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

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Johansen T.

Studia Mathematica

2011

tom 205

nr 2

101-137

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.

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1 Studia Mathematica

1 Johansen T.

1 2011

Wyniki wyszukiwania

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