Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p(⋅)→q(⋅)-theorems were proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x₀ and to infinity, under an additional condition relating the weight exponents at x₀ and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1 are also improved in the same way.     

Interpolation theorems for variable exponent Lebesgue spaces

When Hardy-Littlewood maximal operator is bounded on Lp(⋅)(Rn) space we prove θ[Lp(⋅)(Rn),BMO(Rn)]=Lq(⋅)(Rn) where q(⋅)=p(⋅)/(1−θ) and θ[Lp(⋅)(Rn),H¹(Rn)]=Lq(⋅)(Rn) where 1/q(⋅)=θ+(1−θ)/p(⋅).     

Sobolev's inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity

Our aim in this paper is to deal with the boundedness of maximal functions in generalized Lebesgue spaces Lp(⋅) when p(⋅) satisfies a log-Hölder condition at infinity that is weaker than that of Cruz-Uribe, Fiorenza and Neugebauer [D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003) 223-238; 29 (2004) 247-249]. Our result extends the recent work of Diening [L. Diening, Maximal functions on generalized Lp(⋅) spaces, Math. Inequal. Appl. 7 (2004) 245-254] and the authors Futamura and Mizuta [T. Futamura, Y. Mizuta, Sobolev embeddings for Riesz potential space of variable exponent, preprint]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent.     

Weighted Hardy and potential operators in the generalized Morrey spaces

We study the weighted p→q-boundedness of the multi-dimensional Hardy type operators in the generalized Morrey spaces Lp,φ(Rn,w) defined by an almost increasing function φ(r) and radial type weight w(|x|). We obtain sufficient conditions, in terms of some integral inequalities imposed on φ and w, for such a p→q-boundedness. In some cases the obtained conditions are also necessary. These results are applied to derive a similar weighted p→q-boundedness of the Riesz potential operator.     

New Orlicz-Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω∈(0,1] and ρ(t)=t−1/ω−1(t−1) for t∈(0,∞). In this paper, the authors study the Orlicz-Hardy space Hω,L(Rn) and its dual space BMOρ,L^*(Rn), where L^* denotes the adjoint operator of L in L²(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood-Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz-Hardy space Hω(Rn) for and the corresponding fractional integral L−γ for certain γ>0 maps Hω,L(Rn) continuously into , where is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω(t)=tp for all t∈(0,∞) and p∈(0,1).     

Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup

Under the standard assumptions on the variable exponent p(x) (log- and decay conditions), we give a characterization of the variable exponent Bessel potential space Bα[Lp(⋅)(Rn)] in terms of the rate of convergence of the Poisson semigroup Pt. We show that the existence of the Riesz fractional derivative Dαf in the space Lp(⋅)(Rn) is equivalent to the existence of the limit . In the pre-limiting case we show that the Bessel potential space is characterized by the condition ‖α(I−Pε)f_‖p(⋅)≦Cεα.     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

满足H(m)条件的奇异积分算子的交换子的有界性

主要讨论了满足H(m)条件的奇异积分算子与Lipschitz函数的交换子在L~p和Hardy空间的有界性,并把这个结果应用于与薛定谔算子相关的Riesz变换.     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L_p(R^d) (in the case p > 1), but (in the case when 1/p(·) is log-Hölder continuous and p- = inf{p(x): x ∈ R^d > 1) on the variable Lebesgue spaces L_p(·)(R^d), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator M_s^γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Hölder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator M_s^γδ is bounded on the space L_p(·)(R^d) (or the maximal operator is of weak-type (p(·), p(·))).     

On local summability of Riesz potentials in the case Reα>0

The rangeI ^α (L _p ) of the Riesz potential operator, defined in the sense of distributions in the casep≥n/α, is shown to consist of regular distributions. Moreover, it is shown thatI ^α (L _p ) ?L _p ^loc (R ⁿ ) for all 1≤p<∞ and 0<α<∞. The distribution space used is that of Lizorkin, which is invariant with respect to the Riesz operator.     

Packing,counting and covering Hamilton cycles in random directed graphs

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L ^p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L ^p (R ⁿ , (φ _iρ (x))² dx) where φ _iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

在L2(Rn)尺度下的Riesz位势空间上的最优恢复

考虑了Ｒｉｅｓｚ位势空间在给定网格上赋值的多元函数的重构问题,并且得到了L₂(Rⁿ)中由Ｒｉｅｓｚ位势确定的一些函数类上的精确结果。     

On the regularity and stability of Pritchard-Salamon systems

Let Σ(S(⋅),B,−) be a Pritchard-Salamon system for (W,V), where W and V are Hilbert spaces. Suppose U is a Hilbert space and F∈L(W,U) is an admissible output operator, SBF(⋅) is the corresponding admissible perturbation C₀-semigroup. We show that the C₀-semigroup SBF(⋅) persists norm continuity, compactness and analyticity of C₀-semigroup S(⋅) on W and V, respectively. We also characterize the compactness and norm continuity of ΔBF(t)=SBF(t)−S(t) for t>0. In particular, we unexpectedly find that ΔBF(t) is norm continuous for t>0 on W and V if the embedding from W into V is compact. Moreover, from this we give some relations between the spectral bounds and growth bounds of SBF(⋅) and S(⋅), so we obtain some new stability results.     

Maximal function and Riesz potential on p-adic linear spaces

For maximal function and Riesz potential on p-adic linear space ? _p ⁿ we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential I ^α(f) exists at point b, then b is L ^q Lebesgue point for some q.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

Orthogonally additive polynomials on spaces of continuous functions

We show that, for every orthogonally additive homogeneous polynomial P on a space of continuous functions C(K) with values in a Banach space Y, there exists a linear operator S:C(K)→Y such that P(f)=S(fn). This is the C(K) version of a related result of Sundaresam for polynomials on Lp spaces.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.