JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 126 / 1997

Studia Mathematica 126 (1997), 67-94 DOI: 10.4064/sm-126-1-67-94

#### Streszczenie

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\$.

#### Autorzy

• Sérgio Luís Zani

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek