Continuity of Marcinkiewicz integrals on homogeneous Morrey-Herz spaces

Let μmΩ,b be the higher order commutator generated by Marcinkiewicz integral μΩ and a BMO(Rn) function b(x). In this paper, we will study the continuity ofμΩ and μmΩ,b on homogeneous Morrey-Herz spaces.     

Convergence of mock Fourier series

For certain Cantor measures μ on ℝⁿ, it was shown by Jorgensen and Pedersen that there exists an orthonormal basis of exponentialse ^2πiγ·x for λεΛ. a discrete subset of ℝⁿ called aspectrum for μ. For anyL ¹ functionf, we define coefficientsc _γ(f)=∝f(y)e ^−2πiγiy dμ(y) and form the Mock Fourier series ∑_λ∈Λ^cλ(f)e ^2πiλ·x . There is a natural sequence of finite subsets Λ_n increasing to Λ asn→∞, and we define the partial sums of the Mock Fourier series by We prove, under mild technical assumptions on μ and Λ, thats _n(f) converges uniformly tof for any continuous functionf and obtain the rate of convergence in terms of the modulus of continuity off. We also show, under somewhat stronger hypotheses, almost everywhere convergence forfεL ¹. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

Boundedness of commutators related to Marcinkiewicz integrals on hardy type spaces

In this paper, the authors study the boundedness of the operator [μΩ, b], the commutator generated by a function b ∈ Lipβ(Rn)(0 ＜β≤ 1) and the Marcinkiewicz integrals μΩ, on the classical Hardy spaces and the Herz-type Hardy spaces in the case Ω∈ Lipα(Sn-1)(0 ＜α≤ 1).     

L p -Boundedness of Marcinkiewicz Integrals with Hardy Space Function Kernels

We give the L ^p -boundedness for a class of Marcinkiewicz integral operators μ_Ω, μ^ast; _{Ω, λ} and μ_Ω, _s related to the Littlewood-Paley g-function, g ^* _λ-function and the area integral S, respectively. These operators have the kernel functions Ω∈H ¹ (S ⁿ⁻¹), the Hardy space on S ⁿ⁻¹. These results in this paper substantially improve and extend the known results. Received August 25, 1998, Accepted July 6, 1999     

Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

Littlewood-Paley operators and Marcinkiewicz integral on generalized Campanato spaces

Littlewood-Paley operators and Marcinkiewicz integral on generalized Campanato spaces ε^ρ,Φ are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ε^ρ,Φ.     

Bernstein�CWalsh Type Theorems for Real Analytic Functions in Several Variables

The aim of this paper is to extend the classical maximal convergence theory of Bernstein and Walsh for holomorphic functions in the complex plane to real analytic functions in ℝ ^N . In particular, we investigate the polynomial approximation behavior for functions F:L→ℂ, L={(Re z,Im z):z∈K}, of the structure F=g[`(h)]F=g\overline{h}, where g and h are holomorphic in a neighborhood of a compact set K⊂ℂ <sup>N</sup> . To this end the maximal convergence number ρ(S <sub>c</sub> ,f) for continuous functions f defined on a compact set S <sub>c</sub> ⊂ℂ <sup>N</sup> is connected to a maximal convergence number ρ(S <sub>r</sub> ,F) for continuous functions F defined on a compact set S <sub>r</sub> ⊂ℝ <sup>N</sup> . We prove that ρ(L,F)=min {ρ(K,h)),ρ(K,g)} for functions F=g[`(h)]F=g\overline{h} if K is either a closed Euclidean ball or a closed polydisc. Furthermore, we show that min {ρ(K,h)),ρ(K,g)}≤ρ(L,F) if K is regular in the sense of pluripotential theory and equality does not hold in general. Our results are based on the theory of the pluricomplex Green’s function with pole at infinity and Lundin’s formula for Siciak’s extremal function Φ. A properly chosen transformation of Joukowski type plays an important role.     

A note on certain square functions on product spaces

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition.     

Recurrence in unipotent groups and ergodic nonabelian group extensions

LetT be a measure-preserving and ergodic transformation of a standard probability space (X,S, μ) and letf:X → SUT _d (ℝ) be a Borel map into the group of unipotent upper triangulard ×d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined byf, in terms of the asymptotic behaviour of the distributions of the suitably scaled mapsf(n,x)=(fT ⁿ⁻¹·fT ⁿ⁻²…fT·f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H₁(ℝ)=SUT₃(ℝ), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H₁(ℤ)=SUT₃(ℤ). The author was partially supported by the FWF research project P16004-MAT.     

Orlicz-Hardy spaces associated with operators

Let L be a linear operator in L2(Rn) and generate an analytic semigroup {e-tL}t 0 with kernel satisfying an upper bound of Poisson type, whose decay is measured by θ(L) ∈ (0, ∞). Let ω on (0, ∞) be of upper type 1 and of critical lower type p0(ω) ∈ (n/(n + θ(L)), 1] and ρ(t) = t-1/ω-1(t-1) for t ∈ (0, ∞). We introduce the Orlicz-Hardy space Hω, L(Rn) and the BMO-type space BMOρ, L(Rn) and establish the John-Nirenberg inequality for BMOρ, L(Rn) functions and the duality relation between Hω, L(Rn) and BMOρ, L...     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

A BMO estimate for the multilinear singular integral operator

The behavior on the space L∞((R)n) for the multilinear singular integral operator defined by TAf(x)=∫Rn Ω(x-y)/|x-y|n 1(A(x)-A(y)-(△)A(y)(x-y))f(y)dy is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO((R)n). It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L∞((R)n), TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO((R)n).     

Remarks on g-functions on H^n

In this paper, the authors study some properties of Littlewood-Paley g-functions gψ（f）,Lusin area functions Sψ,α（f） and Littlewood-Paley gψ^＊,λ（f） functions defined on H^n, where α,λ 〉 0 and ψ, f are suitable functions. They are the generalization of the corresponding operators on R^n.     

Weighted endpoint estimates for commutators of Marcinkiewicz integrals

Let μΩ^mb be the commutator generalized by μΩ, the n-dimensional Marcinkiewicz integral, and b ∈ BMO（R^n）. The author establishes the weighted weak LlogL-type estimates for μΩ^mb when Ω satisfies a kind of Dini conditions, which improves the known result essentially.     

On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

We consider complex-valued functions f∈L ¹(ℝ₊), where ℝ₊:=[0,∞), and prove sufficient conditions under which the sine Fourier transform [^(f)]_s\hat{f}_{s} and the cosine Fourier transform [^(f)]_c\hat{f}_{c} belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.     

Equazioni ellittiche non variazionali a coefficienti continui

Riassunto Si considera un'equazione ellittica non variazionale a_aD^au=f a coefficienti continui in un apertogW di ℝⁿ e si dimostrano certe proprietà locali delle soluzioni forti dell'equazione medesima nell'ipotesi che f appartenga allo spazio di Morrey L ^2,λ(Ω) o allo spazio di John-Nirenberg ℰ(Ω). Analoga indagine è svolta supponendoΩ=ℝ ₊ ⁿ e supponendo che u verifichi sull'iperpiano x_n=0 condizioni di Dirichlet omogenee. Ricerca svolta nell'ambito dei Contratti di Ricerca del Comitato per la Matematica del C.N.R. Entrata in Redazione il 17 febbraio 1970.     

Compact embeddings of besov spaces involving only slowly varying smoothness

We characterize compact embeddings of Besov spaces B _p,r ^0,b (ℝ ⁿ ) involving the zero classical smoothness and a slowly varying smoothness b into Lorentz-Karamata spaces L<sub>p,q;[`(b)]</sub> {L_{p,q;\overline b }}(Ω), where is a bounded domain in ℝ <sup>n</sup> and [`(b)]\overline b is another slowly varying function.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

一类粗糙核参数型Marcinkiewicz积分

In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q＞ 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1＜s≤∞ are given. The Lp(Rn) (2≤p＜∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively.