Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier

Troels Roussau Johansen. "Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier." Studia Mathematica 205.2 (2011): 101-137. <http://eudml.org/doc/285477>.

@article{TroelsRoussauJohansen2011,
abstract = {The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_\{R\}$ associated with the Jacobi transform through the defining relation $\widehat\{S_\{R\}f\}(λ) = 1_\{|λ|≤R\}f̂(t)$ for a function f on ℝ is shown to be bounded from $L^\{p\}(ℝ₊,dμ)$ into $L^\{p\}(ℝ,dμ) + L²(ℝ,dμ)$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^\{p₀,1\}(ℝ₊,dμ)$ into $L^\{p₀,∞\}(ℝ,dμ) + L²(ℝ,dμ)$. In particular $\{S_\{R\}f(t)\}_\{R>0\}$ converges almost everywhere towards f, for $f ∈ L^\{p\}(ℝ₊,dμ)$, whenever (4α + 4)/(2α + 3) < p ≤ 2.},
author = {Troels Roussau Johansen},
journal = {Studia Mathematica},
keywords = {Jacobi transform; disc multiplier; Lorentz spaces},
language = {eng},
number = {2},
pages = {101-137},
title = {Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier},
url = {http://eudml.org/doc/285477},
volume = {205},
year = {2011},
}

TY - JOUR
AU - Troels Roussau Johansen
TI - Almost everywhere convergence of the inverse Jacobi transform and endpoint results for a disc multiplier
JO - Studia Mathematica
PY - 2011
VL - 205
IS - 2
SP - 101
EP - 137
AB - The maximal operator S⁎ for the spherical summation operator (or disc multiplier) $S_{R}$ associated with the Jacobi transform through the defining relation $\widehat{S_{R}f}(λ) = 1_{|λ|≤R}f̂(t)$ for a function f on ℝ is shown to be bounded from $L^{p}(ℝ₊,dμ)$ into $L^{p}(ℝ,dμ) + L²(ℝ,dμ)$ for (4α + 4)/(2α + 3) < p ≤ 2. Moreover S⁎ is bounded from $L^{p₀,1}(ℝ₊,dμ)$ into $L^{p₀,∞}(ℝ,dμ) + L²(ℝ,dμ)$. In particular ${S_{R}f(t)}_{R>0}$ converges almost everywhere towards f, for $f ∈ L^{p}(ℝ₊,dμ)$, whenever (4α + 4)/(2α + 3) < p ≤ 2.
LA - eng
KW - Jacobi transform; disc multiplier; Lorentz spaces
UR - http://eudml.org/doc/285477
ER -