Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type

In this paper, the author establishes the embedding theorems for different metrics of inhomoge-neous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. As an application, the author obtains some estimates for the entropy numbers of the embeddings in the limiting cases between some Besov spaces and some logarithmic Lebesgue spaces.     

Degenerate differential operators in weighted Hölder spaces

A differential operator ?, arising from the differential expression $$lv(t) \equiv ( - 1)^r v^{[n]} (t) + \sum\nolimits_{k = 0}^{n - 1} {p_k } (t)v^{[k]} (t) + Av(t),0 \leqslant t \leqslant 1,$$ , and system of boundary value conditions $$P_v [v] = \sum\nolimits_{k = 0}^{n_v } {\alpha _{vk} } r^{[k]} (1) = 0.v - 1, \ldots ,\mu ,0 \leqslant \mu< n$$ is considered in a Banach space E. Herev ^[k](t)=(a(t) d/dt) ^k v(t)a(t) being continuous fort?0, α(t) >0 for t > 0 and \(\int_0^1 {\frac{{dz}}{{a(z)}} = + \infty ;}\) the operator A is strongly positive in E. The estimates , are obtained for ?: n even, λ varying over a half plane.     

Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hölder-Zygmund weighted spaces

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in Hölder-Zygmund spaces on ? ⁿ whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the Hörmander class S _1,δ ^m , 0 ≤ δ < 1 (slowly varying symbols in the class S _1,0 ^m ). We show that such operators are compact operators or Fredholm operators in weighted Hölder-Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.     

Multipliers and Herz type spaces

Hrmander condition for boundedness of multiplier operators will be replaced by a weaker condition described by certain weighted or non-weighted Herz spaces. Some results on boundedness of multiplier operators are then established. As direct corollaries of main theorems in this paper, several celebrated results on boundedness of multiplier operators will be improved or deduced.     

Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

We show that for 1 < p < ??, weight w ?? A _p , and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T _? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _? and sufficient conditions for two-weight inequalities in L ^p for T _?. Our proof does not rely upon extrapolation.     

Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Let(X,p,μ)d,θ be a space of homogeneous type,(?) ∈(0,θ],|s|＜(?) andmax{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞.The author introduces the new Triebel-Lizorkin spaces (?)_∞q~s(X) and establishes the framecharacterizations of these spaces by first establishing a Plancherel-P(?)lya-type inequalityrelated to the norm of the spaces (?)_∞q~s(X).The frame characterizations of the Besovspace (?)_pq~s(X) with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p≤∞ and 0＜q≤∞and the Triebel-Lizorkin space (?)_pq~s(X)with|s|＜(?),max{d/(d＋(?)),d/(d＋s＋(?))}＜p＜∞ and max{d/(d＋(?)),d/(d＋s＋(?))}＜q≤∞ are also presented.Moreover,the au-thor introduces the new TriebeI-Lizorkin spaces b(?)_∞q~s(X) and H(?)_∞q~s(X) associated to agiven para-accretive function b.The relation between the space b(?)_∞q~s(X) and the spaceH(?)_∞q~s(X) is also presented.The author further proves that if s＝0 and q＝2,thenH(?)_∞q~s(X)＝(?)_∞q~s(X),which also gives a new characterization of the space BMO(X),since (?)_∞q~s(X)＝BMO(X).     

Embeddings between weighted Copson and Cesàro function spaces

In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality

$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$

where p₁, p₂, q₁, q₂ ∈ (0,∞), p₂ ≤ q₂ and u₁, u₂, v₁, v₂ are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p₁ and p₂ and possibly different inner weights v₁ and v₂ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

     

Regularity of solutions of boundary value problems for parabolic equations of arbitrary order in weighted Hölder spaces

We consider initial-boundary value problems for a uniformly parabolic equation of arbitrary order 2m in a noncylindrical domain whose lateral boundary is nonsmooth with respect to t. We assume that the lower-order coefficients and the right-hand side of the equation, generally speaking, grow to infinity no more rapidly than some power function when approaching the parabolic boundary of the domain, all coefficients of the equation are locally Hölder, and their Hölder constants can grow near that boundary. We construct a smoothness scale of solutions of such problems in weighted Hölder classes of functions whose higher derivatives may grow when approaching the parabolic boundary of the domain.     

Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces

We investigate spectral properties of integral operators of the form

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

The Calderón-Zygmund decomposition and interpolation on weighted Hardy spaces

We apply discrete Littlewood-Paley-Stein theory, developed by Han and Lu, to establish Calderón-Zygmund decompositions and interpolation theorems on weighted Hardy spaces H _w ^p for ω ∈ A _∞ in both the one-parameter and two-parameter cases.     

Minimality and (weighted) interpolation in Paley–Wiener spaces

We investigate links between minimality, Carleson condition, and (weighted) interpolation in Paley–Wiener spaces. In particular, we show that the Carleson condition on a sequence Λ together with minimality in Paley–Wiener spaces ${PW_{\tau}^{p}}$ implies the interpolation property of Λ in ${PW_{\tau+\epsilon}^{p}}$ , for every ${\epsilon > 0}$ . This result does not, surprisingly, require uniform minimality.     

Dual spaces of C∞ (X)

ForX a locally compact Stonian Space, letC _∞ (X) denote the universally complete Riesz space of all extended-real-valued continuous functionsf onX for which {x∈X| |f (x)|=∞} is nowhere dense. In this paper the dual spaces ofC _∞ (X) (i.e. the spaces of order bounded; of σ-order continuous; of order continuous linear forms onC _∞ (X), and the extended order dual ofC _∞ (X) denote here byC _∞ (X)^ρ (introduced by W.A.J. Luxemburg and J.J. Masterson)) are characterized. It is shown thatC _∞ (X)^ρ can be identified in a canonical way with the inductive limitM _q (X) of the Riesz spaces of all normal Radon measures defined on the dense open subsets ofX. More generally, ifY is a locally compact space thenM _q (Y) is the extended order dual of the inductive limit of the Riesz spaces of all real-valued continuous functions defined on the dense open subsets ofY. IfX is locally compact and hyperstonian, then it is proved thatC _∞ (X) andC _∞ (X)^ρ are isomorphic, and a criterion forC _∞ (X)^ρ to be the universal completion of the space of order continuous linear forms onC _∞ (X) is given.     

Boundedness of composition of operators associated with different homogeneities on weighted Besov and Triebel–Lizorkin spaces

In this paper, we introduce weighted Besov spaces and weighted Triebel–Lizorkin spaces associated with different homogeneities and prove that the composition of two Calderón–Zygmund operators is bounded on these spaces. This extends a recent result in Han et al, Revista Mat. Iber.