and investigate dependence of the solutions on the initial conditions.     

Nonnegative radial solutions to a class of elliptic equations involving the Pucci's extremal operator

In this article we use Leray-Schauder degree to consider the existence of nonnegative radially symmetric solution for the nonlinear elliptic equation in BR, y=0 on ∂BR, where denotes the Pucci's extremal operators with parameters 0<λ?Λ and BR is the ball of radius R in RN, N?3. As an application we can obtain the results to equation , where and 0<q<p.     

Existence of positive solutions for some nonlinear parabolic systems in exteriors domains

We prove some existence results of positive continuous solutions to the semilinear parabolic system , in an unbounded domain D with compact boundary subject to some Dirichlet conditions, where λ and μ are nonnegative parameters. The functions f, g are nonnegative continuous monotone on (0,∞) and the potentials p, q are nonnegative and satisfy some hypotheses related to the parabolic Kato class J^∞(D).     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type

Let 0<γ<1, b be a BMO function and the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for in the context of the spaces of homogeneous type. The first one establishes that, for A_∞ weights, the operator is bounded in the weighted Lp norm by the maximal operator Mγ(Mm), where Mγ is the fractional maximal operator and Mm is the Hardy-Littlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator is bounded from to , where [(m+1)p] is the integer part of (m+1)p and no condition on the weight w is required. From the first inequality we also obtain weighted Lp-Lq estimates for generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.