Wyniki wyszukiwania

1

Two-weight norm inequalities for maximal functions on homogeneous spaces and boundary estimates

100%

Luís Zani S.

Studia Mathematica

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1997

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tom 126

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nr 1

67-94

EN

Let D be an open subset of a homogeneous space(X,d,μ). Consider the maximal function $M_φ f(x) = sup1/φ(B) ʃ_{B∩∂D} |f|dν$, x∈ D, where the supremum is taken over all balls of the form B = B(a(x),r) with r > t(x) = d(x,∂D), a(x)∈ ∂D is such that d(a(x),x) < 3/2 t(x)$ and φ is a nonnegative set function defined for all Borel sets of X satisfying the quasi-monotonicity and doubling properties. We give a necessary and sufficient condition on the weights w and v for the weighted norm inequality (0.1) $(ʃ_D [M_φ(f)]^q wdμ)^{1/q} ≤ c(ʃ_{∂D} |f|^p vdν)^{1/p}$ to hold when 1 < p < q < ∞, $σdν = v^{1-p'}dν$ is a doubling weight, and dν is a doubling measure, and give a sufficient condition for (0.1) when 1 < p ≤ q < ∞ without assuming that σ is a doubling weight but with an extra assumption on φ. Another characterization for (0.1) is also provided for 1 < p ≤ q < ∞ and D of the form Y×(0,∞), where Y is a homogeneous space with group structure. These results generalize some known theorems in the case when $M_φ$ is the fractional maximal function in $ℝ^{n+1}_+$, that is, when $M_φ f(x,t) = M_γ f(x,t) = sup_{r>t} 1/(ν(B(x,r))^{1-γ}) ʃ_{B(x,r)} |f|dν$, where $(x,t) ∈ ℝ^{n+1}_+$, 0 < γ < 1, and ν is a doubling measure in $ℝ^n$.

2

Pointwise multipliers for reverse Holder spaces

100%

M. Buckley S.

Studia Mathematica

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1994

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tom 109

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nr 1

23-39

EN

We classify weights which map reverse Hölder weight spaces to other reverse Hölder weight spaces under pointwise multiplication. We also give some fairly general examples of weights satisfying weak reverse Hölder conditions.

3

On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

100%

Rönning J.-O.

Studia Mathematica

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1997

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tom 125

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nr 3

219-229

EN

Let P(z,β) be the Poisson kernel in the unit disk 𝕌, and let $P_{λ}f(z) = ʃ_{∂𝕌} P(z,φ)^{1/2+λ} f(φ)dφ$ be the λ -Poisson integral of f, where $f ∈ L^p(∂𝕌)$. We let $P_{λ}f$ be the normalization $P_{λ}f/P_{λ}1$. If λ >0, we know that the best (regular) regions where $P_{λ}f$ converges to f for a.a. points on ∂𝕌 are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_0f$ toward f in an $L^p$ weakly tangential region, if $f ∈ L^p(∂𝕌)$ and p > 1. In the present paper we will extend the result to symmetric spaces X of rank 1. Let f be an $L^p$ function on the maximal distinguished boundary K/M of X. Then $P_{0}f(x)$ will converge to f(kM) as x tends to kM in an $L^p$ weakly tangential region, for a.a. kM ∈ K/M.

4

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates

63%

Bui T. A., Cao J., Ky L. D., Yang D., Yang S.

Analysis and Geometry in Metric Spaces

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2013

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tom 1

69-129

EN

Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order (qL/l(φ))′, where (qL/l(φ))′ denotes the conjugate exponent of qL/l(φ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space Hφ;L(X), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of Hφ,L(X) is also obtained. Furthermore, a sufficient condition for the equivalence between Hφ,L(ℝn) and the classical Musielak-Orlicz-Hardy space Hv(ℝn) is given. Moreover, for the Musielak-Orlicz-Hardy space Hφ,L(ℝn) associated with the second order elliptic operator in divergence form on ℝn or the Schrödinger operator L := −Δ + V with 0 ≤ V ∊ L1loc(ℝn), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L−1/2 is bounded from Hφ,L(ℝn) to the Musielak-Orlicz space Lφ(ℝn) when i(φ) ∊ (0; 1], from Hφ,L(ℝn) to Hφ(ℝn) when i(φ) ∊ ( [...] ; 1], and from Hφ,L(ℝn) to the weak Musielak-Orlicz-Hardy space WHφ(ℝn) when i(φ)= [...] is attainable and φ(·; t) ∊ A1(X), where i(φ) denotes the uniformly critical lower type index of φ

Ograniczanie wyników

3 Studia Mathematica

1 Analysis and Geometry in Metric Spaces

1 Bui T. A.

1 Cao J.

1 Ky L. D.

1 Luís Zani S.

1 M. Buckley S.

1 Rönning J.-O.

1 Yang D.

1 Yang S.

1 2013

2 1997

1 1994

Two-weight norm inequalities for maximal functions on homogeneous spaces and boundary estimates

Pointwise multipliers for reverse Holder spaces

On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates